Saharon Shelah- Classification theory and the number of non-isomorphic models

CLASSIFICATION THEORY

AND THE NUMBER OF NON-ISOMORPHIC MODELS

STUDIES IN LOGIC AND

THE FOUNDATIONS OF MATHEMATICS

VOLUME 92

Editors

J. BARWISE, Stanford H. J. KEISLER, Madison

P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam

N O R T H - H O L L A N D

A M S T E R D A M . N E W Y O R K * O X F O R D . T O K Y O

CLASSIFICATION THEORY AND THE NUMBER OF

NON-ISOMORPHIC MODELS

REVISED EDITION

S. SHELAH

The Hebrew University, Jerusalem, Israel

1990

NORTH-HOLLAND

AMSTERDAM'NEW YORK.OXFORD.TOKY0

ELSEVIER SCIENCE PUBLISHERS B.V Sara Burgerhartstraat 25

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Library of Congress Cataloging-in-Publication Data Shelah, Saharon.

Classification theory and the number of non-isomorphic models/S. Shelah. -- 2nd ed.

mathematics; v. 92) p. cm. -- (Studies in logic and the foundations of

Includes bibliographical references.

1. Model theory. I. Title. 11. Series. ISBN M - 7 0 2 W 1

QA9.7.S53 1990 51 1’.8--dc20 8S29756

CIP

First edition : 1978 Second edition : 1990

ISBN : 0 444 70280 1

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1990

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CONTENTS

Contents . . . . . . . . . . . . . . . . . . . . . . . . . v

Introduction. . . . . . . . . . . . . . . . . . . . . . . xi Introduction to the revised edition . . . . . . . . . . xv

Acknowledgements . . . . . . . . . . . . . . . . . . . . ix

Open problems . . . . . . . . . . . . . . . . . . xvii Added in proof . . . . . . . . . . . . . . . . . . xxiii Notation . . . . . . . . . . . . . . . . . . . . . . . . . xxxl

CHAPTER I . PRELIB~INARZES . . . . . . . . . . . . . . . . 1

Q 0 . Introduction . . . . . . . . . . . . . . . . . . . . . Q 1 . Preliminaries and saturation . . . . . . . . . . . . . .

1 1 9 8 2 . Order. stability and indiscernibles . . . . . . . . . . . .

CHAPTER I1 . RANKS AND INCOMPLETE TYPES . . . . . . . . . 18

Q 0 . Introduction . . . . . . . . . . . . . . . . . . . . . Q 1 . Ranks oftypes . . . . . . . . . . . . . . . . . . . .

18 21

Q 2 . Stability. ranks and definability . . . . . . . . . . . . 29 Q 3 . Ranks. degrees and superstability . . . . . . . . . . . . 41 Q P . The f.c.p., the independence property and the strict order

property . . . . . . . . . . . . . . . . . . . . . . . 62

CHAPTER I11 . GLOBAL THEORY . . . . . . . . . . . . . . . 82

Q 0 . Introduction . . . . . . . . . . . . . . . . . . . . . Q1.Forking . . . . . . . . . . . . . . . . . . . . . . . 84

82

Q 2 . The finite equivalence relation theorem . . . . . . . . . 94 Q 3 . The instability spectrum . . . . . . . . . . . . . . . . 101 Q 4 . Further properties of forking . . . . . . . . . . . . . . 108

V

vi CONTENTS

f 6 . The fist stability cardinal . . . . . . . . . . . . . . .

$ 7 . Instability . . . . . . . . . . . . . . . . . . . . . . f 6 . Imaginary elements . . . . . . . . . . . . . . . . . .

CHAPTERIV . ~ E M O D ~ . . . . . . . . . . . . . . . . f 0 . Introduction . . . . . . . . . . . . . . . . . . . . . f 1 . The set of axioms . . . . . . . . . . . . . . . . . . . f 2 . Examples of F’s . . . . . . . . . . . . . . . . . . . f 3 . General properties of F-primary models . . . . . . . . . f 4 . Prime models for stable theories . . . . . . . . . . . . f 6 . Various results . . . . . . . . . . . . . . . . . . . .

CHAPTER V . MOBE ON TYPES AND SATURATED MODELS . . . .

f 0 . Introduction . . . . . . . . . . . . . . . . . . . . . f 1 . Orthogonality. regularity and mhimelity of types . . . . . f 2 . Dimensions and orders between indiscernible sets . . . . . f 3 . Weighted dimensions and superstability . . . . . . . . . f 4 . Semi-regular and semi-minimal types . . . . . . . . . . f 6 . Multi-dimensional theories . . . . . . . . . . . . . . . $ 6 . Cardinality-quantifiers and two-cardinal theorems . . . . . f 7 . Ranksrevisited . . . . . . . . . . . . . . . . . . .

CHAPTER VI . SATURATION OF ULTBLPRODUCTS . . . . . . . . $ 0 . Introduction . . . . . . . . . . . . . . . . . f 1 . Reduced products and regular filters

f 3 . Constructing ultrafilters . . . . . . . . . . . . f 4 . Keisler’s order . . . . . . . . . . . . . . . .

elementary claases . . . . . . . . . . . . . . f 6 . Saturation of ultralimits . . . . . . . . . . . .

. . . . . . f 2 . Good filters and compactness of reduced products .

f 6 . Saturation of ultrapowers and wtagoricity of

. . . .

. . . .

. . . .

. . . .

. . . . yuuudo- . . . . . . . .

CHAPTER VII . CONSTRUCTION OF MODELS . . . . . . . . . . $ 0 . Introduction . . . . . . . . . . . . . . . . . . . . . f 1 . Skolem functions and generalizations of eaturativity . . . .

122 130 137

150

150 152 157

174 183 204

223

223

230 240 249 267 284 289 305

321

321

324 333 345 370

379 390

397

397 400

UONTENTS vii

Q 2 . Generalized Ehrenfeucht-Mostowski models . . . . . . . Q 3 . On the f.c.p., uniform trees and ID(T)I > IT/ = KO . . . . Q 4 . Semi-definability . . . . . . . . . . . . . . . . . . . $ 6 . Hanf numbers of omitting types . . . . . . . . . . . .

CHAPTER VIII . THE NUMBER OF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CUSSES . . . . . . . . . . . . . .

$ 0 . Introduction . . . . . . . . . . . . . . . . . . . . . $ 1 . Independence of types . . . . . . . . . . . . . . . . . $ 2 . Unsupmtable theories Q 3 . Saturated models and the w e h = lTll . . . . . . . . .

. . . . . . . . . . . . . . . .

$ 4 . Categoricity. etlturation and homogeneity up to a cardinality

CHAPTER I X . CATEUOBJCITY AND THB: NUMBER OB MODELS IN ELEMENTARY CLASSES . . . . . . . . . . . . . . . . . .

QO . Introduction . . . . . . . . . . . . . . . . . . . . . Q 1 . Supratable theories and categoricity . . . . . . . . . . Q 2 . On the lower parts of the spectrum . . . . . . . . . . .

CHAPTER X . CLASSIFICATION FOR F&-SATURATED MODELS .

$ 1 . Preliminaries . . . . . . . . . . . . . . . . . $ 2 . The dimensional order property . . . . . . . . . . $ 3 . The decomposition lemma . . . . . . . . . . . . $ 4 . Deepness . . . . . . . . . . . . . . . . . . .

$ 0 . Introduction . . . . . . . . . . . . . . . . .

$ 5 . Deep theories have many non-isomorphic models $ 6 . Infinite depth . . . . . . . . . . . . . . . . .

. . .

$ 7 . Trivial types . . . . . . . . . . . . . . . . .

CHAPTER XI . THE DECOMPOSITION THEOREM . . . . .

$0 . Introduction . . . . . . . . . . . . . . . . . $ 1 . Stationarization . . . . . . . . . . . . . . . . $2 . The axiomatic treatment $3 . Specifying the axiomatic treatment

. . . . . . . . . . . . . . . . . . . .

411 419 426 432

440

440 444 455 464 47 1

479

479 481 497

508

508

509 512 520 527 533 548 550

557

557 557 561 572

viii CONTENTS

CHAPTER XI1 . THE MAIN GAP FOR COUNTABLE THEORIES . 590

. . . . . . . . . . . . . . . . . $0 . Introduction 590 $1 . On FA" and F{ . . . . . . . . . . . . . . . . . $2 . Stable systems . . . . . . . . . . . . . . . . 598

$3 . On good sets $4 . The otop/existence dichotomy . . . . . . . . . . 608

$5 . From the (Xo. 2)-existence property to the (A. 2)-existence property . . . . . . . . . . . . . . . . . . .

$6 . The book's main theorem . . . . . . . . . . . . 620

591

603 . . . . . . . . . . . . . . . . .

616

CHAPTER XI11 . FOR THOMAS THE DOUBTER . . . . . . . 622

$0 . Introduction . . . . . . . . . . . . . . . . . 622 $ 1 . Can the models be characterized by invariants ? . . . . 623 $2 . On having many models. no one elementarily embeddable

into another . . . . . . . . . . . . . . . . . 627 $3 . On the Morley conjecture . . . . . . . . . . . . 634 $4 . ](Ha. T) for a large enough . . . . . . . . . . . . 643

~ P ~ P N D I X . . . . . . . . . . . . . . . . . . . . . . . . 653

$ 0 . Introduction . . . . . . . . . . . . . . . . . . . . . 653 5 1 . Filters. stationary sets and families of sets . . . . . . . . 653 5 2 . Partition theorems . . . . . . . . . . . . . . . . . . 659 5 3 . VarioUereeults . . . . . . . . . . . . . . . . . . . . 666

Historicalremazka . . . . . . . . . . . . . . . . . . . . 673 Referen- . . . . . . . . . . . . . . . . . . . . . . . 684 Index of definitions and abbreviations . . . . . . . . . . . . 691 Index of symbols . . . . . . . . . . . . . . . . . . . . 703

ACKNOWLEDGEMENTS

My reseamh in this area started with my thesis, and I thank M. 0. Rabin for his kind guidance. I thank the Israel Academy of Science for partially supporting my research, aa well aa the United States- Israel Binational Science Foundation (grant 11 10) and the NSF (grants 14PH747 and MCS-08479). Much of the proofreading and checking waa done industriously by M. Abramsky who waa my research assistant from 1973 to 1976, and for this I am grateful to him, aa well as to L. Marcus who did a similar work previously. They are responsible for the fey paragraphs which are in good English. I owe a special debt of gratitude to D. Ehrman, J. Alon and mainly D. Sharon for typing, correcting and recorrecting the manuscript. Lastly, I thank J. Bddwin, G. Cherlin, S. Koppleberg, D. Monk, M. Rubin and P. Schmidt for detecting various errors and inaccuracies.

ix

This Page Intentionally Left Blank

INTRODUCTION

The aim of this book is to represent works of the author on claesi6ica- tion and related topics. The author, in a moment of insanity, believed this would be the easiest way to represent his work.

There is no point in trying to convince you that I think the book is important; since otherwise I would not have spent my time writing it. Anyhow, I have said it in the introduction to [Sh 75~1, which is supposed to be a propaganda for this book. There is also no point in explaining how the subject evolved because I say almost everything in the introduction to [Sh 741. Let us note only that the founding stone is [Mo 651 (there are historical notes at the end of the book).

So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting ycuueelf to the reading, and solving the exercises till you know it by he&. Unfortunately, I suspect the reader is looking for advice on how n& to read, i.e. what to skip, and even better, how to read only some isolated highlights.

If you are generally not interested in model-theory but are interested in ultrafilters and ultraproducts, you can read Chapter IV, Sections 1, 2 and 3. There, many claesical results are represented but also some new ones. We prove that there is a A-good not A+-good ultrafilter iff A is regular (improving Keialer, cf. [Ke 651); show that p < 2” A pwo = p iff there is an ultrafilter D over A and ni < w such that n n,/D = p, thus anawering a question of Keisler (cf. [Ke 0781). We also answer one of the questions from [CK 731 (due to Keisler): there is a regular ultrafilter D over A > No, D not good, but n n,/D > No * n n,/D = 2”. Also an HI-incomplete D over I is A-good iff for every order J, JI/D is A-saturated.

Also Chapter VII is quite isolated. The first section deals with a generalization of A-saturated, so that such models exist in more cardinalities. The second section presents a generalization of Ehren-

xi

xii MTaODUCTION

feuchtMoetowski models, and wume only the knowledge of Skolem functions. The third section uses the second, and some definitions from previous chapters, and pretends to show the generalization of Section 2 is useful, so e.g., we have a model generated by a tree of indiscernibles and this is important for unsuperstable theories. The fourth section again requires no prerequisite, and there we prove that for any theory, and p 2 2ITl, it has p-universal, p-stable models of arbitrarily high cardinalities. In Section 5 we present the theorems on Hanf numbers of omitting types.

Chapter VIII is devoted to the number of non-isomorphic models, but for this we use only some definitions (or rather equivalent conditions explicitly stated and used as alternative definitions). Also Chapter VII, Section 3 is used in Chapter VIII, Sections 2 and 3. Most of Chapter VIII, Section 1 is quite easy (and has real results) and there is not much dependence.

In Chapter VIII, Section 2 we have somewhat heavier material; you can start by reading Theorem 2.2(1): if T is unsuperstable, h > IT1 is regular, then T has 2” non-isomorphic models of power A. Chapter VIII, Sections 2 and 3 are aimed at proving the same for any h 2 I TI + 24,; but Chapter VIII, Section 3 and the later parts of Section 2 me, in their present form, suitable for your too arrogant students (but no guarantee).

Section 7 of Chapter I11 is independent, and contains results on the possible K(h) = sup(lS(A)I : IAI s A} and related problems. Also the combinatonal appendix needs no prerequisite, but it is not so

interesting. If you arrive here I hope you understand that you should really start to read from Chapter I.

Now Chapter I, Section 1 is a restatement of some classical theorems (compactnese and LowenheimSkolem which are proved later, elementary chains, saturated models, etc.). Now the reader should know what are firat order theories, formulas and satisfaction; and no more; but I would be very interested to meet a successful reader who has not read before a significant part of [CK 731, or [Sa 721. We here a h make some conventions.

Chapter I, Section 2 contains central definitions (stability, indiscernibility) and some theorems on existence of indiscernibles and the connection between unstability and order.

Chapter 11, Sections 1 and 2 are easy and central. Here ranks are introduced and stability of formulas and theories are really introduced, i.e., we give many equivalent conditions.

Chapter 11, Sections 3 and 4 are quite peripheral and independent. In Chapter 11, Seotion 3 we investigate more deeply ranks, and connect

... INTRODUOTION Xl l l

them with superstability. Only Claim 3.12 of Chapter I1 is used in Chapter 11, Section 4.

In Chapter 11, Section 4 we investigate some more properties of formulm and theories: the finite cover property, the strict order property and the independence property. Almost the only use later is in the investigation of Keisler orders and s&~ration of ultrapowers, in Chapter VI, Sections 4 and 6.

In Chapter 111, Sections 1, 2, 3, 4 and 6 are very necessary for what follows; Sections 1-4 can be read only successively, Section 6 depends mainly on Chapter 11, Sections 1 and 2. Unlike Chapter I1 we concentrate here on complete types and investigate forking, and in Sections 1-6 deal with stable T only. Forking is introduced in Section 1, investigated in Section 2, used in Section 3 to (almost) give the stability spectrum theorem, and in Section 6 really to give it. In Section 4 we me the connection between ranks and forking, use them to prove results on both sides. In Section 6 we extend our models so aa to have names for equivalence classes, this is important when we look for canonical forms. In Section 7 we investigate various properties, like in Chapter 11, Section 4, mostly for unstable T.

Chapter IV deals with prime models. As we have to deal with five kinds of prime models, we give an axiomatic setting (in Section 1) fmd which examples satisfy which axioms (in Section 2, the results sum up in a table at the end). In Section 3 we prove theorems in this axiomatic setting (prime models exists, they realize only isolated types, etc.). Section 3 uses only Chapter IV, Section 1, but in Chapter IV, Section 2 extensive use of Chapter 111, Section 4 is made. The fourth section is somewhat heavier; we here concentrate on stable theories, and give characterization of prime models (implying uniqueness). Only a few lemmas from it are used later.

The fifth section contains scattered results, most of them do not require much knowledge. The main theorem says, e.g., that for countable stable T, if a primary model (i.e. a prime model constructed step by step) exists, the prime model is unique. We also prove a theorem on the existence of a model of T omitting < 2ITI complete types.

Chapter V contains some of the deepest results of the book. It uses Chapter I V but usually not in a deep sense, the reader would better go to its detailed introduction.

Now in Chapter VI we turn to ultraproducts, Sections 1,2 and 3 do not refer to anything. In Sections 4, 6 and 6 we use Chapter 11. The firat half waa already reviewed. In the second half we investigate

xiv I"F&ODUOTION

Keisler order on theories ([Ke 671) and get quite a complete picture (to complete the picture we should know more on unstable theories without the strict order property), and find, quite accurately, how saturated am ultraproducts and ultrdimits.

Chapters VII and VIII were already reviewed. Chapter VIII, Section 4 is to a large extent a summary of results, and has some not so hard theorems. In Chapter IX, Section 1 we prove the categorioity theorems, relying

on various previous resulta (including Chapters V and VIII). We also prove some results on the number of models in elementary claases (in Chapter VIII we concentrate on pseudo-elementary claeeee), and when a model can be represented as a union of strictly increasing chain of length 6.

In Chapter IX, Section 2 we deal with the number of countable models of a superstable T; and what can be the number of F&,-models of T in He (it is 1, IaI' (p s 21Tl), or 2 2la1), and just models when T is totally -transcendental.

The Appendix contains the combinatorid results we need.

Note. (1) Each chapter hae its introduction, it may be wise to look at it again during the reading.

(2) Exercises am scattered randomly among the sections. Some are the result of the public pressure due to the prejudice that examples clarify notions, others me variants of theorems I want to mention, or remnant of obsolete proof. And some were the result of my preferring to give an exercise with a generous hint rather than a theorem with a thin proof.

(3) The change in the name of the book is not incidental, but a change in point of view during the yeam in which the book waa written, explained in [Sh 76~1.

The reader may ctlso want to know which of my papers becomes obsolete by this book. So this is the fate of [Sh 69a], [Sh 70a], [Sh 7Ob] [Sh 711, [Sh 71b], [Sh 71d], [Sh 72a], [Sh 741 and [Sh 7481. Also [Sh72] except Section 2 which deals with the uncountable caw, [Sh 72b] except the purely combinatorid part and the theorem on L,,,,. But in [Sh 71b], the proof of the last theorem is not covered. We did not deal here with the results on non-elementary claases

([Sh 691, Section 6; [Sh 69~1, [Sh 701, [Sh 76a] and [Sh 7601).

INTRODUCTION TO THE REVISED EDITION

In this edition four new chapters (X, XI, XI1 and XIII) have been added. (In addition, many corrections have been made, including many which previously were in the “Added in Proof’’ part of the First Edition.)

The additional chapters present the solution to countable first order T of what the author sees as the main test of the theory (XII, 6.1).

In Chapter X we introduce the dop (dimensional order property) and show that it is a very meaningful dividing line for superstable theories: if it holds, T has many &-saturated ( = Fgo-saturated) models; if it fails, we have a structure theory: every &-saturated model M of T is Fio-prime over a non-forking tree of models (M,,: qeI c u’A) (each M,,F;fo-prime over 9). If the tree I were to have an infinite branch, the theory is deep and still has many models. Otherwise, we can compute the number of #,-saturated models. This chapter is an improved version of [Sh 83a], [Sh 84bl. Mainly the IE case is explicated and the facts on trivial (regular) types are gathered in one section.

In Chapter XI we prove the needed decomposition theorems assuming

(*) there are prime atomic models over MI U 4 when &, 4 M,, A& {illl,&&} is independent over 4.

Chapter XI1 is the crux of the matter. We prove that the negation of (*) above implies that in models of T we can define a relation which order a large subset of ‘“w( ; the definition being aR6 iff a type p ( z , a, 6) is omitted.

Chapter XI11 is intended to exemplify that Theorem XII, 6.1 fulfills its aim. For this we consider several questions which are

xv

xvi INTRODUCTION TO THE REVISED EDITION

solved using its partition to cases. In Section 1 we deal with “when La,,-equivalent models of cardinality A are isomorphic ”.

In Section 2 we deal with computing IE(A, T). Considering what was done in Chapter X, the main problem is to show IE(h, T) = 2A when T has otop. Usually, the reader thinks this is immediate, forgetting that elementary embedding does not necessarily preserve the omission of types.

In Chapter XII, $3, we prove the Morley conjecture : for countable first order T, No < A < y * I ( h , T) N,)I(A, T) = 1) but I(No, T) = No.

In Chapter XIII, $4, we compute I (h ,T) for A large enough (though we do not know if a relevant invariant can be >No but

For a discussion on the significance of the new chapters, see [Sh 85bl.

We thank Leo Harrington for hearing the proofs in Chapter XI1 while they were generated, and the reader may thank him for persuading me to prove the otop there using finite sequences rather than some considerably more complicated and infinite versions.

We thank Saffe and Hrushovski for pointing out the need for corrections, Grossberg for proofreading VIII, and the Israel Journal ofMuthemutics for allowing us to use portions of [Sh 831 and [Sh 83a] which appear there. Last, but not least, we thank Damit Sharon and Alice Leonhardt for typing the new material and corrections.

S. SHELAH November 1987

< 2x0).

OPEN PROBLEMS

A. Completions

Here we deal with problems which just complete theorems in the book. We feel existing methods will suffice.

(1) Complete the computation of the possible function I(A,T) ( A 2 No, T countable complete first order theory). See XIII, $93 and 4.

Though some think this was “the problem”, I could not make myself excited about it. Still it would be nice to know.

The most appealing part is : ( la ) What can SND(T) be? If it is a limit cardinal >KO, what can

Hopefully SND(T) E (w + 1) U {&, (2’~)’} : even so, we shall need

If we are interested in I(&, T), too, we should ask ( lb) Does the assumption I ( N o , T) = No, T superstable, bound the

(2) What can A be from Theorem V, 5.81 (3) Is DP(T) > lTl+ possible ? Remember DP(T) < S(lTI), all successor < [TI+ are possible;

however, cf. IT1 > No imply S(lT1) > /TI+. (4) Suppose A 2 K,(T) (or even A = K,(T)), M is F,”-atomic over

A , and for every I E M indiscernible over A , dim(1, A , M ) < A. Then, is M necessarily F,”-prime over A ?

Even various weakenings are open. You can assume A > IT1 [is regular] or it can be to prove for Ceq (this strengthens the hypothesis - we have more I ’ s ) , or try the parallel for Fi. Note that this is an attempt to improve the characterization theorem IV, 4.14 ; note also that for T superstable this is true (see IV, 4.18).

( 5 ) Suppose T is stable (D(T)( > IT(‘“, (Vx < ,u)[x‘~ < JD(T)I] K

k( T), Z( T) be ?

another cardinal invariant to compute I(A, T) for N, 6 A <

depth of T? [ ( w + 1) is possible.]

xvii

xviii OPEN PROBLEMS

regular ~ K ( T ) . Prove that for every A = A<“ 2 IT1 T has a t least min(ZA, 2.1 non-isomorphic K-compact models of power A.

See IX, 1.20. (6) For a shallow superstable without the dop variety the depth

is < w. (This means that if every completion of an equational theory is

shallow superstable without the dop, then every such completion has finite depth.) See [Sh 861.

(7) Is every unidimensional stable T, superstable ? (8) See questions in Baldwin and Shelah [BSh 851 and [Sh 86aI.

B. Classifying further unstable theories

Clearly not all unstable theories are equally complicated, and we may want finer divisions : Theorem II,4.7( 1) suggest the strict order property and independence property, and IV, $7 the tree property. We shall deal here with some problems that look like reasonable approaches.

(1) What can { (A,p) be: every model of T of power A has a p- compact extension of power A} ?

The interesting case is IT1 < p < A, x<’’ < A < 2 X (see Ex. VIII, 4.5, which is solved in [Sh 81, Th 1,18]), and we can assume Tis unstable but simple (i.e. without the tree property, since otherwise we have an answer). By [Sh 801, a positive non-trivial answer is consistent ; by [Sh 811 if we change the problem a little (restricting the p- compactness to a predicate) this cannot be proved in ZFC. If the idea can be transferred to the usual case, we may still hope for w + 1 cases (with [Sh 811’s example being for 3 and w’s one are those like Tind; the needed analysis may look like [Sh 83al).

(2) Continue the investigation of Keisler order. See VI and [Sh 72, $11. Note that the countable theories left are

those unstable T without the strict order property, and that the simple partition with the tree property seems meaningful, so [Sh 801 seems relevant. It seemed that the inner partition of unstable simple theories (if exists) is different than in (1). Also note that this involves building special ultrafilters, so we may choose another variant of the order (e.g. limit ultrapower M i I W with W HI-complete ; see Keisler [Ke 631).

OPEN PROBLEMS xix

(3) The following definition was suggested by Grossberg and the author.

DEFINITION 3a : Let h * (p)‘” mean that if I , A E M , M t T , IAl < x, 14 = h then some J E I is an indiscernible sequence over A of power

T..X

We can ask,

CONJECTURE 3b : Suppose T is without the independence property, then for some a, IT1 +x) += (p);:’.

This is confirmed in [Sh 86a] for theories that have no monadic expansions with the independence property. Note that we get complete negative results if T has the (**)-independence property, where

DEFINITION 3c: (i) T has the *-independence property, if for some q@,, z,, Z,, 8) for every k < w and w E k there are 6 and (i < k) such that for every distinct i, < i, < i, < k

(ii) T has the (**)-independence property if there are formulas qn(z0, . . . , z,, g,) ( / ( E t ) constant) such that for every W, E “w(n < w ) there are 6,(n < w ) at(i < w ) such that for n < w and i, < . . . < in-,

We may weaken (ii) and still obtain the result. (4) Find what K = { A : T has a universal model of power A} can

be. Note that for “saturated” we know the answer (see VIII, 4.8).

Also, for a strong limit h > ITI, h E K , and if T has a saturated model of power A , then A E K . Surprisingly, some independence results were found: for Tord see [Sh 80a], and for Tind [Sh 841.

xx OPEN PROBLEMS

C. Problems parallel (to the book)

(1) Prove the main gap (etc.) for uncountable theories. [Here and elsewhere we concentrate on the main gap, but expect the parallel to

Note that almost everywhere in the book the countability of the theory is not particularly important. In fact, there is just one point (but a crucial one) in which we use countability: in XII, $4 (the otop/existence dichotomy).

It seems we should try to deal with inevitable types (instead of isolated ones) (see proof of IV, 5.9) or even with just prime (but not necessarily primary) models over suitably chosen non-forking trees of models.

We may concentrate on unidimensional superstable T (this suffices to prove either I(A, T) < Z2lT1 or T) 2 loll). Note that if q(x , a) is weakly minimal, i% U p(M, a) E A E M , then M is Fio-atomic over A.

(2) Prove the main gap for the class of (TJ+-saturated models. Again only “one point ” is missing : the existence of a regular type

in X, $ 3 ’ ~ proof. Note the analysis of non-multidimensional T (in V, $5) and the dimensional continuity property (see X, $2) (which take care of the cases that an increasing union of saturated models do not). We can look also a t other kinds of saturation.

XI11 too.]

(3) Prove the main gap (etc.) for the class

for T countable. See VI, 5.3, 5.4, 5.5 and 5.6. A priori we considered the order of difficulty in increasing order

as: Ch. X (i.e. the main gap for Fgo-saturated models), question (2) above, the present question C3 and Theorem XII, 6.1. The fact that XII, 6.1 was solved before C2 and C3 is due to our greater interest, but we may have been wrong.

A relevant theorem is [Sh 751. Note that a hidden order property for those classes is:

(Pr) In CeQ, for some formula p(x, g , @ for every k, n < o, there are at, (t < n) such that for every t, < t, < n, m, c m2 < n,

OPEN PROBLEMS xxi

I V ( ~ ” “ > at,, @eslk d I v ( ~ ~ “ , a m z , @m,)I,

d I v ( ~ , ~ ~ , , ~ ~ ~ I < N o .

By [Sh 751 if T has (Pr), I(A,K,) = ZA for A > 2*0 (using cardinality quantifiers) .

(4) Prove the main gap for strongly A-saturated models (i.e. for Fi-saturated models strongly A-homogeneous, i.e. if (ai:i < a < A ) r M ( b , : i < a < A ) s M realizes the same type, then some automorphism of the model takes one to the other).

The following problem should be easier to solve. (5) Prove the main gap (etc.), for the class of No-

(6) Classify when we replace “isomorphic” by “L,,,u-

(6a) Classify the possible Scott heights of models of T. See the works of Nadel and [Sh71a]. This does, of

course, have some connection to the Vaught conjecture. (7) We know that { A : T has a rigid model of power A}

may be “bad” (see [Sh 76~1). But what for IT(+-saturated models? What about the number of non-isomorphic ones when a t least one exists? See [Sh 83b] on very partial positive results.

(8) Vaught conjecture, i.e. (for complete countable T) I ( K o , T) > No = I ( N o , T) = 2*0. We can look also at variants of i t : the number of minimal, models or rigid models.

Morley [Mo] proves I ( N o , T) > K, *I(No, T) = 2’0, and the proof applies to many other cases (but this really belongs to description set theory, where stronger theorems have since been proven, e.g. that of Burgess; see [Sh 84bl).

Some people think this is the most important question in model theory as its solution will give us an understanding of countable models which is the most important kind of models. We disagree with all those three statements.

saturated models (T countable).

equivalent ”.

xxii OPEN PROBLEMS

D. Further reaching problem

(1) Define " T has a structure theory over a class K" and classify accordingly.

It seems reasonable to look a t specific K's. A reasonable interpretation is: for some K , A for every model M of T there is a model I E K , and functionsf:M-t"I, g : M - t A such that if a : ~ M ( t = 1,2, i < n), g(a2) = g(a,2) and ( f (a2) : i < n), (f(a,") : i < n) realize the same atomic type in I , then (a: : i < n), (a," : i < n ) realizes the same type in M ; moreover, if F, BeKI, h < h and c,J realizes the same atomic type in I , then

(3a c M ) [f(iz) = F A g(a) = &] iff (3a EM) [f(a) = Z A g(a) = a]. So the main gap says which T has a structure theory over K = { I : I

a tree with do levels and depth <a}, and the decomposition theorem (XIII, 5.1) says which T has a structure over the class of trees with o levels.

The natural next step is to look at the class of linear order or trees.

(2) Classification over a predicate. We assume T has a designated predicate P, and we want to know

how much M + T is determined by M r P . See [PSh 851 (and previous references therein). The author's recent works are [Sh 85d], [Sh 86bI.

Related problems are (2a) What can be

f(A) = SUP{f(h,N) : 1" = u, ~ ( A , N ) = { (M, a ) a e N / ~ : ~ b T, ( 1 ~ 1 1 = A,,MrPM = N > .

(2b) Replace (M, a)aeN by M in (2a) above. (2c) When over M r P M there is a prime model. (3) Classify for $EL,,,, or $€LA+,, , or for abstract elementary

For example: (3a) If $ E LA+,, is categorical in one ,u 2 then it is categorical

in every ,u 2 &(A). (3b) If $€LA+,,, @ as in VII, $2, @ is such that E M ( I , @ )

I= $ for every I , then either for every A 3 a,( I@l) ({EM(I, @)/ E : III = A}( = 2A, or (possibly changing @) for every Al{EM(I, @)/ E 111 = A} = 1.

classes see [Sh %a].

See [Sh 751, [Sh 83a], [Sh 85aI.

ADDED IN PROOF

In order to make the process of writing converge, the author “closed” it to innovations in the summer of 1983 (except that Chapter X, Section 5 , was expanded drastically to include the proof for I E ) . So let us comment on some of the relevant improvements and advances.

1. Uniqueness of prime models (IV, 5.6)

This is old, but still note [Sh 79a] while reading Theorem IV, 5.6 (which says that if T is countable and stable, and there is a primary model over A , then the prime model over A is unique). By [Sh 79a] the assumptions “countable ” and “ stable ” cannot be omitted and an alternative proof of the theorem is given.

2. Regular, semi-regular and simple types (Chapter V )

See several works of S. Buechler, the thesis and some works of E. Hrushovski and Buechler and Shelah [Bush] and Hrushovski and Shelah [HSh]. More on the dichotomy of multi-dimensionality for stable theories (strengthening it) will appear, relevant to problem (12).

3. Chapter VI

S. Koppleberg continues the work on ultraproducts of finite cardinals (and gets quite general sets).

4. Numbers of non-isomorphiclpairwise non-embeddable models (Chapter VIII)

There are improvements and (more) independence results and xxiii

xxiv ADDED IN PROOF

counterexamples restricting possible improvements, see [Sh 80a], [Sh 841, [Sh 85e, Ch 1111, [Sh 88b] and [Sh 891.

The improvements are : (a) We improve the proof of VIII, 3.2 (give new results for non-

first order cases). See [Sh 85e, Ch. 1111. (b) Improving VII, 1.8: if T E T , are complete, countable T , KO-

unstable not stable, then IE(h, T,, T) > a2} for h > KO by [Sh 88b, Theorems 1 and 21.

(c) Improving VIII , 1.2 if T E T,, ID(T)1 > IT1l+, then IE(h, T , , T ) > lD(T)INo for h > JT,I+. See [Sh 88b, Fact 31.

(d) If T is unsuperstable, h > (T,I +KO, T G T,, thenlE(h, T, , T ) = 2A (and general principles), which implies this is proved (see [Sh 831 for many cases and some principles, and [Sh 891 for all).

For independent results : (A) It is consistent that for some unsuperstable countable T , and

T, , T G T, , lTll = K, and I ( K , , T , , T ) = 1 (by [Sh 8OaI). (B) It is consistent that for T = Tord, the theory of dense linear

order, or TZd, for some T,, A = lTll > KO, IE(h,T, ,T) = 1 (by [Sh 80a] T = Tord, h = K,, by [Sh 841 T = TZd, h = K , , or one of many others). Mekler generalizes the results on graphs to classes of universal theories

(C) For a quite comprehensive list of restricting independence results ID(T)I > IT1[, see [Sh 88bI.

5. Decomposition theorem (Chapter X I )

We here prove [by XII, XI 2.4, for (Tt, Ga)]

(*) if T is countable superstable without dop and otop, then for every M there is a non-forking tree (N, : 7 E I ) , M primary over UN,, IIiVJI < 2'0. Moreover, any suitable tree of this form by (i.e. using (Ti,, E,)) can be extended such that the model M is prime over it.

By [Bush 891 this missing axiom for, e.g., (TLo, C,) holds (in the cases we need) so we can apply this pair in Chapter XI11 to improve the cardinals in the results ( 11N711 < KO).

Earlier results do this too. Saffe (in Trento 7/1984) suggests using the notion of elementary

A G,*B if for every ~ E B , finite d and GEA and p such

submodels used in IX, 1 .1 , 1.5 used also in V, $6, i.e.

ADDED I N PROOF xxv

that l=cp[b,a] there is F E B , satisfying l=cp[6’,it] and

and prove (*) for (Tt, c ~ ) . Hence, we can get there IIN711 < No. But he does not prove Ax(C1). Somewhat later, Harrington proves it and the author notes that since trivially [A c ,B*A c,B] it follows that the other axioms which hold for (Ti,, c,) hold for (Ti,, s d ) [see iTI, 3.17(3)] [only for Ax(A5) use (Cl)], hence X, 2.7, X, 2.15, X, 2.16 ilpply. Thus, showing that very little need to be added to XI to get the above improvement, as well as replacing XIII, 1.1, 1.4, $2 “ A > ITI+2Ko” by “ A > ITI}, “(2No)+” by “K1”.

Bytp(6, a), A ) = R”(tp(F,a), A ) ,

Let us turn to the list of open problems.

6. Problem A7

The problem was: “IS every unidimensional, stable T, superstable ? ” Hrushovski answers positively, using the interpretation of groups.

7. Problem C4

The problem was: “Prove the main gap from a Fi-saturated strongly A-homogeneous model”. For A = KO, we compute the possible spectrum functions [Sh 88aI. Also, the related problem of a A-resplendent model, A > IT1, was solved.

8. Problem 0 2 (Classification over a predicate)

This is essentially solved (but using independence results for non- structure, and much is not yet written, see [PSh 851, [Sh 85d] and [Sh 86bl).

9. Problem 0 3

The classification of a universal class (i.e. ME K if and only if every finitely generated substructure of M belongs to K) is essentially solved see [Sh 85e] and [Sh 891, but not completely written.

see a work by Grossberg and the author (on the categoricity of w successive cardinals) in preparation ; and a work by Makkai and the author (and more in preparation).

On the categoricity of

xxvi ADDED IN PROOF

10. Chapter III , Section 5 (on instability and D ( T ) )

Newelski had shown that we cannot in general improve the results there.

11. SND

E. Hrushovski has proved that for countable first order T , SND(T) satisfies CH. We report below on other relevant works of the author.

12. Better invariants

We prove that if T is superstable without dop, for characterizing up to isomorphism He-saturated models we need only finitary invariants; more specifically we can use the theory of the model in the logic with quantification over sets which are the algebraic closure of finite sets [in a""] and has quantifiers on dimension.

13. Multidimensionality

As we say previously the results of V, $ 5 can be improved. In fact we have now a typescript [Sh 891 proving the following. Assume T is stable (first order complete) and it is multidimensional (see V, $ 5 ) , then there is a (stationary) type orthogonal to the empty set [hence : if T is stable in N, and a < p then T has 2 21,-al pairwise non isomorphic F;t -saturated models of cardinality N,].

14. Isomorphic ultrapowers

We proved that i t is consistent with ZFC, that there are countable models M , N which are elementary equivalent, but for every ultrafilter D on w , M"/D and N"/D are not isomorphic, moreover Th(M) = Th(N) does not have the strict order property and even for different ultrafilters we do not get isomorphic ultrapower. This will probably appear in the proceedings of the MSRI 10/89 symposium in set theory. It also seems that, e.g., in some models of set theory, some ultrafilter will not give isomorphic ultrapower in the use of Ax Kochen.

ADDED IN PROOF xxvii

15. Universal models

In a work in preparation by Kojman and the author more is done on characterizing the possible classes { A : T has a universal model of power A } for T first order, e.g., T is countable with the strict order property or unsuperstable and K, < 2'0, then T has no universal model in K,.

16. Resplendency

We note that the following are equivalent for a first order T : (a) T is stable, (b) if M is a ITI+-saturated model of T satisfying the following then it is saturated: for every c i € M for i < ITI, if q is a consistent theory extending Th(M,ci)i<lTI of power < IT[ then (M,ci)i<ITI can be expanded to a model of q.

17. Omitting types

We forgot to mention a work of Hrushovski and the author, to appear in the Israel J . of Math., characterizing quite fully the possible sets {h( T, p ) : p a type} where h( T, p ) is sup { (IM 11 : M a model of T omitting p } for T (countable first order) stable and for T superstable.

18. Universal classes

The appearance of [Sh 891 has now been delayed for several years. In addition to the four chapters which appeared (in [Sh 85e]), a reasonable amount of material has been available in the form of a preprint before those four were sent for printing (i.e., Chs. V, VI, and 111, $6). Meanwhile Ch. VII has been lengthened and rewritten and more material has been added (mainly 111, $7 ) . We shall summarize this below. Note also that works of Baldwin and Shelah on the primal framework are beginning to appear (it is a more general framework of course).

19. Chapter III

Sections 1-5 have been revised (mainly the later parts of $3, e.g., a complete proof that if in EM(1) we can define (in any logic) order

xxviii ADDED IN PROOF

on { ( q ( x t ) : i < p ) : t E I } then for any h 2 p+++ 17’1 we have 2A pairwise non isomorphic models of the form EM(I).

Section 6, a survey and results on black boxes, is mainly in more finished form (but there is the knowledge that, e.g., a club of 6 < u1 pp(&) = 2’8 helps to get better black boxes on (16+l)f).

We add section 7, done later, where it is proved that, e.g., IE(A, T,, T) = 2A when h > ITl\ (even if A is singular). Some of the cases depend on “guessing clubs in ZFC” which has meanwhile found other applications.

20. Chapter IV

work mainly on Axiom Framework 1. This has been revised but nothing novel added ; here and later we

21. Chapter V

In Section 1 we prove that if K is ( < p, < p)-smooth and ( < p+, p)- based and satisfies LSP(p), then K satisfies smoothness, is ,u*-based and satisfies LSP(p*) for every p* 2 p. For p = xK, if the assumption fails we have strong non structure results, hence we assume the conclusion. Having smoothness, we can allow ourselves to work inside the monster model 6 ; tp (@, A ) is now defined as {F(@) : F an automorphism of 6 over A } , (DK, p)-homogeneous is the replacement of y-saturated, NF(M,,M,,M,,M,) is the replacement of: Mi are algebraically closed sets (not necessarily universes of models of the fix first order T) M , c M l c M,, M , C M , EM, and tp(M,,M,) does not fork over M,. Now Sections 2-7 are concerned with “all that we know on stable theories for (TI+-saturated models holds in this context”. Well, some definitions have to be adapted - we deal only with types of the form tp(N,M), where NF(NnM,N,M, a), usually [IN 11 < xK. The proof of “ every type p , dom( p ) of power < p has a p+- isolated extension ” is somewhat more complicated (we have to add variables to the type during the extensions, in order to continue to deal with types of the allowed form only, i.e., of models).

22. Chapter VI

Here we generalize the superstable theory to our context. But first we have to be able to deal with types tp(a,M), i.e., types of elements.

ADDED IN PROOF xxix

We say tp(a,N) does not fork over M < N if for some Nl we have M < Nl and a E Nl and NF(M, N, N, , a). We define stationarization and parallelism too, and prove the expected results (though, e.g., the uniqueness of stationarization is no longer totally trivial). Next K ( K ) is no longer a cardinal but a set of regular cardinals < xK, K E K(K) iff for some (Mi , i < K ) increasing continuous sequence of Ma < 6, and some a E a we have : tp(a,M,) forks over Mi for every i < K . K is called superstable iff K(K) is empty. Under those definitions the theory generalizes - e.g., non superstability implies strong non structure theorems, the union of an <.-increasing chain of (DK,p)- homogeneous models of length 6, cf(6) not in K(K) is (DK, p)- homogeneous, also the theory of regular types. [Also generalizing Chapter X here is straightforward, and we can prove the categoricity theorem, i.e., like Los conjecture.]

23. Chapter V111

The aim of this chapter is as follows. We strengthen the notion of elementary submodels to : M < * N iff M < N and for every type over M , < M with < K variables with llM,II < h which is realized in N , is realized inM too. Our aim is to show that all the good properties still hold (i.e., Axiomatic Framework 1 ) . More exactly, we show it except when the class has some non structure property. Unfortunately one of these gives h (usually) A+ non isomorphic models (rather than 2”). Essentially the point is that we do not know to discard classes which behave like the class of linear well ordering. One of our problems is to get non structure from order. The problem is that the relevant order is not preserved when we pass to Ehrenfeucht Mostowski models. The solution is a theorem on the existence on indiscernibles which is novel also in the first order case (and is explained below), and then use universality. Subsequently we prove non structure when unions are not O.K.

Maybe it is the right place to say again what the point is of the indiscernibility theorem in Chapter I, last section, of [Sh 85el. For a first order T, to get an indiscernible subset J of I of power A, we need a large cardinal (indiscernible set rather than just n-indiscernible set). Of course we get the theorem assuming stability of T, i.e., no order on n-tuples for every n. There we say that if I consists of n- tuples it suffices to assume there is no long order on 2n-tuples (and if we phrased it rightly - on n-tuples). What we have to say on order,

xxx ADDED IN PROOF

stability and existence of indiscernibles, for any model - not necessarily of a stable theory, is said here in Chapter I, 2.12,2.10 [the first says there are indiscernibles in the presence of stability and lack of long order, the second says that the lack of long orders implies stability; the case of order of length w is spelled out in 2.11, and remember the trivial IS(A)I < &,lSv(A)l (or see 11, 2.15)].

The indiscernibility theorem we use in ([Sh 891) Chapter VII says, e.g. for first order stable T, that if ( A t : t ~ P G z ( A ) ) is given, IAJEK, ~ ( 5 " ) < K (where Ys2(h) = { t : t E A, It1 < 2)) then we can find Y G A, IYI =p+ and (Bt: ~ E Y ~ ~ ( Y ) ) such that: (a) (Bt : ~ E Y ' , ~ ( Y ) ) is independent, i.e., for every t , tp(B,, u{B,: ~ € 9 , z ( Y ) , t is not a subset of s}) does not fork over {Bs: s a proper subset of t } , (b) A, c B,, lBtl 6 K, provided that A > a 1 ( p K ) .

Similar theorems hold for the structures { p : p a finite sequence of ordinals < A ) and for the structures {p : p a strictly decreasing (finite) sequence of ordinals < A}.

NOTATION

We use i,j, a, 8, y , 6, 5 for ordinals; k, I, m, n for natural numbers (i.e., finite ordinals); K, A, p, x for cardinals (i-e., initial ordinals, inh i te if not stated otherwise). 8 is reserved for limit ordinals. i < a, K < A mean i E a, K E A, respectively.

A sequence is a function 3 whose domain, Dom 3, is an ordinal, which is also called the length of B and denoted by I@). B [ i ] denotes the image of i under 8, i.e., the ith member of the sequence. B 1 a (or sometimes I I a) is {(i, 8[i]): i E a n I @ ) } . B Q t means I = t 1 I @ ) ; we write B is an initial segment of t . For any set A, "A = {a: Z(3) = a, Range 3 E A}, "'A = UBtaBA. We sometimes write A", A'" instead of aA, "'A, respectively. Sequences of ordinals are denoted by q, v , p, a (U is also used to denote permutations). The empty sequence is denoted by ( ).

Ha is the ath infinite cardinal. 2,(A) = >(A, a) is defined inductively by >,(A) = A, and for a > 0, >(A, a) = zE<a 23(A*B); aa(No) is written aa. A + is the cardinal successor of A; A'" = &s,,<k AN. The cardinality of a set A is denoted by \ A [ .

L denotes a first order language with equality (unless L is defined explicitly, without mention of the equality predicate). Variables are denoted by x, y, z, finite sequences of variables by 3, g, 2, predicate symbols by P, &, R, and E (for equivalence relations), and function symbols by P. Terms of L me denoted by 7 and formulas of L (L-formulas) by tp, #, 8. tp(Z) denotes the pair (tp, Z) where all the free variables of tp belong to Range Z. We identify tp and lltp.

We also use L to denote the set of L-formulas or the set of L-formulas of the form tp(2) so always ILI 2 No. We consider L to have No vrtriables for cardinality considerations but we often add A new variables to L for compactness purposes while regarding ILI as unchanged.

T denotes a complete theory with infinite models in a first order language L = L(27); 1271 = IL(T)I. An L-model 211 is (12111, . . . , RM, . . . , . . . , PM, . . . where 1H1

xxxi

xxxii NOTATION

is a non-empty set (the “universe” of the model), and if R is an n-place predicate symbol of L then RM, the interpretation of R, is an n-place relation over 1M(, and if F is an m-place function symbol of L then FM, the interpretation of F, is an m-place function from to !MI. Since lMl is the universe of M , IlMll is its cardinality and it is called the cardinality of M .

L(M) is the language such that N is an L-model. Models are denoted by M, N and occasionally by 9. A, B and C denote sets contained in some [MI ; a, b and c denote elements of some ]MI and a, 6 and 2 finite sequences of elements; 8-6 denotes the concatenation of 7i and 6. We write iZ E A instead of E “’A and sometimes write E M instead of B E IMI. We often write just ti instead of Range 8, e.g., we write A u iZ instead of A U Range and if I is a, set of finite sequences we write U I or sometimes just I for U {Range 8: si E I}.

If p(Z) is an L(M)-formula E IMI Z(a) = Z(Z) we write M p[a] to mean p[a] is true in M . Th(M) is the set of all sentences true in M. M is elementarily equivalent to N, written M = N, if Th(M) = Th(N).

If P is a unary predicate symbol of L(M) we often write P ( M ) for PM (note that P ( M ) c ]MI) , and if cp(E; 8) is an L(M)-formula, Z E 1M1, Z ( B ) = Z(p), we write p ( ~ ; a) for (6: 6 E 1M1, Z(6) = Z(Z), M I= p[6; a]}.

M is a submodel of N, written M E N, if L = L(M) E L(N), [MI c INI, for every R E L and E IMI, si E RM O E E RN, and for every F E L and 7i E !MI, FM(a) = J”(7i). If Mi c M, for all i < j .c a then the union of the Mi, Ui<a Mf, is the model M such that 1M1 = Ufca [Mi l , Mf E M for i < a, and L(M) = Ut<,L(Mf). M is the L-reduct of N, written M = N L, if L ( M ) = L E L(N), 1M1 = IN!, for R E L, RM = RN, and for F E L, PM = FN. If M is the‘L(M)-reduct of N, then N is called an expansion of M .

A denotes a set of formulas of the form p(2; 8). An m-formula, or p-m-formula, is a formula of the form p(Z; g) or cp(2; a) where Z(Z) = m and we regard p as a sequence of parameters for which we will usually substitute some and obtain p(2; a). We sometimes consider the same formula as an m-formula for several different values of m. A A-m-formula is a cp-m-formula, for some cp E A.

p is a A-m-type over A in M i f (1) p is a set of formulas of the form p(Z; a) where a E A E IMI,

Z = (xo,. . ., x , , , - ~ ) , and p(Z; p) or ~ p ( 2 ; 8) belongs to A. (Note that we identi& p and 1-,~ as mentioned below.)

(2) p is consistent with M , i.e., p is finitely satisfiable in M : for every finite q E p , M C (3Z) A\(PEP p.

NOTATION xxxiii

If we are not interested in A, we just write p is a A-m-type in B. Types are denoted by p, Q and T . We write m-type instead of L-m-type, A-type instead of A-1-type, p-m-type instead of {p}-m-type, and A-( < K,)-type instead of A-m-type for some m < w . But from 111, $ 3 on type [A-type] means (< K,)-type [A- ( < KO)-type].

p is a complete A-nz-type over A in M if it is maximd, that is, if z E A , p ( ~ ; 8) E A , @) = I @ ) , then p(Z; Z) E p or a) E p. z realizes p in M if V(X; a) ~p * M l=pl[a; a]. We identify 7-p with pl.

p restricted to A, written p A, is {?(it; a) ~ p : E A}; p restricted to A , writtenp A, is {?(it; 8) ~ p : p(f; a) E A or -,p(E, 8) E A } ; ~ restricted positively to A, written p t + A, is {p(p; a) ~ p : p ( ~ ; jj EA}. If A = {p} we write p t p and p r + p instead of p {p} and p I+ {p} respectively. The domain of p , Dom p , is the smallest set over which p is a type. We sometimes consider types with an infinite number of free variables.

Let A be a set of m-formulas, i.e., A = {p@; gf): Z(Z) = m, i c 1 A l } cll(A) denotes the closure of A under negation, i.e.,

cl,(A) = (pi(% gf): i < Id]} u {7p&3; &): i c pi} = {$f(% a): i < plwl}.

clf(A) denotes the closure of cl,(A) under conjunction, i.e.,

clr(A) denotes the Boolean closure of A, i.e.,

c l W = { V A $dZ; #if): $&; E c l l ( 4 IwI, lvfl < go}. fEW jEUt

cl,(A) denotes the closure of A under negation, conjunction and permutation of variables, i.e., cl,(A) = @(A') where

A' = {pf(u(Zf)): = iiT&, 0 a permutation of 2,

(strictly speaking of Range Zf), i c Id]}.

We use mainly clg(A). We define pf as p if i = 0 and lp if i = 1, pif("') is p if " . . . " is true

and lp if " . . . " is false. Sometimes when we do not want to write p(Z; a) out in full too many

times we write "let p = p(Z; Z)" in an abuse of notation. When we say a formula" we may mean p, p(E), p(B; 8) or p(B; Z). Letfbe a function. For a set A,f(A) = { f ( a ) : a E A}; for a sequence 3,

( 6

xxxiv NOTATION

Range I E Domf, f(3) is a sequence t , Z(Z) = I @ ) , t[i] = f(a[i]); for a type p , f(p) = {tp(Z;f(a)): tp(Z; a) ~ p } . (In practice there will be no inconsistencies.)

I, J denote orders or index sets. I, J denote sets of finite sequences; this has been reversed in Chapters I-V and IX.

Theorems, lemmas, claims, conclusions and corollaries are numbered together and are often referred to just by their number, e.g., 11, 4.16 means Theorem 16 of Section 4 of Chapter 11. Within Chapter I1 itself the theorem would be referred to as 4.16. Exercises, problems, questions and conjectures are numbered together; they and definitions are referred to always with name attached, e.g., Definition VII, 3.2 means Definition 2 of Section 3 of Chapter VII. Within the chapter itself, the chapter number is not given. However in Chapters X-XI11 definitions are numbered together with theorems.

When we wish to write two almost identical statements we often write only one and insert the variations from the other statement at the appropriate places in round or square brackets, see e.g., Lemma I, 1.12 and Definition I , 2.2(2).

CHAPTER I

PRELIMINARIES

1.0. Introduction

We here introduce our notation, which is quite standard, then, in Section 1, give some of the classical theorems of model theory (compactness, Lowenheim-Skolem, and the existence and uniqueness of saturated models). They are included mainly for the sake of completeness. Some of them are proved later (compactness after VI, 1.3, Lowenheim-Skolem in VII, 1.1-1.3).

In Section 1 we also introduce &-a saturated model such that we restrict ourselves to elementary submodels of it of cardinality < I( all. By this way we do not lose generality, and hopefully improve presentation.

In Section 2 we deal with some problems concerning stability of models rather than of theories (on which we concentrate in the rest of the book). We define tp,(a, A, N), @'(A, M), stability, indiscernibility and splitting; and then investigate the connections between stability, order and the existence of indiscernible sequences: By 2.6 if, e.g., pi = tp(a,, A u U {a,:j < i}) does not split over A, j p , c p , for i < a, then {ai: i < a) is an indiscernible sequence over A. (This will be the way by which we construct indiscernible sets.)

We prove in 2.8 that if Th(M) is stable in A, I E M , 111 > A, IAI I A, A G N, then there is a J E I indiscernible over A, IJI > A. The method here is to find p E Sm(C, N) and B E C, such that for every C', C 5 C' E IN[, p has a unique extension in Sm(C, N) not splitting over B, and when IC'I s A, this type is realized by some E E I.

In 2.9 we prove that unstability implies the existence of ordered I (remember that by 11, 2.13, T is unstable if€ it has the order property).

1.1. Preliminaries and saturation

We shall use freely the: COMPACTNESS THEOREM 1.1: If T , is a set ofsentences,und every finite subset of T , hm a d e l , then T , hae a model.

1

Proof. See Chapter VI after Lemma 1.3.

DEFINITION 1.1: (1) M is anelenzentaryszlbW1 of N , M < N if and for every tioned otherwise, we assume L ( M ) = L(N) . )

E N E 1M1, 'p E L ( M ) , M C 'p[a] e N C 'p[a]. (If not men-

(2) M,, i < a is an elementary chain if i < j < a => M, < M,. (3) M,, i < a is a wntinwMLB elementary chain if it is an elementary

(4) If iK is an elementary submodel of N , N is called an elementary chainandS<a=>M,=U, , ,M, .

extension of M.

LEMMA 1.2 (Tarski-Vaught Test) : M ie an elementary submodel of N iff M ie a m.&moukZ of N , and when 8 E 1M1, b E IN[ , p E L ( M ) , N C p[b, a], then there ie a b' E 1M1 for which N C p[b', a].

Proof. The "only if" is clear. So suppose the second condition is satisfied. We let E [MI, 'p(5) E L and we shall prove by induction on 'p that M C p[7i] o N C p[a].

If 'p is atomic, this holds as M is a submodel of N . If we get 'p by connectives, the result follows by using truth tables.

So we are left with the c&88 p = (3y)$(y, 3). If M C I@], then for some b E 1611, M C $[b, a], so by the induction hypothesis N C $[b, a] thus N k (3y)$(y, a), i.e., N C p[a]. If N C 'p[iZ], then for some b E INI, N C $[b, a], so by assumption for some b' E I M I , N C $[b', a], so by the induction hypothesis M k$[b', a] thus M C ' p [ i i ] . We have shown iK C p[a] o N C p[a] and thus have finished the induction and the proof of the theorem.

THE ELEMENTARY CHAIN LEMMA 1.3: (1) If M,, i < a is an elementary chain, then M, < U, < a M,.

(2) < ie a partial order; and if M < N, then p is an m-type in M iff p ie an m-type ouer 1M1 in N .

If M, 4 N , 1 = 1,2 and Ml C Ma, then M I <Ma, 80 if M, < N and Mf E M , f o r i < j < a,tlrenU,,,M,<N.

Proof. (1) Let M = U, < a M,; so by definition, M, c M for i < a. We shall prove that for every i < a, E lM,l, p(3) E L ( M , ) , hi C 'p,P] e 211, C @I; by induction on p.

The only non-trivial case is p = (3y)$(y, 3). If Mt C p[a], then for

CH. I, 8 13 PRELIMINARIES AND SATURATION 3

some b E lM,1, Mi C #[by a] hence by the induction hypothesis M C #[by a] so M C (p[a]. If M C p[Ti], then for some b E 1M1, M C #[by a]; but as lMl = ul<cr lMil for some j, i I j < a, b E IM,I. By the induction hypothesis M , C #[b, a], hence M , C (p[zi], so, aa M , < M,, M , k ~ [ a ] .

(2) Immediate.

DOWNWARD LOWENHEIM-SKOLEMTHEOREM 1.4: I f A E 1M1, then M irag an elementary subntodel N (L(N) = L(M)), A E INI, IlNll = I 4 + ILL

Prmf. See VII, 1.1, 1.2 and 1.3.

LEMMA 1.6: (1) If a, E 1M1 for i < a, then (M,.. . ,a , , . . .),<=<

(2) If M c N , Th(M) G Th(N) and in M every element i8 an indi- ( N , . . . , a,, . . . ){ <(1 iifs iK < N .

vidual constant, then M < N .

Proof, Immediate.

LEMMA 1.6: If p i8 a type in M , then in 8 m e elementary extenSgon N of M , i8 T d i Z e d and IlNll I 11M11 -k IL(bl)l.

Proof. By 1.5(1) we can assume every element of M is an individual constant (so we can assume p is over 0 ) . Let c be a new individual constant and

Tl = Th (N) " { d c ) : d4 E PI.

As p is finitely satisfiable in M , every finite subset of T , has a model. Hence by the compactness theorem, T, haa a model N , and by the downward Lowenheim-Skolem theorem we can assume llNll I I T,I = )1M)1 + IL(M)J. As N is a model of Th(M), if for every individual constant c , E L(M) we identify c y and c:, then M will be a submodel of I?. By 1.6(2), M < N and by the definition of T,, cN realizes p .

DEFINITION 1.2: (1) A model M is A-saturated if every type in M over some A G 1M1, IAI < A, is realized in M .

(2) A model M is A-compact if every type in M of cardinality < A is realized in M .

(3) A model M is 8uturated (compact) if it is IlMll -saturated (-compact).

4 PREIJXD?ARIES [OH. I, 8 1

THEOREM 1.7: (1) 8wppse llMll + JL(M)I I A = A'", lMl infinite, then M ha a K - C O ~ ~ ) C G C ~ elementary ext& of cardinality A.

( 2 ) 8-e lldlll + IL(M)( I A = A<", Lo E L(M) , M infinite and for every m thme are I A conzlplete L,-m-typea over 0 in M . Then M ha an elementary extension of power A, w h e L,-reduct ie K-saturated.

Proof. (1) We shall define by induction on a s A models Ma such that: (i) Mo = M and for limit 6 s A, (ii) 1Mal # lHa+ll, M a < M a + l , IIMa+,II I A + IIMaII;

= U a < d Ma;

(iii) every type over Ma of cardinality < K is realized in Ma + ,. We can easily prove by induction on a s A, that < a * M b < M a

(by 1.3) and that llMall I A; t~ lMal # lMa+,1 clearly lliKAII = A. NOW if p is a type in MA, lpl < K , then for some a < A, p is over lMa( (as lpl < K s of A because A<" = A). As M a < M A , p is in Ma, hence realized in M a + , SO (as Ma < M a + , < MA) p is realized in MA. SO M A is K-compact, IIMAII = A and M <MA (as M = N o <MA). SO it Suffices t0 define the &fa's; SO let No = dl and

Now suppose Ma is defined, and we shall define Ma+, . Let (I)i: i c A} be a list of all types in Ma of cardinality < K (as llMll + IL(M)I I A = A<", their number is s A; by allowing repetitions we get i c A). M7e define an elementary chain Mk, i I A, such that M: = Ma

(possible by 1.6 and 1.3). As 1M1 is infinite, also 1M;I is infinite, hence qa = {x # a: a E [ M i l } is finitely satisfiable in M i ; hence by 1.6, M i has an elementary extension of cardinality I A which realizes qa, and we call it Ma+, . So clearly M , < M ; < M , + , , lMa+,l # lMal, and every type in Ma of cardinality < K is realized in M a + l . We have finished the definition of the Ma's, thus the proof,

= u # < d &fa.

11Mk11 I A, Mt = u i < d NL (for limit 8 5 A) and Mk" IX3aliZeS pi

(2) A similar proof.

DEFINITION 1.3: Let L(M) E L(N). An (M, N)-eZementarry mapping is a (one-to-one) function f, Dom f E IMI, Range f c IN!, such that for every a,, . . . , a,, E Dom f and formula tp(xl, . . . , 2,) E L(M), M C Cp[a,, . . . , a,] o N C tpV(a,), . . . , f(a, ,)] . When no confusion can azise we omit " ( M , N)".

Remark. Note that if M = N, then the empty function is an ( M , N ) - elementary mapping. We denote elementary mappings by f, P and g.

OH. I, 8 13 PRELJMINARIES AND SA"RA!L'IOW 5

DEP'INITZON 1.4: (1) M is ieonwrphic to N, Written M z N, if L(M) = L(N) and there is an (M, N)-elementary mapping f with Dom f = 1M1, Range f = IN]. f is called an M m p h i m n from M onto N.

( 2 ) f is an auhwrphim of M if f is an isomorphism from M onto M.

DEP'INITZON 1.6: (1) M is K - h o n t o g e n e o w , if for every (Af, M)-elementary mapping f , lDom f I < K , and every a E IMI, there is an (bl, M)- elementary mapping g which extends f , Dom g = {a} u Dom f.

( 2 ) M is etron~ly whontogeneow if for every (My M)-elementaxy mapping f , lDom f I < K , there is an automorphism of M extending f.

homogeneous. (3) M iS 8trOnglY K - S d U Y d d if it iS K - & W & b d and SkOIlgly K-

Proof. The proof is, in fact, identical to the proof of 1.7, provided we prove the analog to 1.6 :

(*) If f is an (My M)-elementary mapping, a E ]MI, then there is an elementary extension N of M, and an extension g of f , which is an (N,N)-elementary mapping and Dom g = {a} u Dom f .

Proof of (*). Let Dom f 2: {a:: i < a}, and f(at) = b:, and

23 = {d% bl(l,, * ' * Y b:,,,): i(l)Y * * * < a, M != d a , a,(,,, * * * Y a:(&

We want to prove that p is consistent, i.e., finitely satisfiable. Clearly p is closed under conjunctions, so it suffices to prove, for ~ ( z , btcl,, . . . ) E p that bl C (3z)(p(2, bi(l), . . .). By the definition of p , M C v[a, . . .I, hence iK C (3z)v(z, a t ( l ) j . . .); so aa f is an elementary mapping, M C (3z)cp(z, bicl,, . . . ). So p is consistent, hence by 1.6, p is realized i,n some elementary extension N of M ; say by b E INI. Let us exfendfto g by letting g(a) = b. It is easy to check that N, g satisfy our demands.

DEFINITION 1.6 : ( 1) M is h-univereal if for every N elementarily equivalent to M and A c IN1 , IAI 5 h there is an (N, M)-elementary mapping f, Dom f = A. M is universal if it is IIMII-UnivemI.

(2) M is ( < h)-univereal if M is p-universal for every p < A.

6 PRELIMINARXES [OH. I , 3 1

THEOREM 1.9: (1) EVWY A-saturated nzodel A-mw. (2) If A > (L(M)I, then M is A-saturated ifl M is A-mpac t . (3) Every A-saturated model is A-honzogeneoua and A-universal. (4) Every ( c No)-univer8al, A-homogeneous model is A-saturated. ( 6 ) If p I A, then every A-saturated model is p-saturated, every h-

cumpact nzodel is p - m p a c t , and every A-universal model is p-universal. (6) Let A > No be a limit cardinal. If M is p-saturated (p-homogeneous)

for every p c A, then M is ~-8aturated (A-homogeneous). (This does not hold for universality but holds for compactness.)

(7) If M is A-saturated (A-mpact ) and Lo E L(M), then M 1 Lo is A-saturated (A-compact).

(8 ) Every strongly A-homogeneous model is A-homogeneow.

Proof. (1) Immediate. (2) Immediate. (3) For this it suffices to prove:

CLAIM 1.10: Let A E B E IMI, IAI < A, IBI I A, f an ( M , N ) - elementary mapping, Dom f = A. If N is A-saturated OT N = M is A-homogeneoue, t h n we can extend f to an ( M , N)-elementary mapping 9, Domg = B.

Proof of 1.10. Let B = {b t : i c IBI}, B, = A u {bt: i < a} (so lBal c A for a < IBI) and we shall define by induction on a 5 IBI ( M , N ) - elementary mappings f a ; fo = f, fd = Ut<afr, fa+l extends fa and Dom fa = B,, so clearly g = f i B l is the desired mapping. For a = 0, a limit the definition is trivial. Iffa is defined, N = M is A-homogeneous; the existence of a suitable fa+ follows by the definition of homogeneity. If f a is defined, N is h-saturated, the existence of is. proved like (*) in the proof of 1.8.

Proof of 1.9 (continued). (4) Let N be ( < No)-universal and A-homogeneous; and we shall prove it is A-saturated. We prove this by induction on A. So let A E IN I , [ A I c A, p a type over A in N and we shall prove that p is realized in N . By 1.6 there are M and a such that N < M a E [MI and a realizes p. There is an (M, N)-elementary mapping f, Dom f = A u {a} [if A is finite-by the hypothesis; if A is infinite- N is I A1 -saturated by the induction hypothesis, hence N is IA I -universal by 1.9(3)]. Then g = f 1 A is an ( N , N)-elementary mapping, and so g , = g - l is an ( N , N)-elementary mapping. As N is A-homogeneous,

OH. I, 8 13 PRELIMINARXES AND SATURATION 7

IA I < A, some (N, N)-elementary mapping ga extends g1 and Dom ga = Dom g, u { f (a)}. It is easy to check that g2( f (a)) realizes p . (6) Immediate. (6) Immediate. (7) Immediate. (8) Immediate.

THE UNIQUENESS THEOREM 1.11 : If M = N , My N are saturated and have the same power, then My N are isomorphic. Moreover, any ( M , N)-elementary mupping f with domain of curdinulity < 11iW11 can be extended to an isomorphiam from M onto N.

Proof. Let A = 11M11 = IlNIl, [MI = {a,: i < A}, IN1 = {b,: i < A}. Now we define by induction on a < A, (M, N)-elementary mappingsf,, such that: f o = f (ortheempty function), f d = lJ,<d f,, lDom f a + , - Dom f,l I 1, lDom f a [ I lDom f o l + 1.1 < A; and a, E Dom faCr+,, b, E Range fa ,+a. This is possible by 1.10; 80 fA is the required isomorphism.

LEMMA 1.12: Let M be a A-cumpact [A-eaturated] model. If p is a ( < KO)-type (or even a type with I A free variablu) in M , lpl < A [IDompI < A], t h p is realize& in M .

Proof. Easy, so left to the reader.

Remark. We use Lemma 1.12 very frequently but without explicit mention.

Conventions. We work within ZFC set theory in the standard way, i.e., all the symbols and formulas of a language L are sets of heredita,ry power < ILI.

We assume also that there exists an inaccessible cardinal I? (i.e., a regular cardinal I? such that (VA < 4(2A < I?)) bigger than all the cardinalities we shall deal with. Note that (R(I?), r) is a model of ZFC set theory (R(I?) is the set of all sets of hereditary power < E ) .

Every theory T will have a saturated model 6 = 6(T) of cardinality I?, (exists by 1.7) and, unless otherwise stated M , N will be elementary submodels of Q of cardinality < I?; A, B, C subsets of 161 of cardinality < I? ; a, by c , d elements of 161; and “type” will mean “type in 6”. We will not distinguish between individual constants and elements of 6.

These conventions are convenient as now all formulas and sets of formulas of L are elements of (R(k ) , r), and M c N o 211 < V , and in writing M C ~ [ C X ] we can omit M. We shall deal with a fix complete T (unless stated otherwise).

8 PREIJMINARIEIS [CH. I , 8 1

All our important properties (such as rank, forking eto.) are preserved under automorphisms of B and therefore a h under elementmy mappings as an elementary mapping is a (B, B)-elementary mapping and can be extended to an automorphism of B by 1.11. We do not state and prove these preservation theorems as they are trivial.

The assumption on E is justified as (1) It does not, in fact, add any extra axiom of set theory as a

(2) Any model of T of cardinality < t? is isomorphic to some M < B. (3) I f M < S , llSll < t?,thenSisisomorphicover 1M1 tosomeN<Q

(“over IMI ” means an isomorphism which is the identity on ]MI) . If we assume in addition GC (the axiom of global choice) we can take

B to be a saturated class-the union of an elementary chain {MA: A a cardinal, M A is A-saturated} of models of T. It is known that ZFC + GC is equiconsistent with ZFC. See p e 711 and [Ga 761.

hypothesis to our theorems.

EXERCISE 1.1: Prove that every finite model is A-saturated (for any A).

EXERCISE 1.2: Give an example of a (< A)-universal model which is not A-universal for every A.

CONCLUSION 1.13: I f M i g a saturated model, T = Th(M) G T I , T, wnktent, ITl! s IlMll = A, then M can be expanded to a nodel of T,.

Proof. We can assume T , is complete. If T , has a saturated model of cardinality A then its restriction to L = L( T) is also saturated by 1.9( 7) and the theorem follows by 1.11. In generd let N , be a A + -saturated model of T,, N the L-reduct of N,. Define inductively on a < A, N:, N i such that: N i < N , , N: < N , IlNiII = A, < a implies IN;] c IN;\ G [ N i l , and N: is saturated. Now N o = Ucr<A Nz has cardinality A; and when A is regular is trivially saturated. For singular A by VIII, 4.7 T is stable, K ( T ) I cf A, so 111, 3.11 implies N o is saturated (no vicious circle arises in this proof).

EXERCISE 1.3: Prove 1.13 directly.

EXERCI8E 1.4: Call M P-saturated, P a one-place predicate, if every typep over A G IHI, IAI < ~ ~ H ~ ~ , such that P(z) €13, is realized in H.

(1) Prove that if M is saturated, N is P-saturated, L(N) E L(M), llMll = 11N11, M 1 L(N) = N, then there is an (N, M)-elementary mappingf: IN1 + lMl,f(PN) = PM.

CH. I, 5 21 ORDER, STAB- AND MDISCERNIBLES 9

(2) Suppose L(T) has the one place predicate P and the two place predicate c, and for every model M of T and a,, . . . , a,, E PM there is b E PM such that M C ( V ~ ) ( Z E b = V;=l z = a,). Prove that if for a < S < A+, IIM,II = A, P(M,) = P(Mo) , M , is P-saturated, then

(3) Prove that if a theory T has a model My 11M11 > IPMI 2 KO A = A < A > IT1 + No, then T has a model N , llNll = A + , IP(N)I = A. (Show first that we can assume w.1.o.g. L(T) has a predicate E aa in (2), and then define inductively saturated models 211, (a < A + ) of T , llMall = A, P(M,) = P(Mo). For a = 0, 1 use the msumption, for limit 6 use (2) and (1) and for a = f i + 1 remember M , z Mo.)

U, < d M , is P-mturated.

1.2. Order, stability and indiscernibles

DEFINITION 2.1: Let A c !MI. (1) tPA(6, A, M ) = {'p(3; a)t: a E A , t E (0, l}, 'p Ed, M C 'p[6; B13. (2) #?(A, M ) = {tPA(6, A, M ) : 6 E "IMI}. (3) If A = L we omit it, if m = 1 we omit it, if M = B we omit it,

and if A = $4, M = 6, we omit them.

Remurk. If tp(B, A ) = tp(6, A) and E E A, then tp@-5, A) = tp(6-E, A).

DEFINITION 2.2: (1) The model M is stable in (A, A ) if for all A c 1611, IAI s A and m < W , IS?(A, M)I s A.

(2) The theory T is sktble in (A, A ) if every model of T is. We sometimes say "M[T] is (A, A)-stable", instead of "M[T] is stable in (A, A)" .

(3) IfA = L we omit it. (4) T is stable if there exists a A in which T is stable. ( 6 ) T is superstable if there exists a A such that T is stable in all

p 2 A.

LEMMA 2.1: If A is regular, A c IMI, IAI < A, but ISm(A, M)1 2 A, then there ie aJinite B c 1611, IBI < my such that 1S(A u B)I 2 A.

Proof. By induction on m. Form = 1, trivial. Now assume the claim for m and look at Sm+'(A, M). If IS"(A, M)I 2 A we are through by the induction hypothesis. Soassume ISm(A, M)1 < A. For allq ES"+,(A, M ) define

q* = {(3zm)$(zo, . . . y 2,; a): $(so, . . . y 2,; a) E q}.

10 PREIJMINARIES [CH. I, 8 2

Clearly q* is a (consistent) m-type over A and haa a unique extension q+ E P ( A , M). Now I@+: q ~ 8 " + l ( A , M)}1 s I P ( A , M)1 < A, so by the regularity of A, for some p E P ( A , M) l{q E P + ' ( A , M ) : q+ = p}I 2 A. In other words p has 2 A extensions in 8"+ l(A, M). Thus for any (bo, . . . , b,,,-,) rea,lizing p in M !&(A u {bo, . . . , bm-,})I 2 A.

Remurk. If M is (IAI+ + No)-homogeneous we can strengthen the above conclusion to J8(A u B, M)l 2 A.

COROLLARY 2.2: T is stable in p iff for every A, IAI s p

s P. IB(A)I

Proof. p+ is a regular cardinal.

DEFINITION 2.3: (1) (2: i < a} is a A-n-indiscernible sequence Over A if for every io < . - - < in-' < a, jo < - - < jn-l < a, and permutation u of (0, . . . , n - l},

t p d ( 8 ( O ) - . . . -af'(m-l), A) = t p d ( g j U ( O ) - . . . majUCn-l) 9 A ) .

If i < j < a * # # a' we call the sequence non-trivial. (2) If A = L we omit it; if the sequence is A-n-indiscernible for all n

we say it is A-indiscernible; if A = 0 we omit it; if A = {Q} we write Q

instead of A. For simplicity we treat I = {a*: i < a) aa if it were ordered by < . We always aasume Z ( 8 ) = Z(7io) = m. I is A-( <n)- indiscernible if it is A-m-indiscernible for all nz < n. We sometimes write (A, n)-indiscernible instead of A-n-indiscernible.

DEFINITION 2.4: (1) {a*: i < a} is a A-n-indkcernible set over A if for any two sets of n distinct ordinals < a, {io, . . . , in- l}, { jO, . . . , j.- l},

tpd(sjfoh. . .n@"-l, A) = tpd(ajo". . A) . If i < j < a 8 # iij we call the set non-trivial.

(2) We adopt the same conventions as in Definition 2.3(2).

LEMMA 2.3: (1) If A c A, and I = {#: i < a} is a A,-n-indiscernible set [sequence] over A, then I is a A-n-~ndiscernible set [sequence] Over A.

(2) If I is a ~-n-indiscernible set [seq-] over A for all Q E A, then I is a A-n-indiscernible set [sequence] over A.

(3) Let m < w and for di k < m let A, be aJinite set of form& and nk < w. Let a 2 n = max{nk: k < m} [a > n]. Then there is aJinite A such that for any I = {$: i < a}, I is a A-n-indiscernible set [seqt44?nce]

OH. I, 5 21 ORDER, STABILITY AND INDIBCERNIBLES 11

ifl for all k < my I is a A,-n,-indiscernible set [sequence]. ( A &o depends on m = l(@.)

(4) I f {8: i < a}, a 2 w [a a limit ordinal] i s a A-n-indiscernible set [ s e q ~ ~ ] over A and p > a, then we can de$w 8 for a 5 i < p such tluz.t {#: i < 8) is &o a A-n-indbcernible set [seqwnce] over A. (6) If (it = a) E A, then a trivial indiscernible set or sequence lrae only

one element.

Note. The requirement in (4) that a be a limit ordinal is n e v : take a = w + 1, A = {(p(z, y)}; and let CE Ccp[at,a'] 8 i < j < w + 1, (E k (+)cp(a", y ) ; then {ai: i < w + 1) is a A-2-indiscernible sequence which cannot be extended.

The proof is left to the reader.

DEFINITION 2.6: Let I be an W t e set of finite sequences, all of the tame length, (p(Zo,. . . , P - l ; g ) a formula, and E a sequence. cp(ito, . . . , 3-l; a) is connected and antisymmetric over I if for m y n distinct sequences sio, . . . , 3-l from I there is a permutation u of n such that kcp[7iU(O), , , . , 7iU("-'); E] and there is a permutation u of n such that k+Ti"(O', . . . , Zj'(n-l). Y 51.

THEOREM 2.4: (1) If I is an inJinite set of sequence.3 of the same length, A , A and n arejhite, then I iraS an inJinite subset {8: i < w} which is a A-n-indiscernible sequence over A.

(2) IfA,A,narefinite,thenforeveryk < wthereisl = 1(1Al,A,n, k ) < w such thd every set of sequences (of the same length) of cardinality 2 1 iraS a subset of cardinality 2 k which b a A-n-b&mrnible sequence over A.

Proof. (1) follows from Ramsey's theorem and (2) from the finite analog of Ramsey's theorem. (See 2.1 of the Appendix.)

DEFINITION 2.6: (1) The type p (A l , A,)-splits over the set A if there me 6, E such that tpAl(6, A ) = tpAl(E, A) but there is Q E A, such that v(Z; 6), +I; E) El).

(2) p splits over A if p (L, L)-splits over A'.

Remark. Clearly if p splits over A 2 B, then p splits over B.

LEMMA 2.6: For 1 5 n < y I w let A,, be sets of formulas closed under permutations of variables. Let I = (8: i < a} and A, = U (8: j < i}

12 PRELIMINARIES [OH. I, f 2

u A . Assume that for all i < a, 2 I n + 1 < y , p r = tp(tif, A,) does not (A,, A,+,)-split Over A and that p j A, E pi A,, for all j c i < a, 1 I n c y. T h e n I is a A,-n-indiscernible sequence Over A for 1 4 n < y.

LEMMA 2.6: L e t M be stable in h and assume that in M :

(*I There is n o n c o and increasing sequence and ~ E S " ( A ~ , M ) such that p r A,+1 splits over A, for all i c A.

If A E !MI and I E "1611, III > h 2 IAI then there is I' E I , II'l > h such that I' is an indiscernible sequence Over A .

Proof. First we shall show:

(**) There exist B, C, A E B E C E 1611, ICl I A, and p ES"(C, M ) such that

(1) for all C', C G C' s IN[, IC'l I Xp has an extension p' ES"(CI) realized in I - "C' which does not split over B (in particular p does not split over B), and

E C such that tp(E, B) = (2) for all E' E /MI there is tp@', B).

Proof. Let i(0) c i(1) c . . c i(n - 1 ) < a, j (0 ) c j (1) < . . . c j (n - 1 ) c a, Q = Q ( Z O , . . . , Zn- l; g ) E A,, and E E A. We must show I=Q[T~"O), . . . , a'("- l ) ; El iff t ~ [ ? i j ( ~ ) , . . . , aj("- I); El. The proof is b y induction on n. For rc = 1 , p, t A, 5 pj(,, t A,, pic,, t A, so the result is obvious. Now assume the result for n. Let = max{i(n), j(n)). By the induction hypothesis $(,In. . . a"n- l ) and af(o)^. . .âj(n-l) realize the same A,-type over A. Now ps does not (A,, A,+,)-split over A so if Q ~ A n + 1 , then

(by the remark after Definition 2 .1) . Thus since pjcn, A, + ,, pi(,) A, +

E 2% IA,+l we get

OH. I, 9 21 ORDER, STABILITY AND INDISUERNIBLES 13

By way of contradiction assume -,(**). We define by induotion tm hmwsing aequence {B,},,, suoh that B, s ]MI, lBfl 5 A as follows: Bo = A , Bd = uf<d Bf for limit 6. Assume Bf is defined. Let 0, 2 B,, C, c 1M1, IC,l 5 A be such that for all B ' E 1M1 there is EEC, with tp(8, B,) = tp(B', B,) (this by the A-stability of M). Since +**) for every p E P ( C , , M) which does not split over B, there is CL contradicting (**)(I) for C = C,, B = B,. That is, C, c Cl, c IMI, ICLl < A, and p has no extension in Sm(CL) which is realized in I - m(Cl,) and does not split over B,. Define B, +, = U {Cl,: p E Sm(Cf, M), p does not split over B,} u C,.

NOW let B E I - "BAY andp = tp(E, BA). We shdl &OW that p 1 B,+, splits over B,, in contradiction to (*). If not, then certainly q = p 1 C, does not split over B,; q E Sm(C,, M); thus C; is defined and C; E B, +,, So p 1 C; does not split over B,. But p 1 C; is redized by B E I - "(C;), contradicting the definition of C;. So p 1 I?,+, does split over B,, a contradiction. "his proves (**).

Now define by induction i? E I for i < A+ : If B j is defined for j < i let Cf = (J { B j : j < i} u C and let p , E P'(0) be an extension of p which is realized by i? E I - "(0) and does not split over B (lCfl I A). By Lemma 2.6 we just have to show that p j E p , for j < i. Let v(Z; 6) Epj . Let 6' E C be such that tp(6', B) = tp(6, B). Since p j does not split over B, v(Z; 6) Epj * v(Z; 6') Epj . Also p j extends p so ?(a; 6') ~ p . But so does p,, hence v(Z; 6') €9,. p , does not split over B so v(Z; 6) Ep, and the proof is complete.

LEMMA 2.7: If T i8 stable in A, then every model of T 8atisJiee (*)from Lemma 2.6.

Proof. By way of contradiction assume that {Af}tLA is an increasing sequence, p ES"(A~), and p 1 A,,, splits over A,. Choose @, i? E A,, , such that tp(6,, A,) = tp(i?, A,) and v,(Z; @), 7pf(Z; 8) E P for some formula v,. Let 7i realizep. Let p = mi+: 2y > A}. Define by induction on Z(q) elementmy mappings P,,, r ] E u r 2 , Dom P, = Ale,,): If Z(q) = 0 P, is the identity on A,. If 1(q) = 6 then P, = u a < d PVla. If l(r]) = i + 1 let P,,-<o> be an arbitrary extension of P, to A,+, . B',,-<,> will be

Q is an elementary mapping and P,,-<,> will be an arbitrary extension of B to A,,,. Let B = u {F,(6fni?): 7 ~ f + l 2 , i < p}. Clearly IBI 5 H,. &<,, Ia21 5 A. For every r ] E '2 extend P, to a mapping Pg with domain A, u {a}. Let p,, = tp(Pg(7i), B). Now assume r ] # Y E u2. Let a = min@: 7[8] + t@]}. W.l.0.g. q[a] = 0. Thus va(Z; P) EP and C(pa[7i; 5'1.

defined as follows: Let Q be such that 0 1 A, = P,andQ(6,) = P,,-<,>(i?).

14 PRELXMINARIES [OH. I, 8 2

Thus b,[P;(a); P;(@)], and cp,(Z; P;(&)) ~ p , . Similarly 7cpa(a(p; Pk(b)) ~ p , , . But Pb(6.) = F,,(F); thus -,q,(f; P; (b ) ) ~ p , , . In other words p,, # pv. Hence ISm(B)I 2 l{p,,: q ~ ’ 2 } 1 = 2u > A 2 IBI.

So T is not stable in A, a contradiction.

THEOREM 2.8: If T ie stable in A, IIl > A L IAI, then there is J c I , IJI > A, which is an indiscernible sequence over A.

Proof. Immediate by the two preceding lemmaa.

Remark 1. J is in fact an indiscernible set over A, see 2.9 and 11, 2.13(1) and (9).

Remark 2. If K is regular, and for all IAI < K and m < w , ISm(A)I < K, and there is A < K for which 2.6(*) holds, then for all A and I with. IAI < K s IIl, there is J c I, IJI 2 K, J an indiscernible sequence over A. The proof is similar.

LEMMA 2.9: If I is a A-indiscernible sequence over A but not a A- indiscernible set over A then there are n < w , cp(Zo, . . . , P-l; 8) E A, and 8 E A such that cp(fo, . . . ,P-l; 8) is connected and antisymmetric over I.

Proof. Immediate.

Remark. For a finite A, see a stronger result: 11, 4.10(4), Appendix 1.7( 2).

Proof. Let K = zOSp<h (IAI” + 2a’) and 8, i < K + be sequences from 1H1 such that the types tpJ8, A, H) are distinct and let y5 = +(p; Z) = cp(3; g). Now define by induction on a I A sets A, c ldll such that

THEOREM 2.10: Suppose cp = cp(P; g ) , l(3) = m, l(g) = n, IS;(A, M)l > xOsjb<h ( I A ( ' + 2a'), 2

(1) k t fl(31, 32 , z3; ill, g3) = [ ~ ( z i ; ga) ~ ( 3 1 ; ?&)I, 1 6 ) = l(iii), i = 1, 2, 3. Then there are in 1 M 1 sequences 8' i < h such thd M C B[Sit; af] o i < j.

(2) Swppose h -+ ( X ) % [e.g., h = x = ESo or x = K,+, h = ( P o ) + , see Appendix, DeJinitions 2.1 and 2.51 then there are in 1M1 sequences Si', 8, i < X a n r E t ~ { O , l ) m h t h a t M C c p [ i ? ; i E ' I t o i < j.

QH. I, § 21 ORDIOR, S T A B m Y AND INDISOIRNIBLES 15

A . = A ; for limit 8, Ad = u a < d A,; and for every p ES;(A,, bl) U

#$(A,, M ) and B s A,, IBI i: 1.1' + No, p 1 B is realized by a sequence from A,+1 2 A,, and 1A.I 5 IAIla+l1. So a < h =- IA,I < XB<,mn(lA I+2)m"lfl hence JA,I < K .

(*)

Now we s h d prove: There is i < K + such that for every a < h and every

over B. B c A,, IBI < 1.1' + 8 0 , t p ~ z , Aa+1, M ) ($9 tp)-splits

Suppose not, then for every i < K + there are a, < A, B, E A , which contradict (*). Clearly we can replace {8: i < K + } by any subset of the same ctrtrdinality. As h 5 K we can aasume a, = a for every i. Then aa a < h the number of subsets of A, of cardinality < 1.1 + + No is ~ Z o r r r < ~ IAaP 5 Zorucn ~ A ~ ~ a + l ~ " I K so we can aasume B1 = B for every i. Let IBI" = p < 1.1' + KO s A. By the requirements on A,+1 there is B E C 4 A,+1. ICl 5 (n + 1)2" in which every p ES$(B, M) is reahed. Clearly IQ(C, M)I 5 21'1" I 2((n+1)*a')" 5 K

so we can assume that for every i < K + , tp,(8, C, M ) = p. Now M

tp,(F, A, AZ) # tp,(E1, A, M ) there is @ E A such that M C &'" a] G

(possible by the definition of C). As tp,(Zl, A,+l, M ) does not (#, 9)- split over B, for I = 0, 1, M C tp[8; iz] = ~ [ 3 ; a']. So M C tp[F; a'] = - q [ E 1 ; a'] but this contradicts tp,(i?, C, M ) 3: p. So we have proved (*) and now we prove:

(1) Define by induction on a < A, sequences a,, 6,, E, E Aaa+a aa follows. Suppose we have defined them for every fl < a. Let B, =

UB<a aBn6,-aF 80 by (*) tp,(@, Aaa+l, W ($, ?)-splits over Be, 80

there a,, b, E Aa,+1, tp&,, B,, M) = tp#(b,; B,, M) but M C ~ [ 8 ; a,] A IF[$; 6,]. NOW choose 5, E Aa,+n which realizes tp,(8, B, u a, u 6,, M ) .

of E,,

iv[Z1; a]. Choose 8' E C such that tp& B, M) = tpy(si', B, M)

So clearly: (i) If f l I a, then M C q[8; a,] A 7tp[i?; 6,] and by the definition

M I= q[8; a,] = tp[Z,; a,], M C ( p [ E f ; 6,] = tp[E,; 6,]; hence M C cp[Z,; a,] A l@a; 6,].

hence (ii) If a < 8, then tplr(aB, BB, M ) = tp&, B,, M), but E, E B,

M C +[a,; z,] = $[6,; E,] or iK c VIE,; a,] = tp[~,; 6,~. So if we let = aan6,*a,, we prove (I).

16 PRELIMINARIES [CH. I , f 2

(2) Let a,, 6,, Z, be as in the proof of (1). For the dehition of A -+ (x):; and why No + (No)!; see Appendix, Definitions 2.1 and 2.1.

Here we define the colouring f of A: if a < < A, f ({a, 18)) = 0 when M C p[E,; a,], and f ({a, F} ) = 1 otherwise. As A -+ (x)& f is constant on (all pairs from) a subset of A of cardinality x, so, by renaming, on x. If M C 7p[Eo; a,], then for a, = a,, F = E, for a < x and t = 1. If M C p[Zo; a,], then for a, < x, M C p[E,; 5B] o u < f l so let 8, = 6,, b = 5, for a < x and t = 0.

< x, M k p[Z,; a,] e a 2 8, so let

CONCLUSION 2.11 : If ]@(A, M ) I > I A I + No, then there are sequences an, En in ]MI (n < w ) and t E (0, l} such that M C ?[El ; a,]' e 1 < n.

THEOREM 2.12: Suppose M is stable in A, and in it there is no ordered (by 8ome formula of L(M)) set of sequences of length p, and p = A, IL(M)l < cf A or p = A + . If I c ]MI, IIl > p, then there is an indiscernible set J G I of length A+.

Proof. Use the ideas of the proof of 2.6 and 2.10, and use 11, 2.16.

EXERCISE 2.1: Suppose A. < A, < < A,, I E IMI, IIl > hl: and ill is stable in A, for i s n, then there is an (n + 2)-indiscernible sequence J c I, IJI = &+.

EXERCISE 2.2: (1) Find a model of power N,, which is stable in No, but there is no indiscernible sequence of length N, in it.

(2) Moreover, for some symmetric tp = p(z, y), there is no p-2- indiscernible sequence of power N, in it. (Hint: Use a dense Specker order for ( l ) , and add a well-ordering to it for (2) (see e.g. [GS 73]).)

(3) In (1) and (2), show that we can choose a model with'a countable, N,-categorical theory.

EXERCISE 2.3: Suppose A c B c C and for every 5 E C there is a 8 E B such that tp(E, A) = tp(6, A). If p ES"'(B) does not split over A, then p has a unique extension in Sm(C) which does not split over A.

EXERCISE 2.4: Generalize 2.12 to (A, A)-stability.

EXERCISE 2.5: Suppose A is regular, M a model, and for every A c /MI, ]A1 < A implies JS"'(A, M)1 < A, and in M there is no ordered set

OH. I, 8 21 ORDER, STABILITY AND INDISCERNIBLES 17

of sequences of length p for some p < A. Then for every I c 1M1, III 2 A there is an indiscernible set J c I, IJI = A.

EXERCISE 2.6: In 2.12 (cf. Exercise 2.6) add an A E IMI, [ A ] < of p( IA I < A) and demand that J is indiscernible over A.

EXERCISE 2.7: (1) SupposeS E S;(A), 9 = cp(3; g), 181 > IAI. Prove that there are li, E A, pk €8 (n, k < o), and t E (0, l}, such that cp(Z; B,$ €pic iff n < k (iff +Z; $pk, of course). (Hint: Generalize 2.10(1) and 2.11.)

(2) Generalize 2.10(2) similarly.

EXERCISE 2.9: (1) Suppose T is stable, A regular, IAI < A =. IS(A)I < A, and p < A =. p" < A. For any lAol < A and elements ah (i < K , a < A) there is a set S c A, IS1 = A, such that for any i(0) < . < i(n) < K , {(af,O), . . . , ap)): a ~S)isindiscernible over A,. (Hint: You can msume T is stable in some p < A by 11, 2.13 or see 111, 4.23.)

(2) Generalize (1) when we replace the condition on T by a condition on M .

EXERCISE 2.10: Let M be an infinite K+-saturated model, and define the model M" m follows:

pf"I = p1 U"pfIY a i f aEM, a[.] if a E "IMI,

F f ( a ) =

PM" = IMI and M E ME.

Show that if M is A-stable, A = A", then M" is A-stable; and if Th(M) is A-stable, A = A", then T h ( P ) is A-stable; but if A < A" or M is not A-stable, then M" is not A-stable; and if A < A", Th(M") is not A-stable.

Also show that Th(M") depends only on Th(M) and K .

EXERCISE 2.8: Let cp = q(Z; g),

h be regular, 5 E A E J M 1 , (A1 < A. Show that if b = 5-En. Ê, I{p E Smn(A, M ) : cpn(F; b) e p)I 2 A, then for some B G (MI, ( B J 5 m(n - l), I(p E P ( A u B): cp(Z; E) E ~ } J 1 A.

CEAPTmR 11

RANKS AND INCOMPLETE TYPES

II.0. Introduotion

Section 2 (and Section 1 on which it relies) are the cornemtones of the book, but they a,re not difficult.

This chapter has esaentidy two interlocked aims; the first is to investigate the family of formulas cp(Z; a)(a E Q)('p = 'p(Z; g) fixed); or

other words the family of the seta 'p(Q, E Q). We try to classify them by complexity, in Sections 2 and 4. Clearly this hae various implications for the theory T. The second aim is to investigate various notions of ranks (e.g., possible values, equality between distinct ranks), we deal with it in Section 1; somewhat in Section 2, and mainly in Section 3.

The properties we shall deal with me the order property ( =instability), the f.c.p. (the hite cover property) the independence property and the strict order property. We say T haa such a property if some 'p(z; g ) has the property (or, equivdently, by our theorems, some 'p(Z;g) has the property). Essentially we prove that the order property implies the f.c.p. and the order property is equivalent to the disjunction of the independence property and the strict order property. Other properties of this sort are suggested in Chapter 111, Section 7, (K&T) = a, K ~ J T ) = 00, essentially). The most important of these properties is stability (=not the order property) with which we shall deal in Section 2. The main theorem (2.2) lists many propertiea equivalent to unstability of a formula, some of them are helpful in proving assertions about stable formulas, and some are helpful in proving assertions about unstable formulas. For unstable formulaa the important properties are: the order property (i.e. for some a,,, &,(n < W ) t: 'pp,, 6,J if€ 2 > k) (this shows the similarity to the " classical " cam-the theory of dense linear order); and for every X there is an A, IAI I; X < lQ(A)l . Among the properties important for stable 'p m: l&(A)I I; IAI + KO (=stability

18

OH. 11, 8 01 INTRODUOTION 19

in every A); P ( p , v, a) s P ( Z = Z, 9,2) < w ; and every cp-m-type over A is definable by a formula over A, of a fixed form.

For theories we get a similar theorem (2.13): T is stable in every A = iff T is stable in some A iff every v(x; 8 ) is stable iff every ~(3; 8) is stable iff T does not have the order property (i.e. no formula orders an infinite set of sequences), ifF every infinite indiscernible sequence is an indiscernible set.

We also prove that if IIl 2 A > IAl, A regular and A is finite, then some J E I is A-indiscernible over A, IJI = A.

In Section 4 we shall deal with the other properties. We say v(Z; 0) does not have the f.0.p. if for some k, < KO we can strengthen the compactness properties of EE to: if p is a set of cp-m-formulas, and every subset of p of cardinality sk, is realized, then p is realized. The “classical~y example of a theory with the f.c.p. is the theory of one equivalence relation such that for every n < w there are (insnitely many) E-equivalence claases of n elements. It appears that for stable T, the possession of the f.c.p. is equivalent to several natural properties (see the f.0.p. Theorem 4.4). In particular if T has the f.c.p., then there are a formula E(z , y; 2) and sequences Z,, such that E(z , y, En) is an equivalence relation with k, equivalence classes and n 5 k,, < No (this shows the a 5 i t y of any stable T with the f.c.p. to the classical example, and is helpful in proving assertions about such theories). On the other hand if T does not have the f.c.p., then Rm[8(3; a), rp, A] = k is an elementary property of a; and there is k;,@ < KO such that for each a, for all A 2 k;,, the value of P [ B ( Z ; a), 9, A] is fixed; and the value of Mlt[8(Z;a),q] is bounded by some ki,, < KO; and for each finite A, there is k, < KO such that any set of (A, m)-formulas p is realized iff each q E p , IqI < k, is realized; and every (A, m)-type has a finite subtype of bounded cardinality which has the same rank Rm( - , A, A).

Let =(A) be the first regular cardinality p such that IAl 5 A =- ]l?t(A)l < p. It appears that the number of possible functions is small: for stable v = rp(Z; 0): n (1 < n < KO), and A + . For unstable v = v(Z; 8): Ded, A and (2”+ ; when they are distinct, rp satisfies the second case iff v(Z; ti) has the independence property. The strict order property of T is equivalent to the existence of a formula ~ ( 2 ; g) defining a quasi- order (i.e. , a transitive, reflexive, anti-symmetric relation) with infinite chains. In Section 4 we also st& to investigate dimensions.

Ranks were invented and investigated for their use in stability. Several kinds of ranks were used, and most of them are particular caaes of P ( p , A , A), on which we concentrate. We investigate them

20 RANKS AND INCOMPLETE TYPES [CH. 11, 8 0

alao when there is no apparent application; more information is obtained in Chapter 111, Section 4, and Chapter 5, Section 7.

Rm(p, A, A) is interesting mainly for A = 2, No, 00 and A = L or A finite. What is the meaning of the rank Rm(p, A , A ) ? For finite p , we can say that it measures the complexity of the family of sets {a: a realizes p u { c p ( ~ ; 6))) for v E A , 6 E 6.

The basic properties are proved in Section 1. The rank is monotonic in the parameters (decreasing for p and A, increasing in A), and every type has a finite subtype of the same rank. When A 2 No we can extend p while preserving the rank, i.e., if p is over A, there is a complete q 2 p, Rm(p, A , A) = Rm(q, A, A); when A = 2, v E A there is no si such that

Rm[p u {cp(?C; A , A] = Rm(p, A , A) for t E (0, 1);

so for A = L, p over A, p has at most one extension to a complete type over A of the same rank. For every A, the maximal number of such complete extensions is, essentially denoted by Mlta(p, A, A).

In Section 1 we also prove that for limit A, Rm(p, A, A) = Rm(p,A, A+). In Section 2 we prove that for finite A, Rm(p, A, 2) < w or

Rm(p, A , 00) = a; and Rm[e(Z; a), A, k] = 1 is an elementary property of a.

In Section 3 we complete the investigation on the stability spectrum for countable theories. If IT1 < 2H0, then T is stable in IT1 ifF T is stable in some A < 2H0 iff T is stable in every A 2 I TI iff R(x = x, L, 2) < 00. Also T is superstable iff T is stable in every A 2 2ITI iff T is stable in some A < AHo iff R(z = x, L, 00) < 00. But the aim of Section 3 is the investigation of R and D.

We prove that Rm(p, A, A) is < 141 + + No or 00 usually (i-e., A = a, or A 5 No). Also ah = Rm(p,A, A) (except for being decreasing) is essentially arbitrary for KO I A I I TI +, has restricted changes for A s No, is fixed for A > (2lT1)+. In fact it is fixed for A > lA l+ + No except when for some p , ! A ] + s p 5 (2A)+, lA l+ < A p + implies aA = a, < 00; and a,, = 00 except possibly when p is a limit cardinal (remember that for limit p , a,, = a,+) so ah (A > Id 1 + + No) has at most three values (see Theorem 3.13 and Exercise 3.16).

Moreover for A = L, A s KO we can compute Rm(p, A, A) from Rm(p,A, 2) (Exercise 3.21); and Rm(p ,A , a), for finite p , is usually directly characterized as the maximal a for which TCl +(A p, h) is consistent for some (A, a)-function h. We deal much with Dm(p, A, A),

OH. 11, 5 13 BANKS OF TYPES 21

which is a version of Rm(p, A , A) (essentially we require that the many contradictory extensions are finite) and we use it to prove assertions about R for A > lAl+ + No and prove that for A > IAl + + No they am usually equal. We also give a method of constructing counterexamples (Exercise 3.20) and many exercises.

IK.1. Ranks oftspea

DEFINITION 1.1 : Let p be an m-type, A a set of m-formulas, A a cardinal (possibly finite), or A = 00. We define the rank Rm(p, A, A) or Rm[p, A , A] by defining inductively when P ( p , A, A) 2 a, a an ordinal.

(1) Rm(p,A, A) 2 0 when p is a (consistent) type. (When p is inconsistent we stipulate P ( p , A, A) = -1.)

(2) Rm(p, A , A) 2 6 for 6 a limit ordinal if Rm(p, d, A) 2 a for all a < 6.

(3) P ( p , d , A) 2 a + 1 iffor al lp < handall finiteq E p thereare types {qi}isrr which are d-m-types (i.e., rn-types whose formulas me all of the form ~ ( 3 ; 8) or +E; a) where ~(3; 5) is in A ) such that:

such that q~ E qi, l~ E q, (or vice versa). In this cme we say that qi and q, are explicitly contradictory.

(i) for i # j there is a formula

(ii) P ( q u q i ,A , A) 2 a for all i I p. (4) Now if Rm(p,A, A) 2 a but not Rm(p,A, A) 2 a + 1 we say

Rm(p, A, A) = a. (It is easy to see by induction on a that Rm(p, A, A) 2 a implies P ( p , A, A) 2 p for all /3 5 a.) If Rm(p, A, A) 2 a for all a we define P ( p , A, A) = 00.

DEFINITION 1.2: If Rm(p, A, A) = a # 00, define the mu2tipZicity: (1) Mltl(p, A, A) is the first power po such that there is a finite q E p

such that q has no more than po A-m-types qi satisfying 3(i), (ii) from Definition 1.1 (po may be finite).

(2) Mlta(p,A, A) is the first power po such that there are no more than po A-m-types qi satisfying 3(i), (ii) from Definition 1.1 for p = q. If Rm(p,A, A) = 00, define Mltl(p,A, A) = Mlta(p,d, A) = a.

If Mltl(p, A, A) = Mlta(p, A, A) we shall just write Mlt(p, A, A). If a statement is true for both Mltl(p, A, A) and Mlt2(p, A, A) we shall write Mltl(p, A, A). If A = KO we may omit it.

Remark. Clearly if Rm@, A, A) # 00, then 0 < Mlta(p, A, A) 5 Mltl(p, d, A) < A and if p is finite, then Mlta(p, A, A) = Mltl(p, A, A). In fact Mltl(p, A, A)+ < A. See Lemma 1.9.

22 RANKS AND INCOMPLETE TYPES [m. II, § 1

Ndation. If p = {8(z; a)} we may write P[8(z; a), A, A] instead of P[{B(z; a)}, A, A]. The same convention applies to Mlt and D (see Definition 3.2).

8pcial Cases. P ( p , A , A) 2 1 if for all p < A and all finite q E p there are Am-types {q,},s,, each of which is consistent with q, but every two are explicitly contradictory. P ( p , A, A) 2 2 if for every p < A and

explicitly contradictory and P ( q u q,, A , A) 2 1 for all i. every hite Q E Z, there Am-types {q,},s,, which pairwise

Example. Let M be a model with equivalence relations {Ea: a < a,} such that a < /3 =+ E, E En, E, divides every equivalence class of En into infinitely many classes, every equivalence dam of 1, is infinite, and if a, = y + 1, then E, has hhi te ly many equivalence classes.

Let Ta* = Th(iK) and let 6 = 6(Ta') be the saturated model of Ta' of cardinality I?. Then for every a E 161 :

(1) Rl[{z = a}, L, a] = 0,

(2) R1[{zEoa}, L, 001 = 1, (3) R'[{zE@}, L, 001 = 1 + a.

Remark. Taw admits elimination of quantifiers and is totally transcendental (see Definition 3.1).

Proof. (1) ClearlyRl [{z = a}, L, a] 2 0, and there are no q,, q1 such that qo u {z = a} and q1 u {z = a} are consistent while q,, q1 are explicitly contradictory, otherwise a would realize qo and ql. Thus R1[{z = a}, L, A] 2 1 for A 2 2, in particular for h = ao.

(2) First we show R1[{zE,a},L,ao] 2 1. Let p be any cardinal. Choose elements a,, i s p, in the 1, equivalence class of a and let r, = {z = a,} u {z # a,: i # j s p}. Clearly they are pairwise explicitly contradictory, but a, realizes {zE,a} u r,. In order that R1[{zEoa}, L, a] 2 2 there must be two explicitly contradictory types p,, pl such that for i = 0 , l R1[{zE,a} u pi, L, a] 2 1. In particular {zE,a} u pi = q, is consistent, i = 0, 1. By Theorem 1.6, extend q,, ql, to complete types q:, q: over the same sets and of the same rank. But E, E E, for all a < a, so q:, q: can be explicitly contradictory only if there is a, such that [z = a,] E q:, [z # a,] E q: or vice versa. But then from the proof of (1) it is clear that R1(q,*, L, ao) 2 1. So R1[{zEoa}, L, ao] I 2.

(3) We leave this &B an exercise for the reader.

a- II, 8 11 RANKS OF TYPES 23

DEFINITION 1.3: p t- q when every sequence realizing p realizes q. If p t- q and q t-p we write p = q: p is equivalent to q.

Remark. For finite types, p I- q is equivalent to (VZ)[A p --f A q], and for infinite types to the condition that for every finite q‘ E q there is a finite p’ E p such that p’ t- q’.

THEOREM 1.1: (1) If p1 t-p2, then Rm(p,,A, A) s Rm(p2,A, A). I n p ~ t k t h ~ this ie the m e w h Pa E p, . (2) If e q d i t y hQZd.8 in (l), t h Mlt’(p,,A, A) s Mlt’(p,,A, A).

Proqf. (1) We prove by induction on a that Rm(pl,A, A) 2 a =- Rm(p2, A , A) 2 a. For a = 0 or limit there is no problem. Assume for a and prove for a + 1. Let R”’(p,,A, A) 2 a + 1 and let qa E pa be h i te , p < A. We shall h d typea {q,},<,, aa required in the definition. Since p , t- p2, clearly p1 t- qa so there is a finite q1 E p , such that q1 t- pa. Choose pairwise explicitly contradictory A -m-types {qi}, ,, such that R”’(ql U qf, A , A) 2 a for all i s p. By the induction hypothesis, Rm(qa u q,,d, A) 2 a for all i s p. Thus R”’(p2, A, A) 2 a + 1. (2) Similar proof.

THEOREM 1.2: FOT all p, A, A, m there it3 a $nite q E p such that l P ( p , d , A) = R”’(q,A, A), and Mltl(p, A, A) = Mltl(q,A, A).

Proof. If q s p , then Rm(p, A, A) s Rm(q, A, A) by the previous theorem. So if the left-hand side is co, we may take q arbitrarily, for example q = 0. If the left side is a # 00, then Rm(p,d, A) ;t a + 1 and thus there is a finite q c p and there is p < A such that there me no pairwise explicitly contradictoryd-m-types @(}, s,, satisfying Rm(q u q,, A , A) 2 a. But that m e w that Rm(q, A, A) 2 a + 1. By 1.1(2) Mltl(p, A, A)

Among the pairs q,pq satisfying the above mentioned condition, choose one with minimal pq. Thus Rm(q,A, A) = a, Mltl(q,A, A) =

I Mltyq, A, A) = pq, say.

Mtl(p, A, A).

LEMMA 1.3: (1) If A , c A,, A 2 K , then R”(p,A,, A) s Rm(p,d,, K ) .

(2) If there is eqdity in (I), t h Mltz(p, A,, A) I Mlt’(p, A,, K ) .

PTOO~. (1) We prove by induction on a that Rm(p,Al, A) 2 a * Rm(p, A,, K ) 2 a. The proof is immediate.

(2) Likewise.

24 RANKS AND INCOMPLETE TYPES [CH. 11,s 1

LEMMA 1.4: (1) If Rm(p,A, 2) = a # oc, then there is no q(Z; 8) E A and sequence

(2) If Rm(p,A, A) = a # a and Mltl(p,A, A) = 1 then the parallel

( 3 ) If Rm(p, A , A) = a # a0 and Mlt'(p, A , A) < No, t h there is no q(Z ; 8) E A and sequence ii m h that Rm[p U {p(Z; a)", A , A] = a and

m h that P [ p u ( ~ ( 0 ; a)t}, A , 21 = a, for t = 0,l .

wiusion still ho&.

Mlt"p u {q (Z ; 8)t}, A , A] = MltJ(p, A, A) for t = 0 , 1.

Prqof. (1) If there were such a p we would choose qo = {q(Z, a)}, q1 = {+Z, a)} and get Rm(p, A, 2) 2 a + 1.

(2), (3) The proof is similar.

Remark. If q E p , p A ESF(A), Rm(q, A , 2) = P ( p , A , 2) = u # 00, then p ! A may be defined in terms of q in the following sense: q ( f ; i Z ) ~ p t A o p ( Z ; j i ) ~ o l , ( A ) , a ~ A , and Rm[qu{q(Z;iE)},A, 21 = a; for, if q(3; 5) ~p 1 A, then q G q u (q(3; a)} E p and the rank of p U (~(3; a)} is between the rank of q and the rank of p , and thus is a. And if p(Z; g) E cl,(A), a E A, q(Z; a) q! p, then +Z; a) E p , since p t A E @(A). Thus the rank of q u {-.q(Z; a)} is a and by the previous lemma the rank of q u {I@; a)} is not a. (Compare Theorem 2.12.)

COROLLARY 1.6: If Q c p , p ESF(A), Rm(p, A , 2) = Rm(q, A , 2) = a # a, then p is the unique type in #?(A) extending q with rank a.

THEOREM 1.6: If p is an m-type over A , A 2 Ho, P ( p , A , A) = a, then there is q €Sm(A), p c q, Rm(q, A , A) = a.

Proof. Denote r = {+Z; a): Rm[{$(Z; a ) } , A , A] < a, E A}. It is sufficient to show that p u r is consistent since then we can find qEBm(A) extending p u r. Thus P ( q , A , A) 5 Rm(p,A, A) = a but there is no finite q1 c q such that Rm(ql, A, A) < a (since then we would have P[{A q,}, A, A] < a and c7A ql) E T c q, thus making q contradictory). So by Theorem 1.2, Rm(q,A, A) = a.

Now we show that p u r is consistent. Otherwise there is a finite r E p of equal rank and there are #:, 8 E A , i = 1,. . ., n, such that Rm[{$& af)}, A, A] < a and r u {+(l, at): 1 5 i 5 n} is contradictory. Thus r I- qh1(Z, al) v - - - v #,,(?$, a"). Now Rm(r, A, A) = a > max, P[{$@, $)}, A, A]. On the other hand, Rm(r, A, A) 5

OH. II, 5 13 RANK8 OF TYPE8 25

CLAIM 1.7: Tor all r, h 2 No,

Proof. Let #, = #,(if; 8,); note that for 1 I; i I; n {#,} u r k {Vr-l #,} u r,andsobyTheoreml.l(l)P[{#,}ur,d, A] 5 R”[{V:,,$,}u r,A,A]. Thus maxi.,Sn P[{#,} V r, A , A] 5 Rfil[{V?=i #i} U r, A].

For the other direction we prove

(*) =[{ y #,} u r, 4 A] 2 B =2. max R”[{#,} u r, A, A1 2 8. 1

The proof is by induction on B. For = 0 (*) says that if {V, #,} u r is consistent, then there is i such that {$,} u r is consistent, and this is correct. When B is a limit ordinal the claim follows directly from the induction hypothesis. We must now show the passage from p to B + 1. Assume by way of contradiction that Rm[{V, #,} u r, A , A] 2 p + 1 but max, P[{#,} u r, A , A] < B + 1.

Since P[{#,} u r , A , A] < B + 1 for every i = 1,. . . , n , there me pi < A and finite q, s r such that there are no { Y , } , ~ ~ , satisfying the demands of Definition 1.1(3). Let q = uXl q,, p = (n + 1) max,p,. A is infinite by aaumption, 80 p < A. Thus there is a finite q c r and there me pl , . . . , pn < A such that for all i there do not exist pairwise explicitly contradictory types {rj},Sr, such that P[{#,} u q u r,, A , A] 2 /3 for all j 5 p,. On the other hand, since P[{V, #,} u r , A , A] 2

B + 1, and p < A, for all finite q E r, there are pairwise explicitly contradictory {rjlfS, , such that P[{V $,} u q u r j , A , A] 2 6 for all j 5 p. By the induction hypothesis with q u r, in place of r , we get that max, P[{$,} u q u rj , A , A] 2 B for all j. Thus for all j 5 p there is i(j), 1 5 i(j) 5 n, such that P[{$,tn} u q u r,, A , A] 2 p. There are p + 1 j ’s and n i’s, 80 there is i, equal to i(j ) for at least max, plo + 1 j ’s (by the definition of p). Then w.1.o.g. i, = i(j) for all j 5 pto, and 80

u q u r,, A , A] 2 /3 for all j I; pie. This contradicts what we obtained previously by the choice of the p,. Thus we have proved (*), Claim 1.7 and Theorem 1.6.

COROLLARY 1.8: (1) If A 2 KO, in DeJinitim 1.1 and 1.2 we can re9lace “explicitly contradictory” by “contradictory”; or by “q u q, u qj

26 RANKS AND INCOMPLETE TYPES [m. 11, 8 1

implierr a wntradktbn for all i , j" and even by " P ( q U qf u q,, A , A) < a fori < j 5 p".

( 2 ) If A is a set of m-formulas and A , = @(A), t h n for every *type p und infinite cardinality A, Rm(p, A l , A) = P ( p , A, A).

LEMMA 1.9: (1) I fRm(p ,d ,A) = a # 00, h 2 KO, then the value ,uo = Mltl(p, A , A) is achieved, that is, for every finite q E p vor q = p ] thre are Am-typea {qf}f<uo satiafjjing Definition 1.1(3).

Proof. (1) The proof is the same for both multiplicities so we just prove for Mltl.

If po < KO or po is a successor cardinal the claim is obvious; so assume po 2 KO, po a limit cardinal. Clearly for every finite q G p for every p < po there are pairwise contradictory A-m-typerr {qfB d B such that P ( q u # , A , A) 2 a. Let A be a set such that for all i < p < po, qf and q are types over A. By Theorem 1.6 for all i < p < po there are types ELS?(A) such that qf c rf and Rm(q U q, A, A) 2 a. Clearly for every p < po, i # j < p implies 9 f # r f ; since qf,qf are contradictory and so 1 { 9 f : i < p < po}l = po since po is a limit cardinal and 1{q: i < p}I = p for every p < po. So omitting duplicates and re- labelling, for every finite q E p there are pairwise explicitly contradictory Am-types {qf}f

(2) By 1.3(1), Rm(p, A , A + ) s Rm(p, A , A) 60 it is sufficient to show by induction on a, that Rm(p, A, A) 2 a =- Rm(p, A, A + ) 2 a. For a = 0 or a a limit ordinal this is obvious. So suppose a = /3 + 1, P ( p , A , A) 2 a, then for every finite q c p , for all p < A there me pairwise contradictory A-m-types {r,&'}f < p such that Rm(q U #, A , A) 2 /3. But A is a limit cardinal so exactly tw in past (1) for every finite q C p there me pairwise explicitly contradictory A-m-types {qf}f such that Rm(q u qf, A , A) 2 /3. By the induction hypothesis, Rm(q u qf, A, A + ) 2 /3 so by Definition 1.1(3), Rm(p, A, A + ) 2 /3 + 1 = a.

LElClMA 1.10: Assume P ( p , A , A) = a # ao, Mlt'(p, A, A) = po. (1) Thre is u set A , IAI < A, m h that for all q E Q ( A ) , if

Rm(p u q, A , A) = a, t h Mlta(p u q, A, A) = 1. (This will also be true for all A' 2 A.)

( 2 ) If KO I A and Rm[r, u {-,cp}, A, A] < a, t h n Rm[r, u {y}, A , A] =

(3) If Rm[p u {cp}, A, A] = Rm[p u {7cp}, A , A] = a and A 2 So, then Mlt'(p, A, A) s Mlt'[p u {cp}, A , A] + Mlt'[p U { ~ c p } , A, A], and equality holds i f cp E A.

(2) P ( p , A , A) = P ( p , A , A+) for A a limit cardhd.

such that Rm(q u qf, A, A) 2 a.

a, Mlt'Ep u {cp},A, A1 = po.

OH. 11, 8 13 RANKS OF TYPES 27

(4) Mltl(q,A) = Mlta(q,A) (i.e., Mltl(q,A, KO) = Mlta(q,A, No)).

Proof. ( l ) , (2) and (3) are left as an exercise. (4) If P ( q , A , No) = 00 this holds by Definition 1.2. Otherwise let

a = R”’(q, A , No), k = Mltl(q, A ) so by the remark after Definition 1.2, k < KO and k 2 Mlta(q,A). For proving the other inequality, choose finite p c q such that a = R”(p, A, No), Mlt’(p, A ) = Mlt’(q, A ) (by Theorem 1.2). By Definition 1.2 there are pairwise explicitly contradictory A-rn-types r, (i < k) such that a = Rm(p u rf, A , No). If for each i < k, a = Rm(q u rf, A , No), then Mlta(q, A ) 2 k so we finish. Otherwise for some j < k, Rm(q u r,, A , No) < a hence for some finite p1, p E pl c q, P ( p 1 u r j , A , KO) < a. As p E p1 E q clearly a = Rm(p,, A , Ho), k = Mltl(pl,A) so as p1 is finite, there are pairwise explicitly contradictoryd-m-typesr: (i < k) such that R”’(pl u r:, A , No) = a. Choose A such that rf, r i (i < k) and q are over A, and for i < k, choose rf such that r: c rf E S ~ ( A ) , Rm(p, u r f , A , No) = a (by 1.6) and similarly r;, such that r, s e E @(A) , a = Rm(p U r;, A, N o ) . The rf (i I k) are distinct (for i(1) # i(2) < k =+ rill, ri(2) are explicitly contradictory; and if i < k, rf = e, then a I Rm(pl U r;, A, KO) hence a I Rm(pl u r,, A, No) contradicting the choice ofp,, r j ) . As < E &?(A), the rf (i I k) are pairwise explicitly contradictory, and a I Rm(p u r f , A, KO) hence Mltl(p, A ) 2 k + 1, a contradiction.

LEMMA 1.11: There i8 AT depending 012 T only S W ~ t h t for A 2 AT Rm(p, A , A) = Rm(p, A , CQ) for every p and A.

Proof. For every O(Z; a) and A, Rm[B(Z; a), A , A] is a decreasing function of A, hence there is A = A(B(Z; a ) , A ) such that for all p 2 A R”’[O(Z; a), A , p] = R”’[O(Z; a), A, A]. It is clear that A(B(Z; a), A ) depends on O(Z; j j ) , tp(a, 0), A only (as rank is preserved by automorphisms of CE). As the number of such triples is I 2ITI, there is & = sup{A(B(Z; a), A): O(Z, a), A } < 00. As for finite p, Rm(p, A , A) = R”’(A p , A, A), and every type has a finite subtype of the same rank (see l . l ( l ) , 1.2), for every p and A, A 2 AT implies R”(p, A, A) = Rm(p, A, A T ) . NOW we can easily prove by induction on a that for A 2 AT, Rm(p,A, A) 2 a implies R”’(p,A, 00) 2 a. Hence for A 2 AT, R”’(p,A, 00) 2 R”’(p,A, A), and 1.3 implies equality.

EXERCISE 1.1: Let f be an automorphism of Q. Show that Rm(p, A, A) = R”’v(p), A, A] for everyp, A, A; wheref(p) = {v(Z; f (a)): ~ ( 3 ; a) ~ p } .

28 RANKS AND INCOMPLETE TYPES [OH. 11, 8 1

EXERCIBE 1.2: Show that for every A, A, that if for no p does R"(p,A, A) = a,, then P ( p , A, A) 1 a, =- R'"(p, A , A ) = 00, for every m-type p.

EXERCIBE 1.3: Show that for all A 2 X,, Rm(p, L, A ) = 0 iff p is (consistent and) algebraic (i.e. realized by only a finite number of sequences).

EXERCIBE 1.4: Show that if T is the theory of KO independent unary predicates, then for all p, Rm(p, L, 00) = 1 o p is not algebraic and for p not algebraic Mltl(p, L, 00) = 2% (So T = ((3%) Ate,,, Pt(~)il(iEs): u E w s w , w finite}.)

EXERCISE 1.5: (1) Show that if 8 is a limit ordinal (or 0), then for all p,

where k, 1 < w and k* = min{j: n5 2 nk}. (2) Generalize Theorem 1.6 accordingly. (Hint: take 1 = 0.)

EXERCISE 1.6: Let k, 1 be natural numbers, k 2 2 , l 2 1. Show that for all a, p and A (( 3) and (4) are rephasings of ( 1) and (2), respectively) :

(1) R"(p,A, k) 2 1.a * Rm(p,A, k') 2 a, (2) R"(p,A, k ) 2 w - a * Rm(p,A, KO) 1 a, ( 3 ) R'"(p,A,k) S l .R'"(p,A,k') + (1 - I), (4) R'"(p,A, k ) < w * R m ( p , A , Xo) + W .

(See also Exercise 3.22(1).)

EXERCISE 1.7: Show that if any one of p , A, or A is finite, then Mlt'(p,d, A) = Mlta(p,A, A). Show that in general equality does not hold.

EXERCISE 1.8: Prove that 1.10(3) may fail for A < 8,.

EXERCISE 1.9: Prove that in 1.10(3) even for e t e A , 'p E d

Mlt'(p, A, A) 1 Mlt"p u {'p}, A , A1 + Mlt'rp u {+, A , A].

EXERCISE 1.10: Show that if Rm(p, A, A ) 2 n, then for some finite A l E A, Rm(p, A l , A ) 2 n for p finite, A finite or 00. Show that those restrictions are necessary.

OH. 11, 8 21 STABILITY, RANKS AND DEFINABILITY 29

II.2. Stability, ranks and definsbility

Now we show that there is no essential difference between A-m-types for Gnite A and p-m-types.

LEMMA 2.1: Let A = {pk@; g k ) : k < 1 < O}, Z(Z) = m and

# A = #,(Z;go,...,gai-i;z,zo,...,zai-i) ai -1

= (z = zk --f p k ( z ; g k ) ) I\ A ( z = zk l p k - l ( z ; g k ) ) k=O k-1

A V Z = zk A A ~ ( 2 = zk A Z = Zn). k<Sl k < n < a l

Then: (1) For every type p and power A, Rm(p, A , A) = Rm(p, #A, A). (2) For every fornzula p(Z; a) (where p E A or lcp E A ) and co # c1

there is a‘, c U {c0, cl} m h that cp(Z; a) = a’) (more exactly, k(VZ)(p(Z; T i ) = a’))). Hence for every A-m-type p over A, IAI 2 2, there is a #A-type q over A equivalent to it (i.e., p = q). Also i f p E rS,m(A), then q E Q A ( A ) .

G 6 and p (p E A or - , p E A ) for which p(Z; a) = 6). Hence for every #A-m-type q over A thereisa A-m-typep Over A equivalent toit. Ale0 q E STA( A ) implies p E @(A).

(3) For eoery 6 m h that k(3Z)#A(z; 6) there ie

Proqf. Let co # cl E A. ( 2 ) and (3) may be proved as follows: -

tpk(Z; a) = #,(Z; co, . . . , co, co, . . . , co, . . . , a, . . . , co, . . . , c,; Y

Yo Yl g k Yat - i -- c1, co, - . - , c1, co, .

In other words, in # A we substituted (co, . . . , co) for gn, n # k, for g k ,

c1 for z and zk, co for zn, n # k. A similar trick works for l p k ( i & a). And on the other hand,

, co). 2, 2 0 , - * . , Zk, Z k + l i * . *,z11-1

# A @ ; a08 * - 9 a21-1; ck , CO, * - 9 cat-1) pk@, a k ) or l p k - I ( Z , a,) or is false (the c,’e are not necessarily distinct). So in this way q may be defined from p and vice vem. Now for the case where p[q] is complete. If p[q] is a complete A-type (#,-type) over A, IAI 2 2, then

(i) for every k, a k E A ,

q pk(% a k ) or q l p k @ , 9).


or

(ii) For every c, c,, 8, E A ,

p I- #&@; a,, . . . ; c, coy. . .)

p I- 7#&I(Z; go, . . . ; c, coy . . .). The proof is straightforward.

Proof of (1). By the above it is sufEcient to show:

Claim 2.1(4): If A,, A ; are sets of m-formulas which are closed under l, and:

there is a h i t e A,-m-type p z such that v(Z; a) = A p;, and p: and p i @ are explicitly contradictory, or

such that p(Z; a) 3 A g and g and Q are explicitly contradictory, or

Then for every type r and cardinal A, Rm(r, A,, A) = P ( r , A,, A).

(iii) For every v(E; g) ELI, and

C,(3Z)v(3; a). (iv) For every I@; a) E A , and there is a finite A,-m-type

b-l(3Z)(p(Z; a).

Remark. If we amume (iii) alone we get Rm(r, A,, A) 5 Rm(r, A,, A).

Proof. By the symmetry of the tlssumptions it is enough to prove for h i t e r that Rm(r,dl, A) 2 8 * Rm(r,A,, A) 2 8. For 8 = 0 or limit this is immediate. So assume it for a and let 8 = a + 1. Let p < A. Since Rm(r, A,, A) 2 a + 1 there are A,-m-types {rJlr. which are pairwise explicitly contradictory and Rm(r U r,, A , , A) 2 a. By (iii) we can define r: = u {c: v(Z; a) E r,}, and r f , r; are explicitly contradictory for i # j. Also r, 3 r:. Thus Rm(r u $ , A , , A) = Rm(r u r,,A,, A) 2 a. So by the induction hypothesis and 1.2, P ( r u r:, A,, A) 2 a. Thus Rm(ryA,, A) 2 a + 1 = 8. This completes the proof of the claim and the lemma.

Remark. We wil l use Lemma 2.1 without explicit mention.

In the following theorem we show that the unstability of v(E;g) is equivalent to many conditions.

THE UNSTABLE FORMULA THEOREM 2.2: The followingpropertiea of v = q ~ ( Z ; g ) , m = l(Z), are equivalent (relative to a given theory T ) .

(1) cp i8 unstable (in every insnite power); i.e., for every A 2 Mo there is A m h that IS;(A)I > A 2 IAI.

OH. 11, $21 STABILITY, RANKS AND DEFINABILPTY 31

(2) cp ie urntable in at leaet one power h 2 X,. ( 3 ) cp lrae the order property; i.e., t k r e are 6, n < w BucIG thad

(4) For every n < o, F(cp, m, n) i s &tent, where for every k < o {cp(z; p)’f(ksn): n < w} is consistent.

r(9, m, a) = {dq; &,P: r) E “2, /3 < a)

( d y we omit the m). (6 ) r(9, a) i s am&ent for every o r d i d a. (0) P ( Z = Z,cp, 00) = 00.

( 8 ) There are A , p , p €@(A), m h thad p is not A-definable (see

(9) There ie 7u) 9 euch that for every A, IAI 2 2, and p EAY;(A), p is

(7) P ( Z = 1, Q, 2) 2 0.

Definition 2.1).

(9, A)-definable (see Definition 2.1).

DEFINITION 2.1: (1) The cp-m-type p is $@; Z)&fiined if cp(Z; a) EZ) - (2) The cp-m-type p is (+@; Z), A)-defidle if there is a Z E A such that

(3) The cp-m-type p is A&$nuble if there is a $ for which p is ($, A)-

(4) The m-type p is A-definable if p r cp is A-definable for every cp.

C$[Z; Z] and Tcp(Z; a) EP * C+[7i; Z].

p is $(a; +defined.

definable.

Remark. If p is a complete cp-m-type over A, then the mows in (1) go both ways for CT E A. We sometimes write “definable over A ” instead of “A-definable ”.

The proof of Theorem 2.2 will be broken down into a series of lemmas and theorems and completed with Theorem 2.12. From now till then all nohtions will be aa in Theorem 2.2.

Obviously, (1) (2).

LEMMA 2.3: (2) 3 (3).

Procf. Let l&(A)I > h 2 IAI 2 KO. By Conclusion I, 2.11 there are sequences 3, n c o such that either

(i) for every k < w , {cp(Z; &)‘(i<k): I c w } is consistent, or (ii) for every k < w, {p(Z; Zi)u(k<i): 2 < o} is consistent. In m e (ii) clearly cp hae the order property. In cme (i) in order to

prove (3) we use the compactness theorem, replacing the Zi by parameters pi and reversing the order of the first n < w of them.


LEMMA 2.4: (3) =- (4).

Proqf. Define an order < on a22 as follows: if 7 r k = v r k, ~ [ k ] = 0, v[k] = 1, then 7 < v ; if 7 t k = v 1 k, Z(7) = k, v[k] = 1 then 7 < v ; if 7 r k = v t k, Z(7) = k, v[k] = 1, then v < 7. It is easily seen that < is a total order. So by the compactness theorem and the fact that 'p has the order property we see that the set

T u {tp(Z,,; gy)if(n<v): Z(7) = n, Z(v) < n}

is consistent for all n < w. So we get that

F('p, n) = ('p(8,; g,,lk)W[kl: q E "2, k < n}

is consistent for every n < w.

LEMMA 2.6: (4) => (6).

Proof. This is immediate since every finite subset of F('p, a) is equal to a subset of some F('p, n), after changing names of variables.

LEMMA 2.6: (6) * (1).

Proqf. Let A be infinite; we must show that 'p is unstable in A. Let p = infb: 2' > A} and let M be a model of F(tp, p) where a,, realizes g,, and E,, realizes 3,. Let A = u {a,,: Z(v) < p}. Cleazly IAI 5 (Za<# 1a21).K0 I p.A.N, = A (since A < 2*, we have p 5 A). For 7 E '2, let p,, be the tp-m-type E,, realizes over A. If Z(7) = Z(v) = p, 7 # v , then p,, # p,,, since if a is the first ordinal such th t 7[a] # v[u], then 'p(3; i i , , l a )~ [a l E p,, and 'p(3; Siy,a)v[al = -'p(E; B,,lcr)'[al E p,,. Thus

If l34I 2 Ib,,: 47) = d l = l{7: Z(7) = PI1 = 2' > A 2 14. This completes the proof of the equivalence of properties ( I ) through

(6) .

LEMMA 2.7: (6) => (6).

Proof. Let a; realize ijp and Z; realize 3, for all a, p E a>2, 7 E a2. Let A be any cardinal. For all 7 E h'2 let p,, = (tp(3; si~l,#c6': B < Z(q)}; p,, is a (consistent) 'p-m-type. Let pv be such a p,, with minimal R"(p,,, 'p, A), say R"(pv, 'p, 4 = a.

CH. II,# 21 STABILITY, RANKS AND DERDTABILITY 33

Suppose a # a. Let @, y < A, be a sequence of zeros of length y and py = ~ ) , - i i - ~ ~ > . The p% are pairwise explicitly contradictory, p,, c py, and P ( p V , 9, A) 2 a, by the minimality of a. So R"(pv, 9, A) 2 a + 1, a contradiction. Hence P ( p v , 9, A) = a, so R"(3 = 8,9, A) = 00. As A was arbitrary, by 1.11 P ( 3 = 3, 9, a) = 00.

LEMMA 2.8: (6) (7).

Proof. Exercise.

Notation. From now on in this section, R(p) or P ( p ) denotes P ( p , 9, 2).

LEMMA 2.0: (1) Let p be an arbitrary m-type. R(p) 2 n iff

q 9 , n) = {#(Zn; q: #(Z; a) Ep, q E n2}

u ( ~ ( 3 ~ ; g n l r ) n [ k l : q E "2, k < n}

is &tent. (Note t7& I'& n) = I'(9, n) = I'{;+(q, n).) (2) For any rn-type p , for all k, n < o, Rm(p, 9, k ) 2 n + 1 iff

{#(Z,; a): v E " + l k , #(Z; a) ~ p }

u {v(Z,,: 2t.f) 3 -,'p(Ep; 2:'): q, p E n+lk, Y E nzrk, Z(v) = rn < n + 1,

v = q t m = p r m , i = q [ m ] ,

j = p b l , i # j}

i8 consistent. P ( p , 9, k) = n, Mlt(p, 9, k) 2 k, < k iff the set of fomnulas given above is inconsiStent, but if the further wnditbna q[O] < k,, p[O] < ko are added, the set ie d e t e n t .

(3) For any m-type p for all n < o, Rm(p, 9, No) 2 n + 1 ifl {$(a,; a): Y €*+I w, #(Z; a) El)}

u {p)(%,,; * j ) = +zp; * f ) : q, p E n+lo, v E Z(v) = m < n + 1,

v = r) t m = p r m , i = q[rn],

j = 4m3, i # j}

ie wn&tt?nt. Rm(p, 9, Ho) = n, Mlt(p, 9, 8,) 2 ko < No iff the set of fotmukce given above i8 i?acmaisM, but i f the further conditbna q[O] < k,, p[O] < ko are added the 8et is &tent.

Proof. (1) Firat suppose R(p) 2 n. By the compaotness theorem it is sufficient to show that l'&, n) is consistent when p is finite. Now we


define 8, for a E k>2, by induction on k 5 n, such that for all 9, q E k2, R(p,) L n - k where p,, = p u {p(Z; a,,,,)W[fl: j c k}. For k = 0, po is just p; so R(po) 2 n. Suppose the definition for k < n is completed, and look at k + 1. Let Z(q) = k; then R(p,,) 2 n - k. Thus there is a sequence a,, such that if pc(o> = p,, u (~(3; a,,)} and pn-(l> = p,, u {+3; a,,)}, then R(pc<,>) 2 n - k - 1 for i = 0 , l . And this is the condition for k + 1. Thus when I (?) = n we get R(p,) 2 0. Thus pn is consistent. Let 4 realize p,,. Taking a,, for Z,, and a,, for g,, we see that FJp, n) is Consistent.

The proof of the other direction is similar and is left to the reader (now the induction is downward).

(2) and (3) have a similar proof.

COROLLARY 2.10: (1) (7) * (4). (2) R(p) 2 w iff R(p) = 00 iff P ( p , p , 0 0 ) = 00 iff there are an,

n < w , m h that for every k < w p u {p(Z; an)u(ksn): n < w} is consietent.

Proof. (1) This is because F{;=;)(p, n) is equivalent to F(p, n).

This completes the proof of the equivalence of (1) through (7). (2) R( p) 2 w implies FJp, n) is consistent for every n, hence TJp, a)

is consistent for every a, so by the same proof aa that of 2.7, P ( p , p, 00) = 00. Trivially P(p,p,00) = 00 implies P ( p ) = 00 implies P ( p ) 2 w . The Iaat "iff" is proved aa in 2.3 and 2.4 for finite p, and then by compactness.

LEMMA 2.11: (1) (8).

Proof. There is an A such that l8;(A)I > IT p E @(A) which are A-definable is

s I{$(#; a): $ € L(T), a € A}I I IT1 * 2 nea,

So some p E #;(A) is not A-definable.

THEOREM 2.12: (1) Assum R(Z = 3) < w. Then there is a formula $(g; Z) euch that forevery A ([A1 2 2), everyp E#;(A) is (y5, A)-&$dZe.

(2) Tor every O(Z; g), Z(3) = m, $nite A , k, and n there is a #(g) euch that: for all 6, P [ O ( Z ; 6 ) , A , k] 2 n if k$[6].

CH. 11,s 21 S T A B a Y , RANKS AND DEFINABfLITY 35

( 3 ) FOY euery finite A , n, tp(Z; g) E A , and O(Z; 2) there is a #(g; Z) w h that: if O(Z; a) ~ p , Rm(p, A , 2) = P [ O ( Z ; a), A , 21 = n, then p 1 tp is #(g; a)-&finuble.

Remark. Theorem 2.12(1) proves (9) =- (7). It is obvious that (8) + (9) so we have (1) + (8) =+ (9) =- (7) and thus (1) through (9) are equivalent, mmpleting the proof of Theorem 2.2.

Procf. (1) Let A be any set, p E 8;(A). We shall show first that p is (#, A)-defined for some 4. By Theorem 1.2 there is a hite q E p such that R(q) = R ( p ) = k (<o). Assume q = {tp(Z; zit)"["]: 1 < n} where 7 is an appropriate 0-1 sequence. For every 8 E A let q(a) = q u {q(Z; a)}. If tp(Z; a) ~ p , then q(a) E p and so R[q(a)] 2 R(p) . But q E q(a) so R(q) 2 R[q(a)]. Thus R[q@)] = k. On the other hand if +Z; a) EP, then R[q u {7tp(Z; a)}] = k and so by Lemma 1.4(1) R[q@)] < k. Thus we have shown that for any E A, tp(Z; a) ~p iff R[q(a)] 2 k. By Lemma 2.9(1), R[q(a)] 2 k3€FqcE)(tp, k) is consistent. Let O(3; a) = A q. Clearly I'd&y k ) is consistent iff k#p; a], where .-

Hence tp(Z-; a) ~p iff k#p; a], E A, andp is (#, A)-definable. Now we show that the choice of # depends only on 9, and not on p. In fact it suffices to prove that there is a finite A such that every p E #;(A) is ($, A)-definable for some # EA. For if A = {#*(#; 2): i < n} then

#(g; Z) = #(g; ZO, . . . y En-1. 9 Z,ZO, - - - 9 Zn-1)

= A 1% z zt + #t(g; zt)I I <n

is the required single formula. By way of contradiction assume there is no such fhite A. Let P be a

new one-place predicate and b,, . . . , b, - new individual constants. Remember Z(Z) = m and let E ( # ) = n in v(E; 8). For any (not necessarily bite) set d of formulas #(g; Z,J, let

I

(where 6 = ( b o , . . ., bm-l)).


Now T , is consistent for any finite A, since by assumption there are A, p , p E~S:(A) and p not (#, A)-definable for any # E A. So letting M C T , A E 1M1, and 6 realize p we get (M, A, b,, . . ., bm-l) C T,.

Thus T A , is consistent where A,, is the set of all formulae of L of the form # ( j j ; 2,) with l ( j j ) = n. Let (M, A, b,, . . ., bm-l) C T,, where A = PM. If p is the 9-type 6 realizes over A, then for no # is p (#, A)- dehable, contradiction. This proves (1).

(2) is clear by Lemma 2.9(2) using Lemma 2.1. (3) is left as an exercise.

THEOREM 2.13: The following propertie8 of T are equivalent: (1) T is unstable. (2) T is unstable in at least one A, hiT' = A. (3) gbme formula Q(X; j j ) is unstable. (4), 8 m formula ~(3; j j ) is unstable, l(3) = m. (6) There is a formula ~(3; jj) and sequence8 (3: n < w}, l(3) = @) =

l(Z), w h that for all n, k < w , CQ[P; ak] o n < k. (6) There is a formula Q = Q ( ~ O , . . . , P-l; a) and an infinite set I of

sequence8 of length l(Zo) = . . . = Z(P-') such that Q is connect& and antisymmetric over I (see Definition I , 2.5).

(7) There is an infinite indi8CemLible sequence (of sequences) which is not an indiscernible set.

(8 ) There is n < w, afinite A , and A m h that for all no < w there is a A-n-indiscernible sequence I over A which is not a A-n-~ndiscernible set over A , and IIl 2 no.

(9) There is a formula Q = v(f0, . . . , P-'; ij) such that for all no < w

there is a set I of rn, sequences of length 1(Z0) = - . . = l ( 9 - l ) and a sequence ii w h that p(Z0, . . . , 9 - l ; a) is connected and antisymmetric over I .

The proof will be delayed until after Lemma 2.16.

LEMMA 2.14: For every A, Q, IS(A)I 2 18:(A)I. I f ~(3; j j ) is umtable in A, then T is unstable in A.

Proof. Exercise.

OH. 11,s 21 STABILITY, -8 AND DEFINABILITY 37

Proof. (1) Let A = {cp,(Z; g): i < IAl). The mapping g: @(A, M) -+

nf < e , ( A , M) defined by g(p) = (. . . , p ! 9, . . . )f, is one-one, since if p # q €#;(A, M), then there is i < l A J and E E A such that cpf(~; a) E p - CPI(E; 8) e 9; thus g(p) z g(q).

Let A, = ISgi(A, M)I. So h 5 nf< A,. By the assumption on A we can find an i for which A, r A, and that is exactly what had to be proved.

(2) Follows from (1) by taking A = ( p l A l ) + .

LEMMA 2.16: Let I = {Sif: i < i,) be a A-n-indiscernible sequence over A which is not a A-n-indiscernible set over A. Then there is #(E; #,2) E A, E E A such that for o < a # 8, P, 501 iff o < a < p, where ga E-pa-pa+l-. . . n a w + n - 1 ( a d n(p + 1) 5 i,).

Proof. By I , 2.9 there is cp E A and E E A such that cp(E1, . . . , En; E) is connected and antisymmetric over I. So by 3.9 of the Appendix it follows that there is such a #.

Proof of Theorem 2.13. By definition, (1) =. (2).

By Corollary I, 2.2, (2) implies that there is an A for which IS(A)I > h(=AITl) 2 IAl. So (2) => (3) by Lemma 2.15(2). By adding dummy variables, (3) =- (4),. By Lemma 2.14, (4), =- (1). So (1) through (4), are equivalent.

Now we show (3) =- (5): (3) implies that cp has the order property, by Theorem 2.2. Hence there are Tin, n < w, such that for every k < w

-n u(ksn): n < w) is consistent. Let bk realize p,, and set Pk = {cp(x; a ) E" = (bn)^Tin. Then Cp[bk, Tin] iff k 5 n. Let # = #(xl, 9; x,, ga) =

p(xl; jja) A x1 # xa. Then k#[Ek; B] iff k < n, and this is (5). (5) (4), (for some m): Aasume (6) and let b = Sian+'. Then for every k {cp(E; B)u(ksn): n < w ) is consistent (since it is realized by Siak). So cp has the order property and thus is unstable by Theorem 2.2.

(6) =- (6) is clear. (6) =. (7): By Theorem I, 2.4, for every finite A, n(cp E A), the infinite

set mentioned in (6) has a subset {af: i < w) which is a A-n-indiscernible sequence over Si. This is of course not a A-n-indiscernible set over a (by (6)). Thus by compactness we get (7).

(7) =- (5) by Lemma 2.16.


(6) =- (9) and (7) 3 (8) are obvious and (9) =- (6) and (8) =- (7) follow by compactness. This completes the proof of the theorem.

THEOREM 2.17: Let T be stable and A be$nite. Then there i s afinite A* m h that if for f i < a I), is the A*-m-type that SiS realizes over A, = A u u (8,: y < p}, Po = p,, and for every PEA* Rm(pB trp) = Rm(po t v) (m = l(7iB)) and pB is (#&j, Z); A)-&efinuble, #,,,from 2.12; t h n {a,: jl < u} is a A-indiscernible sequence over A.

Prmf. Since A is bite, there is an n < w such that any sequence (6,: y < /3} is A-indiscernible over a set B iff it is A-nl-indiscernible over B for all n1 5 n.

Now define a sequence of finite sets A,,, A,,-,, . . . ,Ao by downward induction such that (1) A E A,, (2) each A, is closed under permutations of variables, and (3) if 'p = p(Eo, , . ., 2-l; g) Ed,, l(P) = - - - = l(2-l) = m, then every pm-type p such that P ( p ) = R m ( p A ) is (#, A)- definable for some # = #@,, . . . ,2-l, g; Z) E A,-l. (3) is accomplished by Theorem 2.12(1)-(3). Now choose A* = U,=.,,A,. So in order to prove the conclusion of the theorem it is sufficient to show that {BE: p < a} is A,-i-indiscernible over A for all i s n, since A E A,. Now if y < p < u, then for every T E A * , po 19, c p , IF, p , r 'p and

unique extension in #:(A,) of equal rank, so since I), 'p, p , t 'p A, E

have p , A, for every Q €A*, y < /I < a. So by the definition of A*, tp(EY, A,) r A, E tp (8 , A,) 1 A, for all y < p < a, i I n. It also follows from (3) that tp (8 , A,) does not (A,, A,+,)-split over A for i < n, p < u so the result follows by Lemma I, 2.6.

COROLLARY 2.18: If T is stable and for every p < u 2 w p , is the m-type Rm(p, 1 tp) = P ( p 0 r v), andpo s p,, then {a,: p < a} is an indiscernible set over A.

Remark. We can use many other parallel conditions in place of "for

R(Pv t ?J) = R(Po t 'p) = m p , r d- BY corollary 1.6, Po t v has a

4XAY) and RbO t 94 2 RRP, t q-4 t A,] 2 R(PB t tp) = R(Po t 94 we = ( p , 1 tp)

realizes over A, = A u u (3': y < Is), for every

every 9, Rm(% t y) = R'"(po t v)", e-g., ~ ( P B , L, 2) = *(PO, L, 2).

Proof. By the previous theorem it is an indiscernible sequence over A. So by Theorem 2.13(7) it is an indiscernible set over A.

THEOREM 2.19: Assume T is stable, A jn i te , A regular, I a set of sequences of length m, IAl < A I 111. Then there is J E I , IJI = A, which is a A-indiscernible set over A.

CH. 11,s 21 STABILITY, RANKS AND DEFINABILITY 39

Proof. Let A* be uin Theorem 2.17. Among the A*-m-typesp, lpl < A, which are realized by at least A sequences from I, choose one p , with minimal ( P ( p , 'pj, 2): j < no) (ordered lexicographically, where A* = {'pj: j < no}). We crtn assume that p , is a tspe over A, 2 A, lAol < A. Now we shall define ?2 E I for i < A. Let 7i j , j < i < A be defined, A, = A, u (J {aj: j < i}. l&$(A,)l < A since T is stable and lA,l < A. Thus there is p, E ~ * ( A ~ ) , po E pi, and p f is realized by A sequenoes from I. Clearly for j < no, R"(p,, 'pj, 2) = P ( p o , 'pj, 2). Choose 8 E I - "(A,) realizing pi. By 2.17 and 2.13(8), J = {?2: i < A} is a A-indiscernible set. If A = No, the claim follows from Theorem I, 2.4( 1) and the fact that

as A is finite every set of sequences is a A-n-indiscernible set for all sufficiently large n.

LEMMA 2.20: Buppose v(J; @) is stable. T h there are jinite A, n such that if I is a A-n-indiscernible set of sequencerr of length I ( @ ) and 7i is any sequence of length 1@), then

I { E E I : kp[a; ~] } l < n or ~ { E E I : C7'p[zi; a}] < n.

Proof. By contradiction. Assume that the conclusion is false. Then by compactness there are an and an indiscernible set I such that { E E I: h@; E l t } is infinite for t = 0, 1. Let {tT}n <@ E I. For every w E w, {'p(Z; F)ii(new): n < w } is cornistent (since I is indiscernible). Thus

1 B?( (J .)I 2 2h > No = n<m

This contradicts the stability of 'p.

EXERCIBE 2.1: Show that in Lemma 2.20, for big enough A, (1) we can choose n = R"(E = E , 'p, 2) + 1, (2) we can chooss n = 2"+ where k = Rm(Z = Z, #, 2) and #(Z; 8) =

(3) when I is infinite we can choose n = R"(Z = J, 'p, 00).

'p(% q,

EXERCIBE 2.2: Show that Exercise 2.1 cannot be essentially improved. That is for some stable (even No-stable) theory T, infinite indiscernible set I, 'p and E ,

I@ E I : C'p[E; iz]t})l 2 n for t E (0, I},


but Rm(f = f , I, A) 5 n for every A 2 2 and Rm(Z = 3, $, 2) = pog, A],

P(f = Z, $, A) = 1 for A 2 X,. (Hint: Let T = Th(M), 1M1 = w u {w G o: IwI = n}, P = o, RM = {(n, w ) : n E w } and ~ ( s , y) = R(r , y), I = 0.)

EXERCISE 2.3: Let (I, s ) be an order; B = {E,: 8 E I ) . We define the L1 model HI as follows:

IM’I = d f : f : I + u } ; E f l = {(f, s>: (Vt 5 4f(t) = s(t)};

TI = Th(H’).

(1) Show that TI admits elimination of quantifiers. (2) Let K(I) = inf{K: I contains no increasing sequence of length K}.

Compare this with the example after Definition 1.2. There, ( I , < ) is Show that TI is stable in A if€ A = 111 + A<x.

a*, i.e., (a, 3) .

EXERCISE 2.4: Suppose M is K-compact, p is an m-type in M , 1pl < K ,

a, E 1M1, and vl E L(H), 1 = 1, 2. If for all E E 1M1 realizingp, i=pl[E; a,] o Cv,p; a,], then h p l [ E ; a,] e ttp,[Z; 4 3 for all E E 6 realizing p .

EXERCISE 2.5: Relativize Theorem 2.2 to a formula d(Z; a) (i.e., in (1) and (2), replace “9 stable in A ” by

in (3), add { d ( Z ; E ) } ; in (4) and (6) , add {O(Zn;Z): 9 ~ ~ 2 ) to P, in (6) and (7), replace f = Z by 8(f; a); and in (8) and (9) demand

“IAI s A implies l{p € @ ( A ) : p u { d ( f ; 6) is consistent}l I A’’,

e(z; a) E p .

EXERCISE 2.6: Do the same with Theorem 2.13.

EXERCISE 2.7: For any finite A, n, and 6(f; 5) there is an @)-type q, such that for every E

Rm[O(Z; a), A , No] 2 n iff E realizes q.

(Hint: Use 2.9(3).)

EXERCISE 2.8: Let ~ ( f ; a) = +(g; 3); and show that v(Z; g) is stable iff +(g; Z) is stable. (Hint: Use 2.2(3).)

EXERCISE 2.9: If T is stable, p E S:( IiKI), iK E A, then there is a unique p E q € @ ( A ) such that for every I, R ( p 1 v) = R(q tp), and if p is $(z; a)-defined, E E IHI, then so is q.

OH. 11,s 31 RANKS, DEGREES AND SUPERSTABILITY 41

II.8. Ranks, degrees and superstability

THEOREM 3.1 : For a finite rn-type p the following are equivalent: (1) R r n ( p , A , 2 ) 2 l A l + + So, ( 2 ) R r n ( p , A , A) = a0 where A = (2"0)+, (3) There are ~"(5; g,,) E A and a,, for q E " > 2 such that for every

7 E "'2, 1, U p,, [p,, = {Q,,l,(Z; a,,ln)R[nl: 72 < $(q)}] i8 COWi8t&.

Remark. From 3.1 it follows that (1) and ( 2 ) are equivalent also for infinite p (by 1.2). If in ( 2 ) we replace A by No, the equivalence is still correct, but the claim is stronger with A. If we replace A by A1 > A, the implication (1) * ( 2 ) is not correct.

DEFINITION 3.1. If Rm(Z = 3, L, 2 ) < 00, T is called totally tranecendental.

Proof of 3.1. ( 2 ) == (1) is trivial by 1.3(1).

(3) + ( 2 ) We shdl prove by induction on a that Rm(p u p,,, A , A) 2 a

for every q E * > 2 . As po = p , this suffices. For a = 0 i t follows by the consistency of p,, up; and for a a limit ordinal from the induction hypothesis. So let a = p + 1 , q E " > 2 ; then { p v : q Q Y E "2) is a family 0f2~0d-m-types which are pairwise explicitly contradictory. By 1.1( 1) and the induction hypothesis Rrn(pv u p , A , A) 2 hence Rm(p,, u p , A , A) 2 p + 1 = a.

(1) =- (3) We shall define by induction on n < w subsets U, of lA l+ + No of cardinality lA l+ + KO, formulas Q , , E A for q ~ " ' 2 and sequences for q E ">2, i E U,, such that R'"(p!,, A, 2 ) 2 i for i E U,, 7 ~ ~ 2 where p i = p U {pTlr(Z; @$)'['I: k < Z(7)).

For n = 0 let U, = {a: a < 141 + + No} and p'o = p, so by (1) the induction hypothesis holds. For n + 1, for every i i for q E "2, hence for some Q!, E A and a!,, Rm[& u {tp!,(Z; a!,)'}, A , 21 2 i, t E (0, l}. Hence for some Un+l c {a: a < 141' + No}, lUn+lI = l A l + + Ho,foreverya,pE Un+l, q E "2, v; = Q:. Let for q E "2, Q,, = Q; (for any a E Un+L) and for i E Un+l, is when q E ,>2 and $ when q E ,2. Hence

is consistent so some assignment Z,, H 6,,, gv H iiv realizes it. These av show that (3) holds.

42 RANKS A N D INCOMPLETE TYPES [OH. 11,s 3

Remark. For A = L them conditions are equivalent for difFerent m's by I, 2.1, i.e., (1)1 o (2), o (3), for d 1, my n.

Proof. (3) == (2) trivially, by the same A. (2) * ( 1) Suppose not ( 1). For every p E @(A) there ie a finite qp G p

suoh that P ( p , A , 2) = P ( q p y A y 2) < co (by not (1)). By 1.6, p(1) # ~ ( 2 ) * Q P ( ~ ) # qfis hence

l8?(A)I s I{qp: p E 8?(A)}l s I@: q a finite A-m-type over A}\

S + + KO, a contradiotion.

(1) * (3) By 3.1 we know that (1) implies condition (3) from 3.1. Letting A = U {a,,: r) E @>2}, and q,, be a type in &(A) exfending p,, [we use the notation of 3.1(3)] we see

I&WI 2 k,,: r ) E'211 = P o

because if r) # Y E "2, let n = min{k: r)[k] # v[k]}. Then

V q l k ( Z ; $Ik)nrkJ E I ) n ~ l Q , , l k ( % q,k)"rkJ = q J v l k ( R a"Ik)v[k' E q v .

On the other hand, lAl I; I">21.N0 = No.

CONCLUSION 3.3: (1) If IT1 < 2n0, T ie stable in IT1 (or in some A < 2b), then T ie shbk in every cardinality p 2 I TI. (2) A totally tranrrcencEental T i s stable in every A 2 I TI.

DEFINITION 3.2: We define the degree Dm(p, A, A) ( p , A , A are as in Definition 1.1) by defining inductively when Dm(p, A, A) 2 a (a an ordinal) : (1) D"(p,A, A) 2 0 when p is a (consistent) type (when p is an

inconsistent set of m-formulas, we let the degree be -1). (2) D"(p,A, A) 2 8 (8 a limit ordinal) if D"(p,A, A) 2 a for all

a < 8.

OH. 11, 8 31 RANKS, DEQREES AND SUPER8TABIIJTY 43

(3) Dm(p,A,A) 2 a + l i f fordlp < Aandallfhitarspthereare, a finite q z r , n < w, $(Z; 0) E A, and sequences a{, i 5 p such that:

(i) Dm(q u {$@; at}, 4 A) 2 a, (ii) {$(z; a,): i 5 p} is n-contradictory (or n-inconsistent) over q,

(4) Exactly as (4) of Definition 1.1. i.e., for every w E p, IwI = n, t=-,(3Z)(AiEw $(3; a{) A A q).

Remark. We ignore the degree for A < X,, and will be mainly interested in the c&88 A > ITI+. We can always replace q by A q.

U a,)}, A, A) = y:

(4) If p is an m-type over A, A 2 No, A is given, then there i s q,p E q E &?m(A) m h thud

W p , 4 4 = D"(q, 4 A).

Proof. The proof of (2) is like l . l ( l ) , 1.3(1); the proof of (1) is like 1.2; the proof of (3) is like 1.7 and the proof of (4) like 1.6.

LEMMA 3.5: (1) If X E {R, D}, p a$nite type and 00 > Xm(p, A, A) 2 a, then for some A-m-type q, Xm(p U q, A, A) = a.

(2) If X E{R, D} a = (21Tl)+, Xrn(p, A, A) 2 a, then Xm(p, A, A) = 00. If X = R we need only a = (21*l+No)+.

Proof. (1) We suppose there is no such q, and prove by induction on 8 2 a that q = p vq,, q1 a A-m-type, Xm(q,A, A) 2 a implies

For 8 = a this is the aasumption, for 8 a limit ordinal it follows by induction. For = y + 1 > a, and e.g., X = D, let rl c q be finite and p < A. Then there are a finite r z rl and formulaa $(Z; a,) (i 5 p, $ E A ) n-contradictory over p v r such that

Dm[p u r u {$(Z; at)}, A, A] 2 a

Xrn(q, 4 A) 2 8.

44 RANKS AND INCOMPLETE TYPES [CH. 11, f 3

(they exist as Dm(p u q , A , A) 2 a implies Dm(p U q,d, A) 2 a + 1). So by the induction hypothesis

D"[p u er u {#(% q},4 A1 2 y-

As this holds for any finite er l , Dm(q, A , A) 2 p. Thus Dm(p, A, A) = 00,

a contradiction. The proof for X = R is similar. (2) By 1.2 and 3.4(1) we can assume p is finite; if X = R we can

also assume IT1 = lAl + 24,. If (2) d m not hold, then by part (1) there are finite p, (i < a) such that Xm(p, ,A, A) = i. By 1.1(1) and 3.4(2) we can assume p , = {Ol(3; q)}. By the definition of a, for some i < j < a, 8, = 8, and tp(q) = tp(z,); but Xm(pr,A, A) = i < j = Xm(p,, A, A), a contradiction.

DEFINITION 3.3: Let &(a) be the set of strictly descending nonempty sequences of ordinals < a. A (A, a)-functim is a function

A : &(a) --+ ((~(3, z), #(E; g), n): Q E L, # E A , n < o, Z(Z) = m};

say h(q) = (v,, #,, n,) (for some fixed m). Let 8 = 8(z; a) be an m-formula, p a cardinal, A a (A, a)-function and

U G &(a). Define T,"(8, h) to be the following set of formdm containing free variables of the form &,,

rfce, h) = (33) w; a) A A ~ ~ ~ ~ ( 3 ; ~ , I i . v l i ) A vqlI(~; r , l I , v l ~ I - l ~ ~ ~ ] : { [ 0 c ISIt,)

r ) = ( ) or r ) E U; and v E "p where k = Z(r))}

u {+z)[/\ #,@; iir,v-d A '~~(3; % , v ) ] : lEW

r ) E U, w = p , I w I = n,, v E "p where k=Z(q)- 1

If U = &(a), we write F,*(8, h).

THEOREM 3.6: Let IT1 + 1.1 c cfp. Then Dm[8(3; @ , A , p + ] 2 a if there M a ( A , .)-function h euch that I',*(8(3; ti), h) is wn&tent.

Remark. For the if part no restriction on p is needed.

Proof. (e) Suppose r,*(O(Z; a), h) is consistent, and let it be realized by the assignment #,,v H 6,,,, Z,,v H For r ) E &(a) u {< )} and v E "p, where k = Z(r)), let

pn,v = a)} u {#qli(z; h 1 1 , v t i ) : 0 < 5 J(q)}

tJ {vr,d% ~ , l I . v l ( I - l J : 0 < 1 5 l(r))}.

OH. 11,s 31 BANKS, DEGBEES AND SUPERSTABILITY 45

We oan easily prove by induction on /3 that Dm(pn,v, A, p+) 2 /3 where p 5 r][l(q) - 13 for 2(q) > 0 and D m ( p < > , < > , A , p + ) 2 a. As p0,<) = {O@; a)}, this proves one direction.

(+) We prove by induction on a, for all O(Z; 6) at once. a = 0 h is the empty function. a = 6 (a limit ordinal). So by the induction hypothesis, for every

/3 < a there is a (A, /3)-function hB for which I',*(O(Z; a), hB) is consistent. For Y E &(a) define h(v) = h,ml+l(~). It is easy to check that a com-

bination of the assignments realizing the I',*(O(Z; a), hB) realizes

a = /3 + 1 By the definition there are p(Z; a), n < w and Am- formulas $@; 6,) (i 5 p) n-contradictory over cp(Z;E) such that Dm[p(Z; a) A $(Z; &),A , p'] 2 /3 and p(E; a) k O(Z; 6). Hence, by the induction hypothesis for every i 5 p there is a ( A , /3)-function h, such that I'y*[p(Z; a) A $(Z; 6,), h,] is consistent. Define ( A , a)-functions hf (for i 5 p) as follows: If r] E de(/3), hf(r]) = h,(r]), if r ] = (p) , k'(r]) = (p, $, n> and if r] = (/3)û, u E &(/I), then hf(r]) = h&) (clearly this exhausts all possibilitiee).

If for some i 5 p, Fy*(O(Z; a), h*) is consistent, we me through. Otherwise, for each i some finite subset of it is inconsistent, hence for some finite u(i) c &(a) I';tf)(e(Z; a), hf) is inconsistent. The number of possible u(i) is 5 1.1 + No, and for each finite u the number of possible hf r u i s j I T I . A s c f p > IT1 + IaI,thereisVcp,IVI =psuchtha t for all i E V, u(i) = u, h' 1 u = h* 1 u. W.1.o.g. u is closed undertaking non-empty initial segments. Now using the $(Z; 6J (i E V) and p(Z; 8) we can ewily show that I';(O(E; a), h*) is realized, a contradiction.

r,*(eg; a), a).

46 RANKS A N D INCOMPLETE TYPES [CH. 11, 8 3

Proof. (1) It suilices to prove (1) for finite p, so by the remark to 3.6 it suffices to prove (2). Notice that 3.6 holds when I TI < cf p = a.

(2), (3) By 3.6, for some (A, p)-function h, F,*(A p, h) is consistent.

E(qi) = k, q;[k - 11 > i, h(qi 1 I ) = (vl, n,) for 1 5 E s k. For k = 1 there = (~1,#1, ni, such that I{. < p: h((a)) = < ~ i , $1, nJ}1 = p, and for i i, and h((a)) = (ql, #1, nl). If we have defined for k, there am ( P k + l , # k + l , nk+l such that

NOW we define for i < p, 0 < k < W; v k , #k, n k , 7; such that 7: E &(/A),

< p: = ( P ) k + l , # k + l , nk+l>} l = p;

and for i i, h(q:-(a)) =

( ' P k + l , #k+l, n k + l > * these v k s #k, nk Prove (2), (3)*

THEOREM 3.8: If A > l!Pl+, then Dm(p,A, A) = Dm(p,A, ao).

Remark. If you wonder why we do not require only A > (Id I + KO) + , then see Exercises 3.10 and 3.14.

Proof. Let p = IT1 +, and w.1.o.g. p is {O(Z; a)}. If Dm(p,A, p+) 2 a, then by 3.6 (when a < p) and 3.7 (when a 2 p) there is a (A, a)-function h such that F,*(O(Z; a), h) is consistent. Hence for every A, Ff(O(3; a), h) is consistent; hence P [ O ( Z ; a ) , A , A] 2 a. By the monotonicity of Dm(O(Z; a ) , A , A) in A (by 3.4(2)) it follows that for every A > p, D"[O(Z; a), A , A] = D"[O(Z; a), A, p+] . NOW we can e d y finish the proof by proving by induction on a that D"[O(Z; a), A, p+] 2 u implies Dm[O(Z; a), A, a] 2 a using what we have just proved (for every p).

Proof. We shall define, by induction on k c w, formulas tpi and natural numbers rnk such that

(1) for every k, there are a,,, q E ">p, such that (i) for every q E "p, p,, = p u {v:(Z; 0 < n < w} is consistent, (ii) for everyq E "p, n < w, w c p, IwI 2 mi+,, {&+l(Z; Gv(i)): i E w}

is inconeietent.

CH. 11,s 31 RANKS, DEGIREES AND SUPERSTABILITY 47

Letf(k) = min{n: m: > 2) u {a). (2) For every k, if f ( k ) < w, then f ( k + 1) > f ( k ) or Z = f ( k ) =

(3) if n < f ( k ) , then tpl = &+l. If we succeed in defining them, and define tpk as yz for large enough

k ( f ( k ) > n), then clearly 3.9 is satisfied by g~, = Alsis, ?@; Yj) A

A p. It is also clear that for k = 0 there me such &, mz (by 3.7(3)). So it suffices to prove, that if q!,, m!, are defined for Z I k, 0 < n < w, then we can define &+l, m:+l, 0 < n < w.

Now we can assume there are a,, r ) E O>p, such that (1) is satisfied by v;, m:, (0 < n < a), a,; and

(iii) ifr)E@>p,Z < w , i l I ia s . - - s il < p,jl s j, I . . - s j, ~ then the two sequences

f(k + l), and mf > m:+l,

h n- Tin-<h)-vl * - * ~ V < h ) - v l

realize the same type over 0, which depend on Z(r)) only and

A, = u {a,,: Z(v) s Z(r)) or v Z(q) # r ) ; v E O>p} u Domp.

Remark. (1) We can choose the v’s as void sequences. Hence in particular, (aV<,): a < p ) is an indiscernible sequence over A,.

(2) This is possible by 2.6 of the Appendix and compactness. See VII, 3.6. So we have q$, m: (0 < TZ < w) , and 4, r ) E O’p, such that (i) and (ii) from (1) hold, and also (iii). We must define g$+l, ,:+I.

Iff@) = w, clearly 3.9 holds. So letf(k) < w and r ) E f(k)-lp, and let 1, be a sequence of n ones. Suppose first that

p* = p u {&a; a,,,): n < f(k)}

{&~>+tt(’; ‘Vl-)’ < w, {&k)@ a V < O > ) )

is consistent. Then define


Clearly this definition satisfies our demands. So suppose that p* is inconsistent. Hence it has an inconsistent finite subtype, which we can mume is

Let us define

LEMMA 3.10: For A 2 N,, Rm(p, d, A) 2 Dm(p, d, A).

Proof. We prove by induction on u that Dm(p, d, A) 2 a implies R"(p,LI, A) 2 u. For a = 0, or u a limit ordinal it is immediate. Let a = + 1, and w.1.o.g.p is finite. So by definition for any p < A there are $(Z; 6,) (i d p) , n-contradictory over some finite r 2 p, with Dm[r u {$(Z; 6i)}y A , A] 2 @, hence, by the induction hypothesis, R"[r u {$(Z; gi)}, A , A] 2 @. By 1.6 there are pi E Sm(A), r u {$(Z; 6,)) c pi, and Rm(pf,d, A) 2 8, where A = Uis, bi u Domp. As the $(f ; gi) are n-contradictory over r , for every q E ST(A), I{i: pi 1 LI = 911 < n, hence, as p was arbitrary, Rm(p, d, A) 2 @ + 1 = u.

THEOREM3.11: D"(p,d, A) = Rm(p,d, A)providedthat:d = cl;(d,), h = p + , p is regular; p 2 !TI+ or p = No > lLI1l; and R"(p,d, A) < 00 or T is stable and pi < p (i < Id1[) * n,<,,,, pi .c p .

Proof. By 1.2, 1.1(1) and 3.4(1), (2) we can mume p = {8(Z; a)}, and by 3.10 it suffices to prove by induction on a that R"(p,d, A) 2 u =- Dm(p, d, A) 2 a. For a = 0 or u a limit ordinal this is immediate, so let u = + 1. If R"(p,d, A) < 00, by 3.6(1) we can assume Rm(p ,d , A)

OH. 11, 5 31 RANKS, DEUREES AND SUPERSTABILITY 49

= a. By 1.8(2) asp is finite and P ( p , A, A) 2 p + 1, there are A,*- types pf (i I p) pairwise explicitly contradictory, much that Rm(p u pf, A , A) 2 8. As Rm(p, A, A) is not + 1, I{i: Rm(p u pf, A , A) 2 + l}] < p, so w.1.o.g. Rm(p upf, A, A) = 8 for every i I p. By 1.6 we can assume that pi EST~(A) (i I p) for some A, and of come i # j * pf # p j . Notice that by 1.3, for every Q E A, Rm(p, Q, a) s &"($I, Q, A) < rn hence by 2.10(2) no(tp) =def Rm(p, Q, 2) < w and there is nl(tp) < w such that for no 8, (I < nl(cp)) does

A (gZ)[ A Q(Z; 8k)if(kr1) A d(iE;a)] I < nl(cP) k<nda9

hold.

that Rm(p u pf, A , A) 2 8. As If T is stable, we can choose A and distinct pi E Sfl(A)(i I p) such

Pj < P( j < 1411) * n Pj < P, 1<1&1

for some Q € A l there are p distinct pf Q and w.1.o.g. i < j s p * pi Q # p j Q. NOW Rm(p, Q, 2) < w by the stability of T (2.13, 2.2) so we can proceed as before so no(p), nl(v) < w exist.

Now the proof splits into two cases:

Case I : p = No > lA1]. We can find Q' such that for each n < w,

{?'(if; 61)1f(1=n): I I n} E pfCn), where i(n) < p. (If T is stable Q' is Q or l~ and if Rm(p, A, A) < a, then by 1.6 of the Appendix such a Q', Q' E

cl,(dl) exists.) Let n1 = nl(cp') and for n I n1 let

By Theorem I, 2.4 we can assume that {61: I < w } is a #,-n,-indiscernible sequence over for each n 5 n,. For every I < w let

#(Z; El) = A TQ'(Z; 5k) A Q'(Z; 6 n l + J * k < n l

It suffices to prove that {8(Z; a) A #( i f ; E l ) : I < w } is n,-contradictory, because by our choice pi(nl+l) 1 +(if; ZI), hence Rm[8(Z; a) A +(if; E J , 4, A] 2 8, hence by the induction hypothesis Dm[8(Z; a) A #(Z; E l ) , A, A] >- p, hence by definition Dm[e(Z; a), A, A] 2 /3 + 1 (as A = p + , # E A ) .

Now if I(k) < w (k < n,) me distinct and

W Z ) A [e@; a) A +(Z; El(k))], k<n l

50 RANKS A N D INCOMPLETE TYPES [a. 11, Q 3

hence for every n s nl,

hence by the $,-n,-indiscernibity of 6, ( I < w ) over a,

a contradiction.

Case 11: p > No, so we can assume A = A,. If T is stable and p 2 IT1 +, then by the induction hypothesis, Dm(p, A, A) 2 /3 so by 3.7 and 3.8, Dm @ , A , A) = 00; so Dm(p,A, A) = R"(p,A, A).

Let R"(p, d, A) < GO, then clearly by the above R"(p U p i , d, A ) = /? < I TI+. Now choose for each i < p a finite q, E p, such that R"(p U q,, A , A ) = p. Hence for each i < p, l{j < p : qs E p,}l < p so we can assume that i < j < p implies qs $ p,. As d is closed under conjunction we can replace qt by A qr = pi@; 5,) and as the number of possible pi is I d I < p and p is regular, we can assume that pr = p for every i < p. As p , ~ S , " 1 (A) for i < p, clearly for i < j < p, -p(z; 5,) ~ p , , so the p , rp are distinct.

So whether Rm(p ,A , A) < 00 or T is stable and pf < p (i < lA1l) * nf < pi < p, we can m u m e f i < I TI + and there is p E A such that

@ = {q Es;(A): Rm(p u q, A , A) 2 p} has cardinality zp. We now need:

CLAIM 3.12: Suppose @ E { p € @ ( A ) : p v {8(z; a)} is d s t e n t } and R"[8(Z; a), p, 21 = no < w. Let K be when 1@1 W infinite, and log, log,I@I if is finite. Then we can find r, (i < K ) auch that:

(1) rf W a p-m-type Over A ,

( 3 ) for each i, there is a unique pf E @ such tlrat rf c pi , (4) i # j implia pf # p,. We shall prove this claim later. Now apply Claim 3.12 to our @, and

we get r, (i < p) aa mentioned there and let A r, = p*(Z;Ef). So R"[B(z; a) A p*(Z; E,), A , A] 2 6, but for i # j,

(2) lrfl = no + 2,

Rm[e(z; a) A F*(Z; zf) A p*(z; E,) ,A, A] < p.

CH. 11, Q 3) RANKS, DEGREES AND SUPERSTABILITY 51

For any sequence i, t = (i(O), . . ., i(nl)), i(0) < . . . < i(nl) < p,

721 = n,(cp), let $@; a') = A -p*(~; Eia)) A Q*(R Eitcn,)).

Clearly R"[O(Z; a) A #(E; Z), A , A] 2 j3, hence by the induction hypothesis and 3.6, for some (A, j3)-function A,, I',*[O(?z; a) A $(- x; Z), h'] i s consistent. For each such T define a (A, a)-function I' as follows:

I c n l

If 7 E (wp), then 747) = hl(d, i f 7 = (B) , then h'(7) = (6 , $ 9 no), if 7 = (j3)-v, v E &(/I), then hf(q) = h,(v). If for a suitable T, T*,[O(Z;a), h,] is consistent, then by 3.6,

D"[O(Z; a), A , A] 2 a, so we finish. Hence assume that for each T, for some finite u(i) c &(a), I';A9[O(Z; a), hf] is inconsistent. Clearly the function P defined by P(S) = (u(t), h' 1 u(i)), has 5 I TI possible values. As p > IT[, p is regular, there is y < p such that for any j, L y, 2, < o, il(0) < ... < il(nl - 1) < y (for I < I,) there is ja < y such that for any Z < I,, iI(nl - 1) < j, and

F((i l (0) , . . . , it@, - 11, jl)) = P(idO), . . . , i h - I), jd)

(this is a variant of the Downward Lowenheim-Skolem theorem, see VII, 1.4). So we can define y(I) < y (I < w ) such that y(I) < y(2 + l), and if Z(0) < - - - < Z(nl - 1) < k < w , then

As in Case I we can assume that the formulas

1 (2 < 4 d ( q a) I\ $(z; E<~0)....,7tn1-1).7tn1+I)>

are n,-contradictory, but of course P((y(O), , , .; y(nl - l), y(n, + I ) ) ) = P((y(O), . . . , y(n, - l), y ) ) and let it be (u, h t u) so clearly I'&( O(Z; a), h) is consistent, a contradiction.

Proof of Chim 3.12. We define r, ,pi by induction on i, so suppose i < K , and we have defined r j , p j for every j < i.

Define an equivalence relation Ei on the cp-m-formulas over A: cp(iE; Zl)Eicp(Z; 82) iff for every j < i, cp(Z; al) Epj o cp@; 3,) Epj . If I@] is finite, the number of B,-equivalence classes is 5 2'; hence

1{p E @: ~(3; ~ ~ ) B , v ( z ; a,) * [v(z; a,) EP 0 c~(z; a,) EP]}~ 5 2,',

so there are po E @, cp(Z; ah) E ~ O , +Z; a:) E ~ O , such that cp(Z; a:), cp@; a:) are Ei-equivalent.


We can choose such po, 7i6, $ also for infinite 6. Ef hae < K equivalence c l m a e ~ t h e r w i s e choose representatives tp(E; a,) (j c K ) , and let 6, (j < i ) ream p,. Then by Exercise I , 2.7 and compmtness there are a(n) < K , p(n) < i such that ktp[6BB(1); 7i(I(k)]ii(k*1) for k, 1 4 w ;

contradicting 2.10(2). In the same way we ctan prove that

1{p E 6: tp(z; a,) * [tp(z; a,) * ~ ( 3 ; G) EP]}~ < K ;

so necessarily suitable po, at, a: exist-. Now we define by induction on 1 s no, p', (i) rf = {tp(z; so), tp@; (ii) Rm[{B(B; a)} u rf , tp, 21 s no - E or l{p E @: ri c p}] = 1. For I = 0, po, 3 + are defined and let t( 0) = 1, so clearly (i) and (ii)

hold. If we have defined for I, and I{p E 6: r: c p]l > 1, then for some tp = tp(3; rf u {tpt} c pist E 6 for t E {Oj 1). Clearly there is t(E + 1) E {0,1} so that

t(Z) E (0, 1) so that . . . , tp@; 3 + l ) N 1 ) } E p1 E 6,

m[{e(z; a)} u r! u {p+i)} , tp, 21 < B ~ B ( Z ; a)} u r f , tp, 21 I no - E

and let p' + 1 = p'st('+ 1).

If we have defined for 1 but 1{p E 6; r: E p}[ = 1 let = 3,

Now let r' = r&+ 1. It is easy to check that all the conditions are t(E + 1) = t(E).

satisfied.

THEOREM 3.13: (1) (A) P ( p , A , A) = P ( p , A , w), and

provided that: (B) P ( p , d , A) 2 lAl+ + No implies P ( p , A , A) = 00,

(i) A 2 (214)++ + N 0 , or (ii) A > Id(+ + KO, Rm@, A , A ) < 00 A not singular nor successor of

(2) T h e m 3.11 hkls for all A > p > No (p aa in 3.11). singular.

Proof. (1) W.1.o.g. we can aesume p is finite. If d is finite, (A) follows from 2.10(2) if Rm(p, A , A) 2 w and from 2.9(3) (using compactness) if P ( p , A , A) < w, and (B) follows from 2.10(2).

So w.1.o.g. d is infinite, Id I = I TI, and by 1.8(2) we can m u m e

If (ii) holds, let A = p+, p regular (if A is regular but not a successor by 1.9(2) we can replace it by A+). By 3.11, Rm@, A , A ) = Dm(p, A , A ) and by 3.8, Dm(p,d, A) = Dm(p,A,co) . By 3.10, Rm(p,d,co) 2 Dm(p,A, 00) hence Rm(p,d, co) I Rm(p,A, A) but by 1.9(2), Rm(p, A, co)

d = Cl$(d).

Rm(p, A, A) hence (A) holds. (B) follows similarly from 3.7(1).

OH. 11,s 31 RANKS, DEGREES AND SUPERSTABILITY 53

Let B(Z; a) = A p . We can assume that (i) holds. By 1.3( 1) it suffices to prove (A) and (B) for A = (2IA')+ +. If R ( p , d, A) = co, (B) is trivial, and otherwise 3.11 applies and we can finish as above. So we need only prove (A) on the assumption that R"'(p, d, A) = 00. So Rm(p,d, A) > (21T9+, hence there are A-m-typesp, (i < (21''I)+), pairwise explicitly contradictory such that Rm(p U p , , A , A ) > (2ITI)+, hence Rm(p U p,, A , A ) = 00. By 1.6 we can assume that for some A, p f ~ S y ( A ) for every i. As

(2'4)' = ){p,: i < (2\")+}1 5 n l{p, r cp: i < ( 2 9 + } )

s n I{q E #:(A): Rm(P u !I, 4 4 = m)1,

@Ed

@ € A

for some c p ~ d there me distinct q,;#:(A) (i < (2IA1)+) such that Rm(p u q,,d, A) = CQ. If Rm(p, cp, 2) 2 W, our conclusion follows by 2.10(2) and 1.3 hence we can assume Rm(p, cp, 2) < w. So we can apply Claim 3.12to 4j = {q €#:(A): Rm(p u q, d, A) = co}, andget cp-m-types

= CQ, R"'(p u r, u r,, A , A) <: 00 (for i # j < Id1 +). We can apply this construction to p u r,, etc. Hence we can define for r) E "'(Id1 +) d-m- types r,,, and natural numbers n,, so that:

r, (i < IdI'), 1.11 5 2 + Rm(p, 'p, 2) =def n<> Suchthat Rm(p U r,, A , A)

(0) rwll E r,, when 1 < Qq), (1) R"'(pur,,,d,A) = cofo reachr )~">( (d (+) , (2) IY,,-<,>l = n,,, r o = 0, (3) Rm(pvr ,~<,)ur , , -<, ) ,d ,A) < CQ (r)E"'(Jdl+),i #j < Id]+. Now for r ) ~ " ( l d l + ) let r,, = Ul<er,,ll, so clearly Rm(p U r, , ,d, A)

= m. Let p , Y,, be over A, so by 1.6 there are PE#?(A), r,, s r", P ( p u r w , d , A ) =mforr )~"(Jd l+)hence i f r ) r Z # v E ' ( l d ) + ) , then Y, $ rm. Let A r,, = #,,(f; a,,), now we can prove that for some #,,(Z; &) andK s ldl+, r, = {B(z,; a): v E "K} u {#n(zv; & ) i f ( v l n = ? v E "K, r) E "K, 0 < n < W}

is consistent; and that this implies the consistency of I', for each p, and this implies Rm(p, d, 00) = co. As the proofs are similar to 3.6 and 3.8 we leave the details to the reader. Instead of 3.6 we can use 1.6 of the Appendix for m = 1.

(2) Easy by 3.11, 3.8 and 3.13(1).

THEOREM 3.14: The following are equivalent. (1) Tie superstable, i.e., stable in every large enough A (in fact A 2 21Tl). (2) T is stable in some A for which A% > A. (3) R"5 = 5, L, (21TI)++] < IT!+.

54 RANKS AND INCOMPLETE T Y P E S [OH. 11, $ 3

(4) For ~ o m e m < w , Rm(2 = 8, L, co) < 00.

(5 ) Tiss tabZeandDl ( z=s ,L , ITJ++) < IT[+. ( 6 ) T is stable and for some m < w , Dm(2 = 8, L, co) < 00.

Proof. (1) =- (2) Immediate. (2) =- (6) By 3.9 (for p = 0) and compactness, if ( 5 ) fails, but T is

stable we can find 4, r) E "'A, satisfying (i) and (ii) from 3.9, and if A = lJ {a,,: r) E ">A}, q,, E S ( A ) , p,, E q,,, then lS(A)I 2 l{q,,: r) E "A}I = P o , as the q,, axe pairwise contradictory by (ii); but IAI 5 A, a contradiction.

(6) =- (6) Follows by 3.4(2) (6) =- (4) By 3.13(2). (4) + (1) For every Q = ~ ( 2 ; g), l(2) = m, by 1.3(1), Bm(2 = 8, Q, co)

I Rm(2 = 8, L, co) < 00, hence, by 2.2, ~ ( 8 ; g) is stable. As this holds for every such Q, T is stable. So by 3.13(1), R*(Z = 2, L, A) < co where A = (2ITI)+ +. Now for every A, for each p ES*(A), choose finite qp E p , R*(qp, L, A) = Rm(p, L, A) which is < co. By definition of rank, 1{p ESm(A): qp = q}l s 21TI so

IS*(A)I 5 I{qP: p E S ~ ( A ) } I .21r' = IAI + IT1 + ZITI.

So T is stable in every A 2 2ITl. (3) * (4) Immediate, by 1.3. (6) 3 (3) By 3.11.

LEMMA 3.16: If IST(A)I > IAl + IT/, then for some countable B c A , IS,(A)I 2 IS,(B)I 2 2*0, and there are ~ E A , Q , , E ~ such that for k < w, q Ek2:

(3x1 A Qnli(z, anli)vtzl* l<n

Proof. Choose, by induction on l < w , formulas ~,,(z; a,,) for r) E '>2 such that a,, E A and for every v E '2,

has cardinality > IAI + ITI. Then we let B = (J {a,,: r) ~ " > 2 } , so 1B1 5 No, B c A, IS,(A)I 2 IS,(B)I 2 2 b becaum the types {Q~I&; avIk)v[kl: k < w } are consistent and pairwise contradictory for v E "2.

{ p : p ES, (A) , ~ ~ ~ ~ ( 2 ; Sivlk)v[kl for every k < l }

EXERCISE 3.1: Prove that i f p 2 Id1 + No, p+ 2 A, and Rm(p, A , A) 2 S(p), then R*(p, d, A) = co. [On S(p) see Definition VII, 6.1(3) and Theorem VII, 5.6.1

CH. 11, 8 31 RANKS, DEGREES AND SUPERSTABILITY 55

EXERCISE 3.2: Prove that if p+ 2 A, X = R or X = D and T is stable, X m ( p , A, A) L [(No + I A l ) Y ] + , then X*(p, A, A) = 00 (forX = D use Exercise 3.10).

EXERCISE 3.3: For a formula B = B(Z;a) define the model Qo. Its universe is (6 E Q: 6 C Orb; a]}, and its relations a m

Re = {(Zit . . . y 4): Q C v[Zly . . . y 4, a]} for every q~ E L(T). Assume T is stable or at least that for every q ~ ,

Prove that P [ B ( Z , a), L, A] computed in Q is equal to R(x = z, Lo, A) computed in Go. Moreover, for each A c L there is do c Lo (and vice versa) such that = Ido) and P [ B ( Z ; a), A, A] computed in Q is equal to Rm(E = Z,Ao, A) computed in Qo.

Moreover we can assume that for every v(Z; g) E A there is p o ( f ; go) E

do, and vice versa, such that (i) for every 6 E Q there is go E Q0 such that for every 3 E Go, 5 =

P(e(z; a), v, 00) < a.

@OY * * .>, (E k c p [ ~ ~ , . . . ; 61 A A B[Z,; a] iff Cro C qo[z; 6O],

6 O ) or there is 6 as in (i). 1

(ii) for every 6 O E Coy Cro C (W)

EXERCISE 3.4: Give an example of a theory T such that D(x = x , L, 00)

= 1, R(x = x, L, 00) = 00. (Hint: See 4.8(2).)

EXERCISE 3.5: Give an example of a theory T,, ITaI = 1.1 + KO such that D(x = x , L, 00) = a, R(z = z, L, 00) = 00.

EXERCISE 3.6: Give an example of a theory T, 1111 = A, such that R(x = z, L, p) < 00 iff p > (2"+. (Hint: It has A one-place predicates O n l s . 1

EXERCIBE 3.7: Prove that if A is finite, and for every k < o, P ( p , A , k) 2 n, then P ( p , A, KO) 2 n.

DEFINITION 3.4: (1) Define CR(p, A) as an ordinal or 00, as follows: CR(p, A) 2 0 iff p is a complete type, and CR(p, A ) 2 a + 1 8 p is a complete tspe and for every p < A and finite B c Domp, there are pairwise contradictory complete types qt (i 5 p), p t B E qi, and CR(qi, A) 2 a-

(2) T is transcendental if CR(p, 2) < 00 for every type p .


EXERCISE 3.8: Prove for CR(p, A) theorems parallel to l . l ( l ) , 1.2, 1.3(1), 3.1 and 3.2.

EXERCISE 3.9: Investigate CR(p, A). (Hint: See V, Section 7.)

EXERCIBE 3.10: (1) If T is stable, show that in (3) of the definition of D”(I,, A, A) we can restrict q to the union of r and a finite clg(A)-rn-type.

(2) If in addition A = clg(A) we can replace “{y5(f;a,); i s p} is n-contradictory over q” by “ { y 5 ( f ; a,); i s p} is n-contradictory” (i.e., n-contradictory over the empty set) and thus omit q completely and use just r.

(3) Hence show that if T is stable and A = clg(A) it is sufficient to define a (A, a)-function as a function

A: &(a) -, {(#(f; g), n) : y5 E A , n < w , Z ( f ) = m}

and we a n then replace IT1 by (Id1 + No) in 3.6, 3.7, 3.8 and 3.11.

EXERCISE 3.11: Show that in the previous exercise the stability is indeed necessary.

EXERCISE 3,12: Let p be a finite rn-type, A c L; prove that the following conditions are equivalent when < 2n0, and always (2) * (1) * (3).

(1) R”(p, A , 2) = a, (2) for some A, I{q € @ ( A ) : I, = q}l > IAI + lAl + No, (3) for some A, IAI s KO, [ {q E S ~ ( A ) : p c q}l 2 P o .

EXERCISE 3.13: Relativize 3.3(1) and 3.14 to 6(f; 8) (as.in Exercise 2.5) and check the other theorems to see if they can be relativized similarly.

EXERCISE 3.14: (1) Show it is possible that Dm(f = Z, $, N,) = 00

but Dm(f = f, y5, A) = 2 for every A 2 8,. (Compare with 3.8.) (2) Show that in (1) we can replace two by any a. (3) Show that in any example of ( l ) , T is necesmrily unstable.

[Hint: Let T, be the theory saying: P , Q form a partition of the universe; P,(O < n < w ) a m disjoint subsets of P ; xRy -+ P(z) A Q ( y ) and S(z) = { y : z R y } (i.e., we look at P as a family of subsets of Q); the sets S(z) for z E PI are pairwise disjoint; P,(cO), P,(c,) for 1 E “w, and

CH. 11, f 31 RANKS, DEQREES AND SUPERSTABILITY 57

the c,’s are distinct; and let It be a one-to-one function from O’w into w , for every r ) E n < w and distinct

21 E Pn+1, n S(cnlk) n S(zI) = 0. kSI(n) I < htn)

Now T1 has a model completion with elimination of quantifiers satisfying (1) for 1,5 = zRy.1

EXERCISE 3.15: Show that in Theorem 3.11, we can replace “ p is regular ” by ‘‘ cf p > I TI ” and if T is stable we can replace “ p 2 I TI + , pf < p (i < [ A l l ) * ~ l < l A l l p l < p” by “ p 2 (21Al)+”. (Hint: See 3.13(2) and Exercise 3.10.) For 3.13(1)(B), A 2 141’ + No suffices.

EXERCISE 3.16: Show that Exercise 3.16 is best possible, i.e., in 3.11 we cannot replace “ p 2 (2IA1)+ ” by “cf p 2

[Hint: Assume 2Nn = Et,+,+, for n < w . M is the model with universe K, and the one-place relations Pt,(u < K,, n < w ) such that P:, E {i: K, < i < N,+,}, and {PZ,: u < K,} is independent for each n < w. Let T = Th(M), so IT1 = N,, but

I a0 ifA < No+,.

For each u < Nu+,, or u = 00 we can expand M to M, by one-place predicates such that 2 is replaced by u, ID(T,)I = Ha+,, lTll = N, where T, = Th(M,).]

+ ”.

1 i f A > R ( z = z, L, A) = 2 if A = K,,, or No+a+l,

We can use Exercise 3.20 for more general examples.

EXERCISE 3.17: Prove that in Claim 3.12, when 0 is finite, we can replace “ K = log, log,I@I ” by “ ~ ~ “ 0 I I @ [ ” (see 1.7 of the Appendix).

EXERCISE 3.18: (1) Prove that the following conditions on A 5 x are equivalent :

(A),,, For some T, IT1 I A, ID,(T)I = x. (B) For some Boolean algebra B, llBll I A and on B there are x

(2) If A, x satisfy (B) for a 1-homogeneous B, show that in Exercise ultrafilters.

3.6 we can replace (2*))+ by x+.

EXERCISE 3.19: Generalize 3.9 to Dm(p, A, p+) : (1) for stable T, (2) in general.

58 RANKS AND INCOMPLETE TYPES [CH. 11, $ 3

EXERCISE 3.20: Let R(A, T ) = R1(z = X, L, A), Mlt(A, T) = Mlt(z = z, L, A). Suppose T, is a complete theory in L, (i < K ) , L, pairwise disjoint, (Vz)P,(z) E T,. Let M , be a model of T,, M = z,<, Mi (i.e., IM/ = Ui<, 1M,1, L(M) = U,<,L,, and if R c L i is a predicate, RM = RM1; for simplicity only, assume L has no function symbols) and let T = Th(M). Then

( l ) T, 2 sUPf<K R(A, Ti)* ( 2 ) If A L No, /3 = supi,, R(A, Ti), then R(A, T) = /3 or R(A, T) =

/3 + 1. Also R(A, T ) = /3 + 1 iff A I p+ where

p = 2 (Mlt(A, Ti): i < K , R(A, Ti) = P}.

If R(A, T) = /3 + 1, Mlt(A, T) = 1. If R(A, T) = /3, Mlt(A, T) =

2 {MlW, Ti): R(A, T,) = P}. (3) Let A < No, /3 = sup,<, R(A, T,), /3 = S + k, where k < No S is

limit or zero. Then

(ii) if {i < K : R(A, T,) 2 S } is infinite, then R(A, T) = S + w ,

(iii) if {i < K : R(A, T,) L S } is finite, then R(A, T) < S + w ; and also if k, = 2 {Mlt(H,, !Pi): R(A, T,) 2 S}, 6 + 1 = R(A, T ) , then k, = max{k: Ak I I } (see the next exercise).

(i) B I R(A, T) 5 6 + w ,

Mlt(A, T) = 1,

EXERCISE 3.21: If A c KO, a = Rm(p, L, A) , /3 = Rm(p, L, H,), then w/3 I a < + w , and if kl = a - w/3, ka = Mlt(p, L, A), k = Mlt(p, L, KO), then A k i I k < A k i + l , and k, = max(1: Z.Akl I k}. Also k is the maximal number of pairwise contradictory extensions q of p satisfying P ( q , L, A) L w/3 (see Exercise 1.6).

Show that we can replace “L” by “A” when A = clk(A).

EXERCISE 3.22: Show that the results of Exercises 1.6 and 1.6 cannot be improved. In particular show that:

(i) For every a, and wa I /3 < wa + w there is a totally transcendental theory T, IT1 = la1 + KO, R(2, T ) = ,9, R(K,, T ) = a. (Hint: By induction on /3 using Exercise 3.20.)

(ii) Suppose n < w , H, = A, < A, < * - - < A, = 00, a1 > a, > . . - > a, where for q c 1 < n, A1 is not the successor of a limit cardinal. Then there is a theory T, IT1 I lal[ + p (where p+ L A,,-l) such that R(A, T) = al when A1 I A < A1+l. (Hint: Use Nu, lNul = *>p, Pf;’. = {v E N u : r ) Q v } for r) E O’p.)

UH. 11, 6 31 RANKS, DEGREES AND SUPERBTABILITY 59

QUESTION 3.23: Investigate the connection between R1(x = x, L, A) and Rm(3 = 8, L, A). (Hint: See V, Section 7 . )

EXERCIBE 3.24: Let Q’ be an expansion of Q by K < E individual constants. Show that

Rm(p, L, A) = P ( p , L’, A), Dm(p, L’, A) = Dm(p, L’, A).

EXERCISE 3.25: Show that if P ( p , A , X,) 2 /3 + 1, q c p is finite, A = clg(A), then there are A-m-formulas v,,(8;Z,,), pairwise contradictory, such that

(Hint: Show there are distinct q,, €Bm(A), P ( q u q,,, A , KO) 2 /3 (by 1.6 and 1.9(2)) and then use 1.6 of the Appendix.)

EXERCISE 3.26: Let p be a strong limit cardinal of cofinality 24, (i.e., (Vx < p)2x IAI + [dl + p,

then x 2 2u; so, e.g., ISm(A)I > IAI + Id1 + p implies IBm(A)I 2 2 ~ . In particular, there are A-m-formulas ~ ~ ( 3 ; a,) over A for r ] E *>p, such that for every r ] E *p,

P U {vnlr(% anlr) A l ( p V ( 8 ; &): 1 < W , Y = ( r ] 1 J)-(i), i < $11) is consistent.

formulas tpn = ~“(8; 4) ( r ] E “ ’p) such that for any r ] E *p (2) Prove that for finite p , P [ p , A, (2”) + I = 00 iff there are cl,(A)-m-

P u { ~ n l l A vnl(I-l)*<a>: 0 < < “9 a < dJ - 111

is consistent ifF for some A, G A and A, Idl[ + IAI = p and 1{q E BTl(A): p u q is consistent}] > p iff there are p,, (n < w ) such that Z n < a pn = ~9 and v;(z; 4) for

7 = (<a,, A), * * - 9 <am-, , Bm-J), a1 < Y < tcms

such that for any aI < < p I (0 < 1 < w )

p U {~v:(z; a:) A e(3; 8:;): 1 < w }

is consistent where qr = ((a,, Po), . . . , (aI- 1, /3*-,)).

A D ( l A l + , IAl ,pk ,pn) fails (see Definition VII, 1.11; e.g., if p = Xu+,), then in (2) we can add “iff Rm(p,A, p) 2 lAl+ ”.

(3) If the p,, are as in (2) and for every n < w , for some k < w ,

= p or


EXERCISE 3.27: If T is stable, Dm[O(Z; Ti), A, a303 2 u + 1, then there are n < o, 8 formula $(Z; 8) and an indiscernible set I baaed on G (eee Definition 111, 1.8) such that {I,4(Z; 6): 6 E I} is n-contradictory but Dm[O(Z; a) A $(Z; 6), A, 001 2 a for 6 E I .

EXERCISE 3.28: Dm(p, A, A ) = Dm[p, cl,(A), A] for everyp, A; if T is stable and for each $(Z; #) E A, n < o, #,,(Z; go,. . .) = Al<,, $(Z; &) € A .

EXERCISE 3.29: Show that in Exercise 3.28 both conditions are necessary.

EXERCISE 3.30: Letp, A, be fixed, and for k < o, R"'(p, A , k ) = uk,

R T p u {v*}, A, kl = *. (1) uk = 6 + nk and t$ < S imply (2) If A = clz(A), then for some t E (0, l}, uk 5 t$ + 1. (3) We cannot improve (2) (to ak 5 G). (4) If u: = 8 + n: for t E (0, l}, then nk 5 nt + nf.

= uk.

[Hint: (1) Use Exercise 1.6: (4) We can msume p is finite and define finite A-rn-types p n (q E k , n = nk) such that

(i) pr<,> (i < k ) are explicitly contradictory, (ii) Rm(pn, A, k ) 2 6 for q E n a k , where pw = p u u {pnl l : 1 5 l(q)}.

(u) for some t, Rm[pn u {$},A, k ] = uk, or (/3) let We prove by downward induction on l(q) that

Rm[pn u {cp">, A, k] = 8 + n:Sk, Mlt(pn u {$},A, k ) = 1:sk

then

QUESTION 3.31: Is Exercise 3.30(4) best possible? (This is a problem in finite combinatorics.)

EXERCISE 3.32: Suppose Z(Z) = Z(g) = m and O(Z,g; C) k cp(Z; 7i) A

$(g; 6), q(Z ; G) k (3g)O(Z, 5; C), and for every 2, O(Z; 2; C) is algebraic. Prove that for every infinite A, Rm[v(Z; Ti), L, A] 5 R"'[$(Z; 6), L, A].

CH. 11, 8 31 RANKS, DEGREES AND SUPERSTABILITY 61

DISCUSSION 3.16 : The following generalizes Exercise 3.3, and shows how the work of Poizat can be put in our context.

Let p be a type over a set A. Define 6, = (a, a)asA; Q, is a model with universe {a E C: a realizes p} , and relation R, = {E E Q,: E satia- fies tp}, for any tp = tp(3, a), si E A.

In 11, Exercise 3.2 we deal with the case p a finite type, then Q,, is a saturated model, and so we can translate results on the stable case to the non-stable case. Here Q,, is not necessarily sctturated, but it is like the Q when we deal with kind I1 in [Sh 76~1. As said there, all results on ranks, degrees and also forking goes through, and some caaes are discussed in detail.

Let me elaborate a little more:

CLAIM 1: Q, is a i - h o g e n w w r ; moreover, i f (ar: i < u), (br: i < u) realizes the same quantifier-free formulas in 6, (equivalently, the same form* with parameters from A in a), then there is an autommphisnz f of Q, which taka (ar: i to (b,: i < a) (and, in fac t , is a ratridion of an automorlphh of Q which is the identity on A).

Proof. Find the required automorphism of Q.

Remark. This would hold also for IQ’I = {a: tp(a, A ) E F}, r E &A). Easily:

CLAIM 2: (1) Q, is i-compact for quantifier free form&,

and ranke generalized. ( 2 ) Q, satisfies the assumption of kind 11, hence all recrults on forking

But we may still be interested in the connection between what occurs in Q and in 6,.

ASSERTION 3: (1) A set I E IQ,I is discernible in Qp iff it is indiscernible in Q over A.

(2) Two such injinite sets are equivalent in 6, iff they are equivalent in Q.

(3 ) For B E C E IQ,l, and I Q p l , tp(si,C, Q,) fork over B iff tp(Si, C u A, Q ) fork over B u A.

(4) Similar equivalence holds for stability in a power.

Procf. E.g. (3) p = tp(8, C, 6,) does not fork over B iff some infinite indiscernible set I over C of sequences realizing p has, under auto-


morphism of Q, which are the identity over B, a bound number of images up to equivalence, so we finish.

For usual rank there is no nice translation, but if in the definition we do not take a h i te subtype but deal with the same type, or deal with C P (and A = 0 for simplicity) the situation will be like 11, Exercise 3.3.

II.4. The f.c.p., the independence property and the strict order property

DEBTNITION 4.1: (1) ~(3; g ) has the finite cover property (f.c.p.) if for arbitrarily large natural numbers n there are go, . . . , such that

C T ( ~ Z ) A c~(z; a') k-zn

C ( ~ Z ) A cp(z;ak). but for every 1 < n,

k < n . k # I

(2) T has the f.c.p. if there exists a formula ~ ( x ; j i ) which has the f.c.p.

Note. Here x is a single variable, see also Theorem 4.4.

LEMMA 4.1: The formula v(Z; g ) h not have the f.c.p. iff there is a natural number n Buch that: If I' is a set of ym-formulas and every subset of r of cardinality < n is &tent, then r is consistent.

Proof. Immediate, by the definition.

THEOREM 4.2: (1) If T does not have the f.c.p., then T is stable. (In other words, every umtable theory has the f.c.p.)

(2) If cp(Z; g ) ie umtable, then the formula

#(Z; 2) = #(Z; g1, p, 83, g4) = [Cp(z;gl) = 7(p(~;ga)l A [v(z;g3) = v(~;g4)]

has the f.0.p.

Proof. (1) By Theorem 2.13, as T is unstable, some y(x; g ) is unstable, hence (1) follows from (2).

(2) As v(Z; j i ) is unstable, by Theorem 2.2(1), (3) it haa the order property, i.e., there are Tio, . . . , a', . . . , 1 < w such that for every k < w ,

4 p k = {v(-; --I i i ( k S l ) : 1 < 5 a )

is consistent. Let n be any natural number. For any k < n define Ek = lioniPn7ikn

i ik+l . We claim that q = {#(Z; Zk): k < n} is inconsistent, but for every

OH. 11, 8 41 F.O.P., INDEPENDENCE AND OBDEE 63

1 < n, ql = {$@; ak): k < n, k # 1) is consistent. As n is arbitrary, by Lemma 4.1, this will prove the theorem.

Let us fist prove that q is inconsistent. Suppose 6 realizes q. Then as C$[6; ao], also Cq[6; aO] E 7q[6; F]. On the other hand for every k < n, t~ +[6; ~ k ] , clearly b[6; ak] = q[6; ak+1]. _ .

Hence

kv[6; it01 0 kq[& it'] 0 Cq[5; as] 0. ' * 0 C q [ E ; it.]

0 C--lp[6; aO1,

a contradiction. Therefore q is inconsistent.

q1 is consistent. Now if 1 < n, then clearly a sequence realizing p r + realizes q,; hence

THEOREM 4.3: (1) There is a stable theory with the f.c.p.; and there is a stable theory without the f.0.p.

( 2 ) There are stable t k i e s which are (i) superstable and without the f.c.p., which are also stable in KO,

(ii) superstable with the f.c.p., which are also stable in No, (iii) unsuperstable without the f.c.p., (iv) uneujperstable with the f.c.p.

Proof. Clearly (2) implies (l) , so we prove (2) only. (i) The theory of a model whose only relation is equality.

(ii) Let M be a model with the equality relation, and an equivalence relation, such that for every n there is an equivalence class of cardinality n. Clearly its theory satisfies our demands.

(iii) Let M be a model such that IMI = Oo; the relations of M are equality and for every n the equivalence relation En defined as

En = ((7, v): r ) , v E %, r) n = v 1 n}.

The theory of M is the required theory; see Exercise 2.3. (iv) By combining the two previous examples we can easily construct

such a theory.

THE f.c.p. THEOREM 4.4: Let T be stable. Then the following assertiw are equivalent (where 1 5 m < w , 2 I A I m):

( 1 ) T h a the f.c.p., i.e., some q ( x ; 8) hccs the f.0.p. (2), 8ome q(3; g) has the f.c.p., l(3) = m. (3), Not for every Enite A , is there a k < o such that a set p of A-m-

(P) , For somejnite A , k, and formula O(3; g), Rm[B(?Z; it), A , No] = k formulas is realized iff every q E p , 141 < k is realized.

64 RANKS AND INCOMPLETE TYPES [CH. 11, $ 4

is not an elementary property of ii (that is, there is no formula $(g) such that Rm[B(Z; a), A , KO] = k o h,h[Z]).

(5), There are finite A, k and a formula 0 such that for no 1 < w

Rm[B(Z; a), A, HO] = k * Mlt[B(Z; a), A] I 1. (6)A,m For some finite A, for no k < w does the following hZd: for every

set p of A-m-formulas, there &a q E p , IqI < k, auch that Rm(p, A, A ) =

(7), For some finite A, for no k < w does the following hZd: for every

(€9, There is a fornula; y(Z, g; 2) Buch that Z(Z) = Z(g) = m and (A) for every Z, y(Z, g; 8) is an equivalence relation. (B) for every n and for some En, y(Z, g; Z,,) has 2 n but onlyfinitely many

R"(q, A, 8.

A-m-type p , Rm(p, A, k ) = Rm(p, A, 00).

equident chases.

Proof. We shall prove (1) * (2), * (l), (2), * (3), =s (€9, + (4), =-

[not (6)A,m and not (7),]. Clearly this is sufficient (as (1) does not depend on m and A).

(8)m * (2)m and (3)m * (6)A,m, (5)m * (7)m, (vm)[not (3)m and not (s)ml *

(1) * (2), Trivial. (2), (1) We shall prove it by induction on m = 1(@. For m = 1

this is true by definition. Hence assume we have proved it for m and we shall prove it for m + 1. Suppose it is not true. Then there is y = y(Z; g) = y(zo, . . . , x,; 9) and for every n < w a set r,, of formulas of the form y(Z; a) such that rn is inconsistent but every subset of r,, of cardinality < n is consistent. By Lemma 4.1 there is 1 < w such that if F is a set of formulas of the form y(zo; cl, . . . , c,, a), and every subset of F of cardinality < I is consistent, then r is consistent. Let

)I +(xi, - - - 9 2,; go, - - ~ 1 - 1 ) = (3zo)[ A q(zo, z1, . - 9 zm; g k k < l

and let for every n

r,* = {+(zl, . . . , x,; ao, . . . , for y(3; Tio) E F,,, . . .; y(Z; 8-l) E F,,}. Clearly every subset of r,* of cardinality <n/Z is consistent. Hence by the induction hypothesis, for some sufficiently large n, r,* is consistent. Hence there are cl, . . . , c, which realize I':. Let

r = {y(xo; c1,. . . , c,, a): y(z0, . . . , x,; a) E Fn}. Clearly every subset of r of cardinality I Z is consistent, .and so by the definition of 1, F is consistent; hence there is co which realizes it. So (co, c l , . . . , c,) realizes r,,, a contradiction. So the implication is proved.

(€9, Suppose A exemplifies (3),; so for every k < w there is a (2), * (3), Trivial, A = {y(Z; g)}. (3),

CH. 11, § 41 F.C.P., INDEPENDENCE AND ORDER 65

set pk of A-m-formulas, which is not realized, but every q c pky IqI < k, is realized. W.1.o.g. l )k has minimal cardinality; sop, is finite, lpkl 2 k, and every proper subset of l ) k is realized.

Let n(0) = Rm(Z = B, A, 2), so Rm(p,A, A) I n(0) for everyp and A. Now for each k, we define by induction on Z s n(0) + 1 sets pfc c pk. Let p! = 0, and pfc' be a subset of pk of minimal cardinality such that

-1, pf, exists). There is a maximal Z(0) such that for some n < w, W =

{k < w: Ipfco)l I n} is infinite. So for some n(l) , 4 2 ) < w, 8, and infinite Wl, for each k E W,, Ipf0)I = n(1) and A yfco) = 8(Z; 4) and P[p$O) , A, No] = n(2); necessarily n(2) = n(0) - Z(0).

(n(2) I n(0) - Z(0) by the definition of thepf, andn(2) 2 n(0) - Z(0) by the choice of Z(0)) and n(2) > 0 (n(2) 2 0 for we can choose k > n(1) and n(2) # 0 by the choice of Z(0)).

Clearly JpfZo)+ll tends to infinity with k; and for proving (5), it suffices to prove that Mlf[B(Z; z k ) , A] = Mlt[p$O), A] tends to infinity

I pjZ0)I. Let A be such that all p k are over A. For each Q E pfZ0)+l - pfco), by the choice of pZo)+ l , Rm[p$O)+ - { Q } , A , No] 2 n(0) - Z(O), so there is q,,, ES"(A), pZ0)+l - {tp} c q,,, and Rm(q,,A, No) 2 n(2). But Rm(pfc'o)+l, A, No) I n(0) - Z(0) - 1 so 1~ E q. Clearly the qVys show

(5), + (4)m Suppose A, k, 8 exemplify (6),,,; so there are a,(Z < w)

such that Rm[8(Z; a,), A, KO] = k and Mlt[O(z; a,), d] 2 1. So for some A, I{q E S ~ ( A , ) : P[{8(z; a,)} u q, A , KO] = k}l 2 1, SO if I 2 24n, by 1.6 of the Appendix there are tp;, 8; (i < n) such that R"(pf, A, No) = k,

p ; = ttp;(z; b y ( * = j ) : j 5 i} u p(z; a,)}

pfc c pi+' and Rm(pfc+l,d, NO) I n(0) - 1 - 1 (as R"(pk,dy 80) =

with k. We prove that, more specifically, Mlt(p$O),d) 2 Ip$O)+l I -

that Mlt(pfc",A) 2 lp$o)+ll - Ip$O'I.

andyfEcll(A).SowecanfindZ(n + 1) < wandtp, suchthatfori < n < w

@) = (p,, and let for i < n, A pi(,,, = 8,(Z; EF). If (4), is not true, there are k, t,b such that: for every E, P [ d t ( Z ; a), A , No] = k 4 C@] and similarly for 8,$ .

By a compactness argument we can find a, 8, (j < w) such that:

P [ 8 ( Z ; a), A, KO] = k,

Rm[{8(Z; a)} U {Cp,(Z; 6j)u(j=f): j I i}, A, KO] = k

contradicting the definition of the rank. (a),,, => (8),,, We choose A, k, 8 exemplifying (4),,,, so that k is minimal.

Clearlv for everv 8,. the property P [ 8 , ( S ; Zl), A, KO] 2 k is elemen-


tary (as i t is equivalent to -,V14k R”[e,(Z; a,), A , KO] = I, and by the minimality of k). Moreover also the property “Rm[8(Z; a), A, KO] > k or Rm[B(Z; @ , A , No] 2 k and Mlt[O(Z; a ) , A ] 2 1” is elementary (it is equivalent to the existence of Qt,, E A (i # j c 1) and 6‘,, such that

A R”[{e(Z; a)} u {Ti,,@; &,,): i < E},d, KO] 2 k j < l

where Q‘.,(% h,,) = l Q , , i ( R 6,,‘)). If R”[e(Z; a), A, KO] = k S- Mlt[O(Z; a), A] I I for some I (not

depending on a) it will follow that also Rm[O(Z; a), A, KO] = k is elementary, contradicting our hypothesis (( 4),).

So for every I < w there is 8, such that R”[e(Z; a,), A, No] = k, Mlt[O(Z; a,), A, KO] 2 I. So for some A, (by 1.10(1) and l.lO(4)).

q = {q E ,Y:(A~): R*[{e(z; a,)} u q, A , K,] = k}

has cardinality Mlt[B(Z; a,), A] and q E @, implies Mlt[{O(Z; a)} u q, A ] = 1. By Claim 3.12, for each j c w, there is I ( j ) c w (e.g., 2,’ + 1) and distinct q{ E Qltf) (i < j) and 4 E q{, Ir{l 5 Rm(Z = Z, A , 2) + 2, and

We can find 8* and b: (i c j, j c w ) such that for every i c j, j c 4 c q E @,cn =- 9 = qi.

0, O(Z; a,(,)) A A TI = 8*(Z; 6{, a,(,)), hence

Rye*(%; 6{, a,,,,), A , N,] = k, mt[e*(2; 6{, a,(,)), 41 = 1.

Clearly for every cp E cl,(A) there is a formula +p, depending on A , m, k and 8* only, such that:

and for each F(Z; 5 ) exactly one of &,[E, 6{, iX,(n],97@, 6{, si,,,)] holds. Rm[e*(Z; E{, ai(j)) A ~ ( 2 ; a), 4, No] 2 k * W,JC Z{, 5i,,,,I,

We define an equivalence relation E(ji,, j ia; a): b,, b, are equivalent i f (i) Rm[8*(2; 6,, a), A , No] = k A Mlt[B*(Z; 6,, a), A] = 1 iff

m[e*(z; 6,, a), A , KO] = k A mt[e*(z; 6,, a), A ] = 1.

(ii) if the condition from (i) holds, then for each Q E A w)[+,Jz, 61, a) #cp(g, 62, a)].

It is easy to check that E(&,ji2; a) is an equivalence relation, expressible by a first order formula (for (i) look at the beginning of the proof of (4), =- (8)”), and E(Jl, fla; a,,,)) has c KO but 2j equivalence classes.

(8), =- (2), Let 7 ~ ( Z , ji; Z) be as in (8),,,, and we shall prove that -,Q(Z; ji, Z) has the f.c.p. For each n let Zn be such that Q(Z, ji; En) has ~n but c KO equivalence classes; let a,, . . . , a, (I 2 n) be a set of

CH. 11, $41 F.C.P., INDEPENDENCE AND ORDER 67

representatives, so p = {+Z; a,, 4): 1 I i I Z} is inconsistent but every p' E p , p' # p is realized.

(3), =- (0)A,m Trivial (asp is realized iff Rm(p, A , A) # -1). (5), * (7), Immediate (use a minimal k). We finish by:

CLAIM 4.5: Suppoee not (3), and not (ti),,, for every m. Then

(A) not (0)A,m,

(B) not (7)m*

Proof. (A) By (B) it suffices to prove (A) for any fixed A < KO. By Theorem 2.2(7) and Theorems 1.1(1) and 1.3(1), for allp, R m ( p , d , A) I Rm(Z = Z, A , 2) = no < 0. By 2.1, w.1.o.g. A = {cp}, so it suffices to prove that for every k I no there is n,(k, A) < w such that every cp-m- type of rank < k has a subtype q of cardinality I n,(k, A) such that Rrn(q, v, 4 < k.

Define for t E (0, I}, +t(. . . , Z,, . . . ; g) = cp(Zq; and let -t j +(. . ., P,, . . . ; . . . , z; 9 * * *)nskh .vsk>A; f , l<A

= A {cp(~,; z$') G 7cp(fp; 2)'): v E k > A ; q, p E 'A; k > n = Z ( U ) , v = 7 1 n = p n, i = q[n], j = p[nl}.

For notational simplicity let +t = f ( t ( Z * ; g), + = +(Z*; Z). By Lemma 2.9(2) it is clear that for every cp-m-type q Rm(q,cp, A) 2 k iff {+(Z*; Z)} u {Q(Z*; a): cp(Z; a)t E q} is consistent. Taking in Theorem 4.4(3) A = {+, +O, +l}, m' = Z(Z* Z) the result follows.

(B) Since for every A 2 2, and p :

(ii) Rm(p, cp, n ) = Rm(p , cp, a)

-def Rm = (0 Rrn(p, cp, 4 I no - (- 3, cp, 2) < w , (n I A) implies Rm(p, cp, n ) =

RYP, c p 9 A); it suffices to prove: (*I for every n there are a(n), p(n) < w such that for any set

of cp-m-formulas, p , if for every q c p , IqI I a@),

(Take n2 = max{p(n): nsn,} , so for (A) k = m&x{a(n): n I no}.) We prove it by induction on n. For n = 0, Rm(p, cp, A ) 2 0 iff p is

consistent; hence by "not (3),,," our assertion holds. Suppose we have proved for n, and we shall prove for n + 1. By Theorem 2.12(2) for every q E O'2, k, Z < w there is a formula +[J such that

RYq, cp, Rn)l 2 n, then R"(p, q, 00) 2 n.

Rm[{cp@; Si,)R['': i < Z(q)}, cp, k] 2 Z iff kgJISio, . . . , 82(q)-1].

Let m, = Z(1) where cp = cp(Z; g); and A, = {I#")*": q E a(n)r2}. Let k(n) < w be such that


(i) every inconsistent set of A,-m,-formulas F has an inconsistent subset of power < k(n), and

(ii) there are no a,, Sl such that C&; a,] = 7p[6,; at] for i # j < k(n) [k(n) exists by "not (3)," and Theorem 2.2 and 4.31.

Now let P(n + 1) = 24k(n) + /3(n), and a(n + 1) be such that if r i a a set of p-m-formulas, and for every r' c r of power <a(n + .l) -[I", p, P(n + l)] 2 n + 1, then P[r, p, P(n + l)] 2 n + 1 (exists by what we have already proved for part (A) of this claim) so a(n + 1) depends on P(n + 1).

So it suffices to prove that if p is finite and Rm[p, p, P(n + l)] 2 n + 1, then Rm(p,p,a) 2 n + 1. As Rm[p,p,/3(n + l)] 2 n + 1 there are p-m-types q, 2 p (i < /?(n + 1)) such that P [ q l , p, P(n + l)] 2 n, and they are explicitly contradictory in pairs. As P(n + 1) 2 ,B(n), by the induction hypothesis P ( q , , p y a ) 2 n; so by 1.6, we can assume all the q, belong to some &(A). So, by 1.6 of the Appendix, by renaming, there are cp(Z; a,) (i < k(n)) such that +Z; a,) E qj for i < j < k(n), p(Z; 7i,) E ql, and p(Z; E ql for all i < I < k(n), where t depends on n only. By (ii) of the definition of k(n), for i , j < k(n) p(Z; Zit) E qj 9 i = j. Now define 6, by induction: for i < k(n) 6, = a,; and for i 2 k(n) define 6, so that for every j I i, Rm[q{, p, P(n)] 2 n where qf = p u {p(Z; 6y)ti(Y-n: y I i}. By part (i) of the defhition of k(n) this is possible. By the induction hypothesis, Rm(q{, p, a) 2 n, hence Rm(p, p, a) 2 n + 1. So we have finished the induction, hence the proof of 4.6(B).

THEOREM 4.6: (1) If T does not have the f.c.p. and A hJinite, then there is n,(A) < w such that: if {ay: y < a 2 n,(A)} iS a A-n-indhcernible set over A, B > a, thn we can deftne Zfor a I; y < f i so that {a: y < /?} is a A-n-indiscernible set Over A.

( 2 ) I f A c l M l , V ~ 1 M 1 fory < aandMis(IAl+ + l A l + + lPl + 1.1 +)-cornpact, then we can choose

( 3 ) If A, A are Jinite a < P = w, A c !MI, E IMI for y < a then we can choose cUE(MI for a < y < /3 (under the assumption of ( 1 ) ) .

Proof. ( 3 ) follows from (2), and (2) will be clear from the proof of (1). So we shall prove (1) only.

If n is too large, every set of distinct 8 s is a A-n-indiscernible set. Similarly if m = Z(ao) is too large. So we can prove the theorem for fixed n, m, and still get an nl(A) which is good for all n, m. Let

E [MI for a I y < ,9 (as in (1)).

A* = {p(Zo,.. . , i f " - ' ; 8): p(P, . . . , it"-'* , 8) = +(F(O), . . . , zo(n - 1) J * 8) for some #(Zo,. . . , if"-1* , 8) E A and permutation 0 of n}.

CH. 11, 4 41 F.C.P., INDEPENDENCE AND ORDER 69

As T does not have the f.c.p., by Theorem 4.4(3), there is a natural number n3 = nl(A) such that:

If I' is a set of formulae of the form ~ ( 9 ; Zil, . . . , Z-l, a), cp E cl,(A*) and every subset of I' of cardinality <nl is coneistenf then I' is consistent.

It is clearly sufficient to prove that we mn define aa such that {ay: y < a + 1) wil l be a A-n-indiscernible set over A. For this it is clearly sufficient to find aa which realizes

pa = { c p ( ~ ; W-', . . . , avo, E)*: E E A , yo < - - - < yn-2 < a, t E {0, I}, cp E L I * and Ccp[Z-', Z-2, . . . , Tio, a*}.

For this, it is clearly sufficient fo prove that pa is consistent. By the definition of nl, it suffices to prove that every subset of pa of Casdindity < nl is consistent. Let q be a subset of pa, 1q1 < nl. Clearly in q apptw ~ ( n - l)(nl - 1) P a . So if a > (n - l)(nl - l ) , then there is 7iv, which does not appear in any of the formulas of q, hence i2 realizes q, so q is consistent. So pa is consistent; hence we can define P, hence we mn define iir for a 5 y < 8, by induction, as required.

DEFINITION 4.2: (1) A formula cp@; g) has the independence property if for every n < w there are sequences aI (I < n) such that for every w E n,

r

(2) T has the independence property if some formula cp(x; a) has the independence property.

DEFINITION 4.3: (1) A formula cp(Z; g) has the strict order property if for every n there are (I < n ) such that for any k, I < n

C ( 3 Z ) [ i c p ( E ; a,) A cp(E; a,)] 0 k < I .

(2) T has the strict order property if some cp(x; g) has the strict order property.

Remark. ( 1 ) As in Definition 4.1(2), we can also in Dehitions 4.2(2) and 4.3(2) replace cp(z; a) by cp(Z; 3); in this case we use Theorems 4.11 and 4.16.

(2) We shall show that a theory is unstable iff it has the independence property or the strict order property; but neither of these properties implies the other (see 4.7 and 4.8).


Proof. (1) This follows from (2). It T is unstable, some (p(z; j j ) is unstable (by 2.13); hence by (2) (p(z; i j ) has the independence property or #,,(z; Po,. . . , ijn-l) has the strict order property (for some n, q) . Thus in the first case T has the independence property, and in the second c&88 the strict order property.

Suppose on the other hand that T, and thus some (p(z; i j ) , has one of these properties. If (p(z; j j ) has the independence property, then by (2), (p(z; j j ) is unstable and by Theorem 2.13, T is unstable. If (p(z; 5) has the strict order property, then so does #,, for q = (0) so by (2) again, (p(z; j j ) is unstable and 80 is T.

(2) Suppose (p@; i j ) has the independence property. Then clearly (p(3; j j ) has the order property. So by Theorem 2.2 (p(Z; i j ) is unstable. Now assume that for some q #,, has the strict order property. Then #,, has the order property and thus #,, is unstable. So there is an A such that ISFq(A)I > IAI 2 KO, where Z(Z) = m. For every qES$&4) let tZq realize q, and define

p* = {(p(Z; a)t: ii E A , t E (0, l}, kgl[iiq; a]'},

i.e., the (p-m-type iiq realizes over A.

hence p # p implies q* # p*, so Clearly p* E S ~ ( A ) ; and for p, q E STw(A), p* = p* implies q = p;

This means that (p(Z; j j ) is unstable. So the strict order property of #,, implies the unstability of (p; and also the independence property of (p(Z; j j ) implies (p(Z; i j ) is unstable.

So it remains fa be proved that if (p = (p(3; j j ) is unstable, then it has the independence property or for some q , & has the strict order property. By Theorem 2.2, (p(3; j j ) has the order property, so there are sequences lie, . . . Z7 . . . such that for every k < o there exists Ek such that kv[ek; an] iff k s n. By the compactness theorem and Theorem I,

OH. 11, 8 41 F.C.P., INDEPENDENCE AND ORDER 71

2.4, we can assume that (3: n < w ) is an indiscernible sequence. If for every n < w , w c n,

k(32) A ~ ( 3 ; ~ k ) M ( k e w ) ,

then clearly ~(3; j j ) has the independence property and so are finished. So we assume that there are n < w , w E n such that

[k<n I

k7 (38 ) [ A Cp(3; ak)if(kew)]. k < n

Let IwI = no. There is an u < w such that we can define w, for I I u such that:

( 1 ) ~ ~ = ~ , ~ , = { n - n ~ , n - n , + 1 ,..., n - I } , ( 2 ) for every I I a, lwll = no, w1 E n, ( 3 ) for every 1 < u there is k, < n such that wl +1 = w, v {k, + 1) -

{h} (and 80 k, E ~ 1 , k, 4 wI+1, k, + 1 $ w I , k, + 1 E W I + ~ (we step by step raise w = wo to w,).

the definition of the Sik’8 and w,, We are assuming i=1(33)[Ak<n ~(3; iZk)if(kewO)]. On the other hand by

c(3z)[ A ~ ( z ; a*)if(kewa)]. k < n

Hence there is 1 < u such that

kl(3Z)L A ~(3; (kea)], where 8 = w,, k < n

and

k(31)[ A ~ ( 3 ; ak)if(kE” , where t = w ~ + ~ . kcn 1

Let = k,, and * = *(3; j j , P O , . . ., g - 1 , p + a , . . ., p-1) = A ~ ( 2 ; jjk)if(kes) A ~(3. 9 - j j )

ken k + 8 . 8 + 1

We shall prove that $ has the strict order property. Let y < w , and

of (2: i < w ) , and the definition of $, if p I k < 1 < /I + 2 + y , then (as this holds for k = 8, I = /3 + 1, y = 0) ,

define 6 = @On. . . -a8 - 1^@B + 2 + Y” . . , an - 1 + 7, By the indiscefibfiity

k(3z)[$(P; a’, 6) A --1cp(3; ak)];

RANKS AND INCOMPLETE TYPES [CH. 11, 5 4 72

but (by the same argument)

Hence, observing again the definition of #,

As this is true for every y < w , clearly # haa the strict order property, and it is also clear that # is of the required form. As this waa proved under the assumption that cp(S;g) does not have the independence property, we have proved the theorem.

THEOREM 4.8: (1) There is a theory To,, with the 8 t k t order property and without the. independence property.

(2) There w a theory Find with the independem property and hthout the strict order property. Moreover, there is no form& (p(xl,. . ., x,,; a) which M connected and adqpnmetrh over an infinae 8et.

Remark. Cf. Definition I, 2.5. Of course by 2.13 there are m < w and a formula y3 = #(.", . . *, P-1; 6), Z(P) = * * = Z(P-l) = m such that # is connected and antisymmetric over en infinite set of sequences of length m.

Proof. (1) Let Tord be the theory of the rational order, i.e., dense order without first or laat element. Clearly (p(s; y ) = [a: < y ] haa the strict order property. The checking that it does not have the independence property is left to the reader aa an exercise.

(2) In the language of T , there will be only the equality sign, a one place predicate P(x), and a two place predicate %By. Its axioms will be:

(1) xEy implies T P ( z ) , P ( y ) ; that is, ( V q ) [ x B y + ,P(z) A P(y)] . (2) if P(y ) , then y is uniquely determined by {x: xEy}, and conversely;

i.e.,


(3), For every 2n different elements in l P , xl, . . . , q,, zl, . . . , xn there is a y such that zlEy,. . . , xnEy, 7x1Ey,. . . , lxnEy. That is,

( V q , . . ., xn)(Vxl,. . ., xn) [ A LZk # d A 7p(xk) A ~ p ( x ' ) ] 0 < k s n 0 < 1Sn

-+ ( 3 ~ ) A (%&Y A ~ ~ E Y ) ] . 0 < k s n

(4), The mme as (3)n, interchanging P and l P , xEy and yEx. That is,

(VYl, - * * , Yn)(vY1, * 9 Y") [ A [Yk # Y' A p(Yk) A p(Y')l 0 .c k s n 0 < l sn

+(3Z) A (%By, A 7ZEYk) *

0 < k s n 1 (5) All models of Tina are infinite, i.e.,

(3z0, . . ., z,) A zi # z, for every n > 0. t<jSn

It is not hard to prove that Tina is consistent, by building a model for it. It is also easy and standard to prove it has elimination of quantifiers, and is complete. By this it can be shown that no formula 'p = 'p(Z; 9) has the strict order property. On the other hand, clearly y(x; y) = zEy has the independence property (hence the order P r o P Y ) *

THEOREM 4.9: If 'p@; a) i8 urntable, A < K < Ded A, then there is an A such that IAI 5 A, ISg(A)I 2 K . (See Definition 1.4 of the Appendix.)

Proof. By Exercise 1.4 of the Appendix, there is an ordered setJ, I J I 2 K ,

with a dense subset I, IZl = A. As 'p(Z; 9) is unstable, by Theorem 2.2 'p

has the order property. Hence by the compactness theorem there are BE, 8 E J such that: for every t E J, {'p(Z; B8)if(tsE): 8 E J) is consistent.

Let A = (J {aE: 8 E I } . Clearly IAI s I Z l - Ho = A. For every t E J let pt = {'p(Z; aE)if(t<8): 8 E I } . Clearly p t is a consistent 'p-m-type over A. Let pt c qt E St(A) . Now if al, 82 E J, a1 < 82, then there is t E I , 81 < t < a2, so y(Z; a,) E qa1, -I~(Z; a,) E qsa. Hence a1 # 8 2 implies q8, f

l#t(A)l 1 1{q8: 8 EJ}I = IJI = K > A = IAl.

THEOREM 4.10: (1) If 'p = 'p(Z; a) hag the independence property, then for every A there is an A m h that IAI = A, ISg(A)I = 9'.


(2) If f o r some infinite A, IS;(A)I 2 Ded,lAI, then Q ( Z ; ~ ) has the

(3) If cp(Z;g) has the independence property and Z(g) = k, then f o r

(4) If f o r every n c w there is afinite A, IAI 2 2, euch tlrat I&(A)I 2

independence property. (Ded,+ee Definition 1.4 of the Appendix.)

every n c w there is an A , IAI s nk, I8;(A)I 2 2".

IAI", then ~(3; g) has the independence property.

Proof. (1) Let

r = { (W[ A QP; SJ A A l ' p ( Z ; W)]: LEW LEU

w G A, u c A; w n u = 0; w, u are finite.

As Q has the independence property, r u T is consistent, hence has a model N. Let 6, realize gt and A = Uf<A 6,. Clearly IAI = A. For every (not n e c e d y finite) w G A let pw = {Q@; 6L)ifuEw): i c A}. By the definition of r, pw is consistent, hence there is qw ES;(A), pw c qw. Clearly w # u implies qw # qu, so

IS;(A)I 2 I{qw: w E A}1 = l{w: w E A}l = 2 A

thus proving (1). (2) Let U = {Q@; 6): 6 E A}. The mapping p + p n U from S;(A)

to subsets of U is one-to-one. Hence {p n U: p ES;(A)} is a family of power 2Ded,lAI = DedJUI. So by Theorem 1.7(1) of the Appendix for every n c w there are ~ ( 3 ; 6') E U, i < n, such that for every w E n there is pw E S;(A) such that for i c n ~ ( 2 ; at) epW o i E w. This implies that Q(Z; g) has the independence property.

(3) The proof is similar to that of (1). (4) The proof is similar to that of (2), this time using 1.7(2) of the

Appendix.

THEOREM 4.11 : The following statements about a theory T are equivalent.

(1) Some formula p(x; g) has the independence property, i.e., T hm the independence property.

(2), Some formula Q(Z; ji) ( l ( Z ) = m) hm the independence property. ( 3 ) There are Q ( X ; g) and n < w such that fo r every k c w there is an

(4), There is ~(3; g) , m = I @ ) , 8uch that f o r every n there is afinite A , A , IAI = k , IS,(A)I 2 2k'n.

2 2, p;(A,I 2 pq".

CH. 11, f 41 F.C.P., INDEPENDENCE AND ORDER 75

If for ~0112e A, (vp)b < Ded, A -+plTl < Ded, A], the follo2oing statements are equivalent a8 well.

(6 ) For every p there is an A , IAI = p, IS(A)I 2 2u. (6), For s m A, ISm(A)I 2 Ded,lAl, and (Vp)[tc < Ded,lAl

< D4lAl l .

Procf. Clearly (1) implies (2), (by adding dummy variables to p). By Theorem 4.10(3), (2), implies (4), and by 4.10(1), (2), implies (6),, when assuming the existence of the suitable A.

Suppose (6), holds, and let A = 1.41; x = Ded, A; so by Lemma I, 2.1 for some By 1B1 = IAI, IS(B)I 2 x . As x is regular, clearly by Lemma 2.16(1) for some 'p, 1SJB)I 2 x . Hence by 4.10(2), 'p = p(z; #) has the independence property. So (6), implies (1). By 4.10(4) (a), implies (2),.

By 4.10(1), (1) implies ( 6 ) , and by 4.10(3), (1) implies (3), and trivially (3) implies (4),, (6) implies (6),. So we have (1) =+ (2), + (4), =. (2),, (1) =+ (3) + (4),; and when suitable A exists (1) =- (2), =+

(6), * (l), (1) * (6) 3 (6),. So the theorem is proved, by the consistency results of Baumgartner (see [Ba 761).

DEFINITION 4.4: (1) Kz(A, T) is the first cardinal p such that IAI s A implies lS;(A)I < p. (2) Kr;(A, T) is the first regular or finite cardinal which is 2 KT(A, T). (3) Usually we omit T. If d = L we omit it, ifd = {p} we write just

'p and if m = 1 we omit it.

THEOREM 4.12: Kr;(A) can be only one of the following functions:

constantn (1 < n < w ) , A + , Ded, A, (2"+.

Moreover, emh of these functions is realized (by 8ome p, T ) . I n fact, p is unstable ifl Kr;(A) = Ded, A for all A or Kr;(A) = (2")+ for all A.

Proof. First suppose that p(Z;#) is unstable. Then by Theorem 4.9, A < K < Ded A implies there is an A, IAI s A, IS;(A)I 2 K. Hence K;(A) 2 Ded A. If for every A, Kr;(A) = Ded, A, the conclusion of the theorem holds. So suppose for at lectst one A, Kr;(A) > Ded, A. So by Definition 4.4, there is A, IAI s A, 15;(A)I r: Ded, A. As Ded, A is regular, by Theorem 4.10(2) ' p ( f ; #) has the independence property. So by Theorem 4.10(1) for every A there is an A, IAI s A, ISz(A)I 2 2A,

76 RANKS AND INOOMPLETE TYPES [OH. 11, 8 4

hence Kr;(A) > 2A. But always IS;(A)I 5 2 1 4 + N o , hence Kc(A) I; (2"+. So if p is unstable, Kr;(A) = Ded, A (for every A) or Kt$!(A) = ( 2A) + (for every A).

So suppose p(5; 8) is stable. For every n, if for one A, Kc(A) > n, then there is A, IAl I; A, ISg(A)I 2 n. It is easy to find B E A, IBI 5 KO such that ISg(B)J 2 n; hence for every p, aa p 2 K, 2 IBI, Kr;(p) > n. Hence if for some A, Kr$(A) = n, then for every p, Kr;(p) = n. As for every A, and a, ti realizes a type in 8; (A), clearly IS;(A)I 2 1, son > 1.

So suppose Kr;(A) 2 No for some A. Then for any n there is A,, l@(An)l 2 2". We can define by induction on k < n, sik, ~ [ k ] such that

1{p ES;(A,): p(5; sil)n[ll ~p for every 1 < k}l 2 2n-k and

1 {p@; al)n[lI: 1 < k - 1) u {p(z; a k - 1 ) 1 - n t k - 1 1

is consistent. For simplicity suppose every $11 is zero. Hence {p(?; jjl)if(l<k): 1 I; k < n} is consistent for every n. So for every A, {p(Za; fjn)if(8<a): /3 5 u < A} is consistent. Let sin realize ijn and b realize P, A = U {an: < A}, and par be the p-m-type b realizes over A. Clearly,

IAI = A, pg(A)I 2 l{pa: u < A}l = A.

Hence for every A, Kg(A) 2 A + . As p(5; g ) is stable, IAI 5 A implies ISg(A)I s A, so K;(A) 5 A + . So K;(A) = A + . Clearly in the c m s K;(A) is (2"+, A + , n; Kr;(A) = K;(A).

Let us prove that each of the mentioned functions is Kri(A) for some p and T.

(1) By Theorem 4.8(2) there are T and p(z; y ) such that p(z; y) has the independence property. Hence Kt(A) = (2A)+.

(2) Let T be the theory of the order of the rationals, and p(z; y ) = z < y . Clearly K$(A) = Ded A.

(3) If T is the theory of equality, ~ ( z ; y ) = [z = y ] , then K$(A) = A + . (4) Let T be a theory with an equivalence relation E with n 2 1

equivalence classes, and p(z; y) = [ zEy] . Clearly Ki(A) = n + 1.

THEOREM 4.13: Swppocle #(g; 5) = p(3; jj) and #(jj; 3) does nd have the independence property. Let


or

Then there is n = n(cp) < w such that (1) if {a*: p < a} is a A,-n-indiscernible set, E a 8equence then either

< a: hp[7iB; a]}l < n

I@ < a: C-,cp[ iZ*;E]}I < n;

(2) if {a*: p < a} b a A,-n-indbmible sequence and E b a sequence, then there are 0 = a. 5 a, 5 - - - 5 an = a such that: if i < n, a, 5 jl,

< a,+,, then ~tp[iP'; E ] iff ~tp[7iB'; E].

Proof. (1) As $@; 3) does not have the independence property there is no < w such that

r = { ( 3 p ) A +(y"; ~ I ) U ( ~ € W ) : w = no . I<no 1

is inconsistent. Let n = no.

ordinals pol . . . , fin- ,, yo, . . . , yn- , < a such that Now suppose our conclusion is incorrect. Then there are distinct

hp[$o; E l , . . . , hp[iZ*n-1; E l , C - l c p [ Z i Y o ; E l , . . . , t7cp[ZiYn-1; 51.

Remembering that {a*: p < a} is a An-n-indiscernible set, we can see that taking 8 for f' for I < n, I' is satisfied. Hence r is consistent, contradiction.

(2) The proof is essentially the same.

DEFINITION 4.6: (1) Let I be a A-n-indiscernible set in M (i.e., 8 E I * 8 E 1M1). Then &(I, A, n, M) is the first cardinal p such that there exists a maximal A-n-indiscernible set I,, in M , I c I , , II,I = p (I, being maximal means that there is no A-n-indiscernible set I, in M , I , c I , , Ii Z Id.

(2) Similarly we define dim(I, A, M), dim(1, n, M), with A-indiscernibility, n-indiscernibility instead of A-n-indiscernibility ; and the same with dim(I, A, <n, M), but not dim(1, M).

THEOREM 4.14: Suppose T a h x not have the independence property, A is$nite. Then there are a natural number n* = n*(A) and a finite A* such that: for any n; if I, is a A-n-indiscernible set in M , I, is a A*-n*- hdiscernibk Set in M , 11, n Is1 2 n*, then

n*[dim(I,, A, n, M)]"- l 2 dim(I,, A*, n*, M).

78 RANKS AND INCJOYPLETE TYPES [OH. 11, 0 4

So if one of those dimensions is M t e , then dim(I,, A, n, M) 2 &(I,, A*, n*, M).

Prmf. Remember that by Lemma I, 2.3, for any finite k; A, , . . . , Ak; n', . , . , nk there are finite A, n such that I is A-n-indiscernible if€ it is Ai-nf-indiscernible for every i, 1 s i s k. Remember also that for sufficiently large n, every set of sequences (of length m) is A-n- indiscernible. Thus it suffices to prove the theorem for a fixed n. Let

A = { q k ( Z o , . . ., 3-l): k < k, < w }

where &to) = - - - = Z(3-l) = m. Define, for k < E,, u a permutation of n

$&?, . . . , 3 - 1 - , 5 --O ) = Cpk(?iYO', * . ., 2 9 - 1 ) 1 and let zk = fin.. .-3-1 9 80 4k.u = $k,u(Zk; ZO).

Now no $k,u(Zk; Zo) hae the independence property (by Theorem 4.11, ae T does not have the independence property). So there is i = i(k, U) < w such that

is inconsistent. Define n* = max{i(k, u) + n: k < k,, u a permutation of n}

u is a permutation of n . i. A' = {qk,s(fo; P): k < k,, u a permutation of n,

' p k , @ O ; P) = # k , u ( Z k ; ZO)} u {Zo # g}, I ( @ = m.

We shall now show: If I, is ad-n-indiscernible set in M, I, ad1-( < n*)-

n*[dim(I,, A, n, M)]"-' 2 dim(Ia, A', <n*, M). Then we can find the required A* by Lemma I, 2.3(3).

By the definition of dimension we can assume I, is a maximal A-n-indiscernible set in H of cardinality &(Il, A , n, M); and similarly for I , . So it suffice to prove only that n*II,I"-l L lI,l.

E 11} such that E realizes p iff I, u { E } is a A-n-indiscernible set, c 4 I,. We can wume that p is such that every formula in q contains elementa from ~n - 1

indiscernible eet in H, 11, n I,[ 2 n*, then

Now there is a Aa-m-type q over A = u {a:


sequences of I,. So every formula ~ ( 3 ; 6) in q is realized by every i5 E I, n I , except at most n - 1. Hence

I{E E I,: C&; 6]}1 2 II, n I,I - (n - I) 2 n* - (n - I) > i ( k , U)

where Q is tpk,o or - , Q ~ , ~ . By the definition of dl and i (k , u), clearly I{Z E I , : C - p [ E ; 6]}1 n*~Iln-l~Aal = n**)I,)n-l,thenthereisaZ €1, - I, which satisfies every formula in q, hence I, u {Z} is a A-n-indiscernible set, and aa i5 E I,, Z E 1". This contradicts the choice of I, as a maximal A-n-indiscernible set in H. So [ I a [ I n**II,l"-l so by replacing n* by n** we finish the proof.

THEOREM 4.16: If T &ea not have the independence property and 111 n IpI 2 H,; I , , I , are indiscernible sets, t h n

dim(I,,A, M) + IT1 = dim(I,,A, Y) + IT].

I Idall.

Proof. The same as the previous theorem.

THEOREM 4.16: If s m e ~ ( 3 ; Z) has the strict order property then T has the strict order property.

Proof. We prove it by induction on m = l(3). If m = 1, there is nothing to prove. Suppose we have proved for m and we shell prove for rn + 1 ; for notational simplicity let Q = ~ ( z , 8; Z), 1(g) = m.

Let {az: 1 < w } be an indiscernible sequence such that C(32, g)[lQ(z, 8; 8,) A Q(z, 8; %)I

iff 1 < k (exists by compactness and Theorem I, 2.4). If there is b such that

C(3g)[ iQ(b , g ; %z) A Q(b, 8 ; %+I)]

for every 1 < w , then 8( g ; z, Z) = ~ ( z , 8; Z) has the strict order property (this is exemplified by (b)n6&, 1 < w ) hence we finish by the induction hypothesis. Otherwise, by compactness there is a minimal k < w such that for no b, C(3g)[-,tp(b, g ; Zai) A v(b, g ; 82z+l)] for every 1 < k + 1. Clearly k 2 1. Let

80 RANKS AND INOOYPLETE TYPES [CH. 11, $ 4

(by the minimality of k and the indiscernibility of {a,: 1 < w}) and the same x shows that (ii) holds.

By this and compactness, it follows that $(x; Lo, . . . , Zak-1) haa the strict order property, hence we have finished the proof.

QUESTION 4.1: Suppose ISg(A)I 2 DedJAI; does cp necessarily have the independence property? (See 4.10(2).)

EXERCISE 4.2: Show that 4.6 may fail when T is stable but does not have the f.c.p.

EXERCISE 4.3: Show that we can add to 4.4 the conditions: (9)" There is a finite A such that for every 1 < w there is a A-

indiscernible set I, of sequences of length m, such that 1 S dim(I,, A, 6)

(lo), For some finite A,, A, for no k < w does the following hold: for every A,-m-type p , Rm(p, A,, k ) = Rm(p, A,, a).

(ll)*,,, For some finite A,, A, for no k < w does the following hold: for every set p of A,-m-formulas, there is q E p , 1q1 < k such that R"(p,

EXERCISE 4.4: Show that T has the strict order property iff some cp(Z; g) is a partial non-trivial order, i.e., it is transitive [ ( V Z , j j , Z)(cp(Z; g ) A cp(g; Z) + cp(Z; Z))], irreflexive [(V3)+3; Z)], asym- metric [ (VZ, jj)(cp(3; j j ) -+ +g; Z)] and with arbitrarily large finite chains [for every n, (3Z0, . . . , Zn) A,<,, cp(Z,; ?El+,)].

EXERCISE 4.5: Let T& be the theory with the following axioms:

< KO.

A) = R"(q, A,, A).

( V X Y ) ( X R ~ ~ R x ) , ( V X ) i XRX, (1)

(VX,,...,X~;Y~,...,&)[/\X~ 1 . j # y j + ( 3 z ) ( A z t R z 1 A A J l ~ j R z ) ] .

(2)n Show that also TZd satisfies the demands in Theorem 4.8(2).

OH. 11, 5 41 F.O.P., INDEPENDENCE AND ORDER 81

EXERCIBE 4.6: For stable T, regular A, and a model M the following conditions are equivdent.

(1) There are finite A, m and a A-m-type p over IM), 1p1 = A, such that every q E p, 1q1 c A, is realized in M , but p is not realized in M .

(2) There are finite d, m, A-m-types r , p over ]MI, 1p1 = A, Irl I A, such that for every q c p, Iql < A, r u q is realized in M, but r u p is not realized in M .

(3) There are finite A, I , n such that A = dim(I, A, n, M). (4) There rn A, m andp E AT( IMI) such that M realizes every q c p,

[Hint: For (2) + (3) let p = {pi(Z; Zi): i < A}, and let 6, realize IqI < A, and some r c p, 1.1 = A, but not p itself.

Y u {p@; Zj): j c i}. Now use 2.18 and 4.13(1). For (3) + (4) use

p' = {p@; a): ZE M, p E cll(A) and I{E E J: Cp[E; st]}l 2 KO}

for any J, I c J' E 1611, I JI = A, J G J' enough indiscernible.]

EXERCIAE 4.7: Suppose T does not have the f.c.p. and M omits some A-m-type of cardinality A, and A is finite. Then for some E(3, g ; 2) E L and B E !MI, E(Z, g; E ) is an equivalence relation, and in M it has I A but 2 No equivalence clams.

[Hint: By 2.1 we can assume p = p 1' p and by 4.2(1) T is stable, hence by 2.13 and 2.2 n1 = Rm(p, p, No) c o. We can aasume p haa a unique extension q in 8;( 1611). We prove by induction on n,. Choose a finitep, c p, n1 = Rm(p,, p, KO), let $($; 2) be aa in 2.12(1) and let n* be maximal such that

{r EA;((M(): p1 E r , n* = Rm(r, p, No)}

is infinite, so n* < nl. Let pa be finite, p1 G c q, such that p1 s r E 5:(1M1), Rm(r, p, KO) > n* and r # q implies pa $ r . Let d(2; E) (a E IMI) say: {p(Z; go): $(go; Z)} is consistent, has rank (Rm( -, 9, KO)) = n*, and extends pa (0, E exist by Theorem 4.4). We could have chosen p1 such that n* is minimal. Then

E(Zl, Za; Z) = 8(2,; Z) A 0(Za; E ) A (Vg)[$(g; 21) $(g; Z,)] is as required.]

QUEATION 4.8: Suppose the condition of Exercise 4.6 holds. Is there an equivalence relation over M with exactly A equivalence classes ?

CHAPTER I11

GLOBAL THEORY

III.0. Introduction

The main interest and motivation in this chapter is in uncountable theories. The results of Chapter I1 are sufficient to find the stability spectrum of T (i.e., the class of cardinals in which T is stable) when T is countable. We shall prove:

THE STABILITY SPECTRUM THEOREM: For any stable T, there is

OT T is stable in A iff A = A < K ( T ) + 2Ka. a cardinal K ( T ) I I TI + such that T is stable in A iff A = A<n(T) + ID(T)I

The main idea behind the theorem, is that if T is unstable in A, A 2 2ITI, then there is a sequence of equivalence relations (in T ) E, (i c 6) each E, partitions many equivalence class of A,<, E,, into infinitely many parts, and AI61 > A. (This is almost true, as by 6.13 if No < K < K(T) , K is regular, then there are such Ei (i < K ) . )

For proving the stability spectrum theorem, we seek a notion, which we call " p forks over A" and which satisfies:

(0) K ( T ) is the minimal cardinality K ; such that for every p E Sm(A) there is B G A, IBI < K ( T ) such that p does not fork over B (in fact, K ( T ) I ITI+).

(1) ForeveryB G A, IBI < ~(T),{p~S~(A):pdoesnotforkoverB} has small cardinality (in fact I 2 9

(2) I f p E Sm(A) forks over B, then for some finite C c A, p (B u C) forks over B.

(3) If p E &"(A,), A , (i I 6 ) is increasing, and p A,+1 forks over A, then T is unstable in A whenever Alal > A.

Clearly properties (0), (1) enable us to prove stability results, whereas (2), (3) enable us to prove instability results.

The notion of splitting is almost suitable, but it satisfies (3) only for 82

CH. 111, 01 INTRODUCTION 83

A < 2161. This was the motivation for defining: p strongly splits over B if ~(3, a), -,T@, 6) ~p where a, 6 belong to an infinite set indiscernible over B. This is sufficient to prove the stability spectrum theorem for A 2 2ITI. However, for finer considerations it is not good enough, as there is an unjustified dependence on the parameters appearing in the type, and as it does not necessarily satisfy:

(4) If p does not fork over B, Domp E A then there is q, p E q E

Bm(A), which does not fork over B. All this leads to defining “p forks over B” by: for some A, Domp E A,

each extension of p in Bm(A) splits strongly over B. Notice that for B c A, p E #“‘(A), “ p does not fork over B” is a good substitute for “p and p 1 B have the same rank” and it looks more natural (see Section 4 for the connection).

In Section 1 we define forking, and prove its simplest properties. It is interesting to note that “ p E @“(A) does not fork over A” is relatively problematic, and in its proof we assume T is stable and use the ranks Rm(p, A, KO) for finite A’s. Hence in Sections 1-6 we deal with stable T only. Note that some of the lemmas in Section 1 are needed only to get better results in Section 2.

We say p is stationary over B, if for every A, Dom p E A, there is one and only one q E Sm(A) extending p which does not fork over B. It is interesting to know how much information p should contain in order to be stationary over B (assuming p does not fork over B); and the finite equivalence relation theorem answers it. A formula v(Z; a) is almost over B if some $(Z, p, 6) (6 E B) is an equivalence relation with finitely many equivalence classes, and $(E; p; 6) t- (~(5; a) = ~ ( p ; a). So p E Bm(A) (B s A) is stationary over B iff it does not fork over B, and it “decides” all formulas almost over B. Hence each p E B ( ~ M ) ) is stationary over [MI and we can prove property (1) of forking.

In Section 3 we prove the stability spectrum theorem for A 2 2ITI. We also define the dimension dim(1, A, M) (Definition 3.3) and show that it is well behaved when it is ~ K , ( T ) . We prove that a K,(T)- saturated model is not A+-saturated iff some I has dimension s A. We also prove that A-saturation is preserved under unions of length 6 if cf S 2 K ( T), and that a A-stable T has a saturated model of cardinality A.

In Section 4 we give a quite complete picture of the connection between ranks and forking, being stationary and the multiplicity; and show that complete types over 1M1 have usually multiplicity 1. But more important are the results:

84 OLOBALTHEORY [CH. 111, 5 1

THEOREM 0.1: (1) (symmetry) tp@, A u 6) does not fork over A iff tp(6, A U a) doee not fork over A (8633 4.13).

(2) (tradtivity) if A E B E C, and tp(a,C) dim not fork over B, tp(a, B) does nd fork over A then tp@, C) doee not fork over A (see 4.4).

(3) Let B c A , then tp(6, A u a), tp(iZ, A) donot forkover B u a, B reap. iff tp(7i-6, A) do@ not fork over B (4.14, 4.16).

(4) T h type p does not fork over lbll iff p i8 finitely sati8jiuble in bl (8W 4.10).

This theorem will enable us to introduce in Chapter IV F',-constructions.

In Section 6 we complete the stability spectrum theorem for h I 2ITI, by proving in the appropriate caaes, that there we many independent formulas (over F) using very specific dependence relations.

The idea of Section 6 is that sometimes equivdence claaaee are not leas real than elements, hence when we seek a canonical form, sometimes it is an equivalence class. So we introduce P, which is just like Q except that we have elements for equivalence classes. We get, e.g., that for stable T , for eachp E Lgm( 1N1) there is a canonical'B G IMI, over which p does not fork, B is the algebraic closure of some C, ICl < K ( T ) .

In Section 7 we are interested mainly in unstable theories, and get various results. Some theorems are generalizations of the stability spectrum theorem. We find qra(T) s /TI+ such that K < K,,(T) iff in T there are K independent orders iff for some A, IS"(A)I > Ax for every h < DedlAI. (Thus we get a complete characterization of Kr(h) for countable T, and a better picture in general). We also find when there are > h pairwise contradictory types of cardinality < K over a set A, IAI = A.

Those proofs indicate an alternative proof to the stability spectrum theorem for h 2 2ITI without using forking or ranks (see Exercise 7 .8 ) .

In Sections 1-5, T will be stable and A finite in all unstarred theorems and l e m m . No unstated assumptions are made on T or A in starred theorems and lemmm.

III.1. Forking

Notation. For an index set U let 2, = {xu: u E U} and for a sequence Ti = ( u ~ ) ~ + , let Pa = A type in 2, over B is a consistent set of formulas using only variables from P, and parameters from B. The

type is complete if it is maximal among the types in Z, over B. For an m-type p and a set I of sequences of length m let p ( I ) = { Z E I : a realizes p}. Similarly we define cp(I; 6) for a formula ~(2; 6). Insted of %A we sometimes write just A.

DEFINITION 1.1: tp,(B, A) = {cp(Z6; a): 6 E B,a E A, Ccp[6; a]}, tp*(6, A) = {cp(Ze; 8): si E A , E E 6, C cp[E; a]}.

DEFINITION 1.2: p split8 strongly over A if there is an indiscernible sequence I = {Z: n < w } over A and a formula cp such that cp(Z; ao), -,cp(Z; al) ~ p . (Compare Definition I, 2.6.)

Remark. Clearly ifp splits strongly over A then p splits over A.

DEFINITION 1.3: The formula cp(Z; ti) &idea over A if there we n < w and sequenoes a’, 1 < w, such that

(1) @(Ti, A = tp(8, A), (2) {cp(Z; #); 1 < w} is n-inconsistent (see Definition 11, 3.2(3)(E)).

DEFINITION 1.4: The type p in I , forb over A if there are formdaa cpo(Zo; So), . . ., cpn-l ( - 3” l; Z-l), with Z c 3, for k < n, such that

( l ) p V k < n vk(*; ak) ,

(2) cpk(P; ak) divides over A for dl k < n.

Instead of writing “{cpO(z; a)} forks” we write “cp(2; a) forks”.

Note. p is not in general a type over A; see Corollary 1.3.

LEMMA 1.1*: (1) If Z E g, k$(g; E) --+ cp@; 13) and cp(1; E) divides over A, then #(g; E) d i v k h Over A. (In particular this holds when $ is obtained by adding dummy variublea to cp.)

( 2 ) If p i s a type in Za, @finite) then p forks over A iff there are formula (po(Za; go), . . . , cp’-’(Za; 3 - l ) smh thUtp k V k < n cpk(Za; ak),andcpk(Za; 7ik) divides over A for k c n.

(3) cp(Z; 8) divkh Over A iff there are n c w and a sequence I = {a‘: 1 c w} indiscernible Over A such that ti = iio and {cp(Z; 8): 1 < w} is n - i e t e n t .

(4) If cp(E; a) d i v k h over A then cp(I; a) forb over A. (6) p f o r b over A iff some finite &type q of p forb Over A (so q is a

type in Ia for some finite a).

tp(8, B)) f o r b Over A. (6) tp*(C, B) fwks ovw A iff for 8Ome E E C, tp*@, B) (OT qUiVdently

86 QLOBAL THEORY [CH. 111, 8 1

(7) I f p , c p , f o r i < j < Sa~nop,forksoverA,t laenU,, ,p ,doea not fork over A.

(8) If A c B, q and p are types in the same variables, q t- p [ p E q] and q doecr not fork [aplit ~trongly] over A then p doea not fork [aplit strongly] over B.

(9) I,fpu{$'(Z; a')}forksOverAforl < n < w,thenp~{V, , , ,#(Z;t i~)} forks over A.

Proof. ( 1) is immediate and (2) follows from the parenthetical remark in (1).

(3) If the stated condition holds then clearly cp(Z; a) divides over A. Now suppose cp(Z; a) divides over A. Then there are n < w and sequences a', 1 < w such that tp(si, A ) = tp(2, A), and {cp@; 8): 1 < w} is n- inconsistent. By Ramsey's theorem and compactness there are E l ,

1 < w such that (6': 1 < w} is an indiscernible sequence over A, tp(6', A) = tp(a, A) and {cp(Z; 6l): 1 < w } is n-inconsistent. So there is an automorphism P of Q such that F 1 A is the identity and F(6O) = a. So (P(6'): 1 < w ) is the required sequence.

(4) Immediate. (A converse is Exercise 4.15.) (5) Since i f p t- V,,,, qi then for some finite q c p , q t- V,<,, 'pi.

The rest are immediate.

LEMMA 1.2: Lei p , q be m-types, p is Over A . (1) If q(Z; g ) E A , l(3) = m, cp(3; a) divides Over A, then Rm(p, A, No)

(2) If q 1 p , q forks over A , then there is afinate A , m h that for every

(3)* If cp@; g) E A , q(3; a) divides over A, A 2 2, then

> m p u (94% a,}, A , N o ] .

Pnite A 1 A,, Rm(q, A , X,) < Rm(p, A , 24,).

R"(P, 4, A) < 00 - Rmrp u {q(% a)}, A, A1 < m p , 4, A). Dm(p, A , A) < co + Dm[p u {q(Z; a)}, A, A] < Dm(p; A , A).

(4)* If q 2 p , q f o r b over A, A 1 X,, then there i8 a jhite A , such that for every A 2 A,,

Rrn(;p, 4, A) < + Rrn(q, 4, 4 < R Y p , A, 4 D"(p,A, A) < a0 =+ P ( q , A , A) < P ( p , A , A).

(5)* If in (4) we mame only that q split8 over A, then the conclusion is valid for A = 2.

Proof. (1) By Lemma 1.1(3) there is I = {a': 1 < w} which is an indiscernible sequence over A, a = sio, and {cp(Z;8): 1 < w} is n- inconsistent. Let p i = p u (~(3; 8)} for 1 < w, a = Rm(p, A , X,).

OH. m, 8 11 FORKING 87

Clearly a 2 P ( p l , A , No) = P ( p o , A , No), by Theorem 11, 1.1(1) and Exercise 11, 1.1 and a < m as T is stable.

Suppose a = P ( p l , A, No) by way of contradiction. Let B = A u U I . By Theorem 11,1.6 there are ql E 8T(B) such that Rm(pl u ql, A , No) = a. Notice that p u ql = p l u ql. Now 1{2 < w : p(Z; 8) E qk}l < n 80

l{ql: ql = qk}1 < n. Thus there are infinitely many distinct qk, which are clearly pairwiee explicitly contradictory. Since P ( p u qky A , KO) = a, Rm(p, A , KO) > a, contradiation.

(2) Since q forks over A, let q k V k < n q k ( Z ; ak) as in Lemma 1.1(2) where ~ ~ ( 3 ; Z k ) divides over A. Take A , = {vk(Z; g k ) : k < n} and let d 2 do be finite. clearly q k p U {Vk<,, vk(f; ak)} k p . Thus

R"(q, A, No) 5 P[ p u { v pk@; ak) , 4 No k < n } I

= max P [ p u {$(it; a")}, A , No], k e n

this by Claim 11,1.7. By part (1) just proved, P [ p u {cpk(Z; Gk)}, A , No] < P ( p , A, No). Since n < w we get

Rm(q, A , No) 5 max Rm[p u {vk(E; 7ik)} , A , No] < Rm(p, d, No). ken

(3), (4), (6) are proved similarly.

COROLLARY 1.3: If p $8 a type in it, Over A thenp does not fork Over A .

Proof. Suppose p does fork over A. By 1.1 (ti), ( 1) we can aasume p is an m-type. Taking q = p in part (2) of the above lemma P ( p , A, KO) < R"(p, A , No): contradiction.

Remark. Cleasly if p is a type over A then p does not split strongly over A. This also follows from Theorem 1.6.

THEOREM 1.4*: If p is a type in f , over B which does not fork over A , then there ie a complete type q z p in it, Over B which does not fork over A .

Proof. Let r = {#(ita; 8): +i E U, Ti E B, #(Za; 6 ) forks over A} and define q' = p u {-,#(Za; a): #(Za; a) E r}. We now show that q' is consistent. If not, then it has a finite inconsistent subset

p' u (-,yP(Zaa,; ak): k < n < w},

where p' E p, yP E r. Thus p k Vk<n #". Now forks over A so V k < n #k forks over A by Lemma 1.1(9) so p forks over A by Lemma

88 QLOBALTHEORY [OH. 111, $1

1.1(8), contradiction. Thus q' is consistent, q' is a type in itu over B. Let q 2 q' be any complete type in ?Zu over B.

We now show that q does not fork over A. If it does, then q k V k < n ~ k ( Z a , ; ak), .Ti, E U, where 'pk divides over A . Let q* E q be a

finite subtype such that q* V k < n qk and let O(Za; 6) = A q* (a E u, 6 E B). q is complete so that O(Za; 6) E q. Clearly O(Za; 6) forks over A so (I(%; 6,J E r. So E q' E q, contradiction. Thus q does not fork over A .

COROLLARY 1.6*: Let A , E Bl, A , E B,, C,, Ca be any sets, F an elementary mapping from B, onto B,; F(A,) = A,, tp*(C,, B,) does not fork over A,. Then we can extend F to an elementary mapping F', Dom F' = B, v C, m h that tp*(F'(C,), B, V C,) do@ not fork over A,.

Proof. Directly from the theorem.

THEOREM 1.6: (1) If p strongly splits over A , then p f o r b over A . (2) p f o r b over A iff there is a set B such that p is a type over B and for

every q 2 p (in the same variables) which is mnplete over B, q strongly splits over A .

Remark. B may be chosen such that B - Domp is finite.

Proof. Without loss of generality we may assume that p is an m-type. (1) By definition there are a set I = (3"' n < w } indiscernible over A

and a formula Q such that Q(Z; a0), 7 ~ ( Z ; a') ~ p . Now by Lemma 11, 2.20 there is n(y) < w such that for all 8, I{n < w: C Q [ E ; a"]>l < n(q) or I{n < w : Clp[8;7in]}I < n(q). Define 6" = aannSian+l and $(Z; 6") =

Q(Z; aan) A l ~ ( Z ; Zan+l). Clearly (6": n < w } is indiscernible over A . If w E w , IwI = n(tp), then {#(Z; 6"): n E w} is inconsistent; otherwise, letting E realize this set, I{n < w , C c p [ E ; Z]}l 2 1{2n: n E w}l = n(rp) and I{n < w : Cl~[8; a"]}l 2 1{2n + 1: n E w}1 = n(tp), Contradiction. Thus by Lemma 1.1(3) #(Z; 6O) divides over A. In addition p I- $(Z; 6O) since tp(Z; ao), +%; a,) ~ p . Hence p forks over A.

(2) * By definition, p k V k < n rpk(Z; ak) where tpk(8; Z k ) divides over A. So by Lemma 1.1(3) there are sequences (or sets; it is the same by stability) {a:: i < w } indiscernible over A, = iZk, and there are n(k) such that { q k ( Z ; a:): i < w } is n(k)-inconsistent. Let B = Dom p v {@:k < n , i < n(k)}andletqE8m(B)extendp.Sincep I-Vk<n~k(Z;tZk) we have a k < n such that q k ( Z ; 7ik) E q (remember = ak). Now I{i < w : q k ( Z ; Z : ) E ~ } I < n(k), so there is i o ( sn (k ) ) such that - - q k ( Z ; iZPJ E q. Thus q strongly splits over A.

m, § 11 FORKINQ 89

c Now assume p is a type over B and every q, such that p c q E

Sm( B), strongly splits over A. By Theorem 1.4 if p does not fork over A there is q E Bw( B) extending p which doea not fork over A , contradiction by (1).

DEFINITION 1.6: Let I be an infinite indiscernible set and A any set. Av,(I, A) = {cp(Z; a)': cp(Z; 9) E A , t E (0, l}, E A, and for all but finitely many E E I, h p [ E ; a]'}. If A = L we omit it.

DEFINITION 1.6: The infinite indiscernible sets I , , I , equivalent if there is an infinite J such that I, u J and I , U J m e still indiscernible.

LEMMA 1.7: (1) For every infinite indiscernible set I, Av(I, A ) is a complete type over A. ( 2 ) If I is an infinite indbcemible set over A then B realizee Av(I, A u U I ) ifl I u {E} is indiscernible Over A.

Proof. (1) By Lemma 11, 2.20 for every E A and formula cp Icp(I; a)! < Noor(lcp(I;7i)1 < Ko;thuslcp(Z;a)~Av(I, A)orcp(Z;7i)~Av(I, A). In other words, Av(I, A) is complete. In order to see that Av(1, A) is consistent, notice that every finite subset of Av(I, A) is realized by dl but finitely many members of I.

(2) Left rn an exercise.

LEMMA 1.8: The following d i t i m on the infinite indidcernible seta I , , I , are equivalent.

(1) I , , I , are equivalent. ( 2 ) FOT any set A Av(I,, A ) = Av(I,, A). (3) There is a &el M y 1M1 z I , , I , such that

Av(Ii, = Av(Ia, 1611).

Proof. (1) == ( 2 ) Let J be an infinite set such that I, u J , I, u J are indiscernible. By definition Av(I,, A) z Av(I, u J, A), but since both are complete types over A, they are equal. Likewise, Av(I,, A) = Av(I, u Jy A) = Av(J, A) = AV(I1 U J y A). SO Av(I1, A) = Av(I,, A).

(2) * (3) Trivial. (3) * (2) If (2) does not hold then there are A and a formula p(Z; a)

such that cp(Z; 7i) E Av(I,, A) and lcp(Z; a) E Av(I,, A). By Lemma 11, 2.20 there is n(cp) < o such that for any indiscernible set I* and sequence a*, Icp(I*; &*)I < n(cp) or I-&*; a*)l < n(cp). Choose 6', . . ., 6n(@) E I, such that b[6*; a] (1 s i s n(cp)) and E l , . . . , F(@) E I , such that

90 GLOBAL THEORY [CH. m, 0 1

t-p[i?; a] (1 5 i 5 n(T)). Thus t(3Z) AYL$ (~(6'; 5) A +8; Z)). Since M is a model, 6', i? E lMl there is a* E lMl such that

n W

1 = 1 'A (&'; a*] A -q@; a*]).

Thus (cp(Il; a*)] , llT(I2; a*)] 2 n(T). Hence ~ ( 2 ; a*) E Av(I,, 1M1) and +Z; a*) E Av(I2, IMI), contradiction.

(2) * (1) D e h e by induction on n < o sequences c7L which realize Av(I1, C,,) = Av(I2, C,) where C, = U I, u U I 2 U U {i?: i < n}. The sets I, U {b: n < o} and I, U {b: n < w } are indiscernible by Lemma 1.7(2) and so I, and I, are equivalent.

DEFINITION 1.7: p is stationary over A if (1) p does not fork over A and (2) p has no two contradictory extensions which do not fork over A.

Remark. By Theorem 1.4 an equivalent condition is that for all B such that p is a type over B, p has a unique extension to a complete type (in the same variables) over B which does not fork over A.

LEMMA 1.9*: If A E B, p does not fork over A , p r B is stationary over A , then p does not split over B.

Proof. By way of contradiction assume tp(6, B) = tp(Z, B) and ~(3; 6), ltp(5; a) ~ p . Let P be an elementary mapping such that P B is the identity and P(E) = 6. ( p r B) u {tp(Z; 6)) does not fork over A, since it is a subtype of p. Likewise (p t B) u {-,cp(t; a)} does not fork over A. Thus P [ ( p B) u {+if; a)}] = p 1 B u {-,cp(f; 6)) does not fork over A (since P 1 A is also the identity). But then p t B has two contradictory extensions which do not fork over A , contradiction.

DEFINITION 1.8: The infinite indiscernible set I is based on A if for every set B, Av(I, B) does not fork over A. (The interesting cam is when B 2 A, and this gives an equivalent definition.)

Remark. I is based on U I.

LEMMA 1.10: (1) If p is an m-type over B z A which is stationary over A , B, = B U Uf < , 8, p E p , = tp(Zj, Bj), and p j does not fork over A , then I = {ii': i < i,, 2 o} is an indiscernible set over B based on A.

(2) I n the notation of ( l ) , for all C z B, Av(I, C) is the unique extension of p in P ( C ) which does not fork Over A .

Proqf. (1) By Lemma 1.9 p , does not split over B and for all i < j < i,, p i c p j because p is stationary over A. So by Lemma I, 2.6, stability, and Theorem 11, 2.13(7) I is an indiscernible set over B.

Now we show that I is b a d on A. Let B be any set, and define C = B u B’ u Uief0 3. Inductively define ?if, for io s j < io + w ,

such that p , = tp(?ij, C u Ute 8) extends p and does not fork over A. This is achieved by 1.4.

Clearly p , t B, also eatisfies the above conditions. Thus, aa above, {sij: j < i, + w } is an indiscernible set over B. Now Av(I, B’) c Av(I, C) = Av({Si’: j < i,}, C) = Av({?i’: j < i, + w}, C )

= Av({iZ’: i, I j c i, + w}, C ) = pfo,

since pi0 = tp(8, C) for i , I j < i, + W . But pf0 does not fork over A so neither does Av(1, B‘).

(2) Follows from (1).

LEMMA 1.11: (1) If p E S ~ ( ~ M ~ ) does not fork over A c \MI, and M M (( IA I + 2)ITl) + -saturated, then p is stutionary over A .

(2) If A c B and r, = { p E Sm(B): p doe0 not fork over A}, then IFl[ I (IAI + 2)lT1 for any B.

(3) If A c B and I’a = { p E Sm(B): p doe0 not ~trongly elit over A}, t h n Ira[ I (IAI + 2)ITI.

(4) There are at nmt (IAI + 2)ITI non-equivalent infinite indiscernible set8 based O n A .

Proof. (1) Assume p is not stationary over A. Then there is B 3 and p c ql, qa ES”(B), ql # qa, and q,, qa do not fork over A. Let 6 E B be such that ~(3; 6) € q l , l@; 6) € q a . Now we define an increasing sequence A, c I”, lA,l I (IAI + 2)1TI, sequences 6, E IM( and typesp, E S ~ ( ~ M ~ ) , for i < ((IAI + 2)lTl)+. Let A, = A, and if 6 is a limit ordinal A, = U { A j : j < S}. Assume A, is defined; we shall define 6,, p , and A,+l . Let 6, E be such that tp(6,, A,) = tp(6, A,); there is such a 6, by _saturativity. By the assumption on q,, qa we have that p t A, u (~(2; b,)}, p A, u {ltp(P; 6,)} do not fork over A. Without loss of generality ~ ( 2 ; 6,) ~ p . Let p , ‘be such that p 1 A, u {-vp(Z; 6,)) E p , E Sm( /MI) , and p , does not fork over A (by Theorem 1.4). Take A , + , = A, u 6,.

So we have defined A,, p , and 6,. Now from {6,: i < ((]A1 + 2)IT9+} choose an (infinite) indiscernible sequence over A. If i, < il and gf0, 6,, are in the indiscernible sequence over A then p(3; gf0), - , ~ ( f ; 6,1) ~ p , ,

92 QLOBAL THEORY [OH. UI, 8 2

80 pf, splits strongly over A and thus forks over A (by Theorem 1.6); contradiction.

(2) Follows from (3) since strong splitting implies forking. (3) Define an equivalence relation - between hite sequences from

B as follows: 6 - E if there are a,, . . ., a,,, a, = 6, a,, = E, and infinite indiscernible sets I,, . . . , I,,-1 over A such that a,, ar+1 E I , for all 0 I 1 < n. Let B G B be such that every h i t e sequence in B has an --equivalent sequence in B. The mapping p HI, 1 B' is a one-one mapping from Fa into Sm(B'). (Assume p1 # pa E Fa, p1 B' = pa 1 B'. Let v(Z; 6) €pl, +Z; 6) and 6 - E E B'. If p(Z; E) E p1 then q(Z ; E) E 212 andpa splits strongly over A, contradiction; if -vp(Z; 2) E pl, then p1 splits strongly over A, contradiction.)

By Theorem 11, 2.13 T is stable in (IAI + 2)lTl so (3) will follow if we show that - has 5 ( [ A [ + 2)ITI equivalence classes (since then B' may be chosen such that ISm(B')I s (IAI + 2 ) 9 . If not then there is I = {&: i < ((IAI + 2)lTl)+}, 6f 3, 6, for all i # j, and we may assume I ( & ) = 2(Ej), for all i ,j. By Theorem I, 2.8 I haa a subsequence of the same power which is indiscernible over A. This is a contradiction to the inequivalence of the 6:. This proves (3).

(4) Assume there are { I j : j < ( ( [ A ] + 2)ITI)+} pairwise non- equivalent indiscernible sets baaed on A. Choose M such that IMI 2 A v u {u I,: j < ((IAI + 2)lTl)+} and M is ((IAI + 2)lT1)+- saturated. Let p j = Av(I,, 1M1). By Lemma 1.8 the inequivalence of the I, implies that pf # p j for all i # j. But by definition the fact that I, is based on A implies that p j does not fork over A. So there are > (IAI + 2)lTl complete types over which do not fork over A, in contradiction to (2).

This completes the proof of Lemma 1.11.

CONCLUSION 1.12: For every sequence 6 and set A there is an indiscernible set I over A and based on A , such that 6 E I . If M is a ( [ A [ + KO)+- saturated mode2 such that A v 6 c IMI then we may take I c 1M1. If tp(6, A) doe% not fork over B G A then we can c h e I 8 0 that I is based on B.

Proof. Let Ml be a (( [ A I + 2 ) 9 + -saturated model such that lMl I z A v 6. tp( 6, A) does not fork over A by Corollary 1.3. Thus by Theorem 1.4 there is a complete type r z tp(6, A) over lMll which does not fork over A. By Lemma 1.11( 1) r is stationary over A. Now by repeated use of Theorem 1.4 and by Lemma 1.10(1) we may build an indiscernible

CH. In, 5 11 FORKING 93

set (over lM1l) of sequences realizing r based on A. Call this set {E,: i < w}. Now tp(6, A) = tp(Eo, A) so there is an automorphism P of Q which is the identity on A and such that P(Eo) = 6. Without loss of generality P(E,) E [MI for all i < w , so we may take I = {P(E,): i < w}.

If tp(6, A) does not fork over B G A then we can choose r so that r does not fork over B and then I is baaed on B.

Remark. See 6.3(6).

EXERCISE 1.1: Show that in Lemma 1.7(1) it is possible to replace the assumption that T is stable by the (weaker) rtssumption that T does not have the independence property.

Exercise I.IA.Another&ampleis: T = Th(N),N= (IN, E , C , P ~ ) ~ , ~ where E is an equivalence relation over N with infinitely many equivalence classes, each infinite, PI 5 c/E, and every Boolean combination of the P, has infinite intersection with c/E. Let A = 0, p = 0, q = {xEc} and A = {P,(x): 1 < n} u {E(x; y)}. Clearly p u q fork over A, but R(p, A, 2) = R(p u q, A , 2) = n when n > 0. For 2 I h < KO, we use N = ("1, E, e, Pt,a)l<n,n<h where N is as above but (Pt ,m: m < A ) is a partition of c/E, and {P,,M,Jx): 1 < n} u {E(x, c)} is not algebraic for every 7 E %.

EXERCISE 1.2: For an indiscernible sequence I = {a,: i < S}, 8 a limit ordinal, define the average of I over A as (~(3; 6): 6 E A, for all sufficiently large i, l=v[a,; 6]}. Check the theorems corresponding to the ones proved here when T does not have the independence property but is not n e c e s d y stable.

EXERCI8E 1.3: Show that Corollary 1.3 is not in general true for unstable T.

Example 1. ill contains two one-place predicates P, R, and one 3-place predicate E such that every element a of P is defined by E(z, y, a) which is an equivalence relation over R, and all possible equivalence relations over R appear precisely once in the list {E(x, y; a): a E P}. Let {x: E(x; b,, a,)} have infinitely many E(z, y; a,-,)-equivalence classes for I = 0, 1 and R(s) k Vl<a E(z, b,; a,).

Example 2 (without the independence property). Let 0, be a connected graph without circuits in which every vertex has 00 neighbours. E(x, y, z ) = "the paths (x, z), (y, z ) have a point in common apart from z.

94 QLOBALTHEORY [OH. 111, 8 2

Remark. Even if in Definition 1.3 we demand, "{ti,,: n < w } is an indiscernible set", the example falsifies 1.3.

m.2. The finife equivalence relation theorem

Notation. FEm(A) will denote the set of formulas 'p = 'p(Z; 3; a) such that I (%) = l(8) = m and ti E A, and 'p is an equivalence relation with a finite number of equivalence classes, FE(A) = Um<a,FEm(A). Elements of FE(A) will also be denoted by E(Z;g) ; n(E) will be the number of equivalence classes of E.

DEFINITION 2.1: (1) The formula 'p(Z; 6) is almost over A if there is E(Z; 8) E FE(A) auch that (V?@j)[E(S; 9) --f ('p(Z; 6) = 'p(8; 6))]. In this case 'p is said to depend on E. Notice that if 6 E A then 'p@; 6) is almost over A.

(2) The type p is almost over A if every formula in p is almost over A. (3) stp(a, A) = the strong type of 7i over A = {E(Z; a): E(Z; 8) E

FEm(A)}. Similarly define stp*(C, B) for a set C.

Remark. realizes stp(si, A ) and stp(i5, A) I- tp(a, A). Notice also that stp(a, A) is almost over A, and if atp(a, A) = stp(6, A) and 'p(Z; a) is almost over A then k'p[a; a] = 'p[6; a]. If realizesp, p is almost over A, then stp(a, A) t- p.

LEMMA 2.1*: (1) FEm(A) is closed under conjunction, i.e., if 'pi@; 5; a,) E

FEm(A), i = 1, . . . , n then 'p(Z; 8; a) =def ATzl 'p,(Z; 8; a,) E FEm(A). Hence stp(8, A) is closed under conjwnctiow.

(2) If E(Z; 8) E FEm(A) and E ' ( P 3 ' ; 8-8') =def E(Z; g), then E' E FE"(A) (where n = m + l(x'), l (Z ' ) = Z(8')).

(3) If S = (xo, . . ., x,,,-~), 8 = (yo, . . . , Y,,,-~), 0 is a permutation of 0, . . . , m - 1 and E(Z; 8) E FEm(A) then

Proof. Immediate.

LEMMA 2 . P : (1) The set of formulas which are almost over A is closed under all the connectives and quuntiifiers, under the addition of dummy variables, and under permutation of variablea.

(2) The number of formulas which are almost over A is, q to logical

OH. In, 8 21 THE FINITE EQUTVALENCE RELATION 95

equivalence, at most I A \ + I TI. Thus the number of non-equivalent types which are almost over A is 5 2IAI + I T I .

(3) If stp(Z, A ) = stp(6, A) then there is an a u t m p h i a m F of Q such that F(8) = 6, F t A is the identity, and F preserves the formulas which are almost over A. We can replace a, 6 by sets.

Proof. (1) The cases A and are easy to see ( A by the preceding lemma), as are dummy variables and permutations of variables. Now let tp(zo, Z; 6) be almost over A ; i.e., there is E(zo, 3; yo, ij) E FE(A) such that E(zo, Z; yo, jj) k (cp(zo, f ; 6 ) = tp(yo, 8; 6)) . Let tp*(Z; 6) = (3z0)tp(z0, Z; 6). Define

E*(% Y) = ( V ~ O ) ( ~ Y O ) W ~ O , 3; Yo, 8) A (VYo)(~zo)E(zo, 3; Yo, g).

It is not hard to see that E* E FE(A) and that tp* depends on 2*; in fact n(E*) I 2n(B). Note that the above proof works even when Z = g is the empty sequence.

(2) and (3) immediate.

LEMMA 2.3*: (1) The formula tp(Z; a) is almost over A iff r = {tp(Z; F(a)): F is an aubnwrphiam of Q which is the identity on A} contains only a jnite number of nonequivalent formuha iff r contains < IlQll non- equivalent formulas.

(2) tp@; a) is equivalent to a formula over A iff dl the formulas of r above are equivalent.

Proof. (1) The third statement implies the second by compactness and the inverse implication is immediate. Also the direction * in the first "iff" is clear by definition. Now for the other direction, if r has only a finite number of non-equivalent formulas, then there is n such that the set Fl = {O(g,; a): O(&; a) E tp(a), A, i < n} u {l(VZ)[tp(Z;jj i) = ~(3; g,)]: i < j < n} has a contradiction. Let n be minimal. Then we can assume that there is a contradiction in

96 GLOBALTHEORY [CH. 111, g 2

NOW J%; %) = (Vg)[O(g; 2) --+ ( q ( Z l ; g) = v(Z2; @)I E FE(A), n(E) I 2"-l, and ~ ( 2 ; a) depends on 1.

( 2 ) Similar.

LEMMA 2.4*: If E(Z; 8) b an equivalence re2atio-n almoflt Over A , then there i8 an equivalence relation E' Over A which refim E; Le., (Vq)(E'(Z; a) -+ E(Z; g)), 8wh that if E ?uu finitely many equivalence c l ~ s s e s , then 80 does E'.

Remark. This lemma says it is pointless to replace "E over A" by "E almost over A" in Definition 2.1(1). See Exercise 2.3.

Proof. By Lemma 2.3(1) there are E,, i < n, equivalence relations such that every automorphism P of 6, such that P A is the identity, takes E to one of the E,, and for every Ef there is such an P. Thus every automorphism of 6 which is the identity on A yields a permutation of the E,. Thus Ef(Z; g) is preserved (up to equivalence) under all automorphisms of 6 which are the identity on A . So by Lemma 2.3(2) E'(Z; g) = A,<,, Ef(Z; g) is over A , and certainly refines 1. I f E has k equivalence classes so does each E,, hence E' has at most kn < No equivdence classes.

LEMMA 2.6: Let I = {a,: 1 < w } be an indbcernible 8et which b based on A . Let v ( Z ; g ) be a formula and f o r all n < w define q,,(~;a") =

Over A . (The size of n depends on tp only.)

Proof. By Lemma I I , 2 . 2 0 there is n such that for any indiscernible set I, and for all of length 1(Z), Iv(a, I ) ! < n or 17(p(a, I)I < n. Thus for any 6, bn[6; a"] * there is w c 2n - 1 , 1.11 = n such that bq[6; si,] for a l l i ~ w * lq(6,I)I >n=+llq(6,1) l < n=+q(6;8)EAv(IY6)=+lq(6,I)I 2 KO * 17q(6,1)( < n *there is w G 2n - 1, IwI = n such that Cq[6; Zf] for all i E w * bvn[6; an]. So bn@; a") o ~ ( 6 ; g ) E Av(1, 5).

Now by Lemma 2.3( 1) if qn(Z; an) is not almost over A , then there are {P,: i < ((]A1 + 2)lT1)+}, Ff 1 A is the identity, F, (P) = 4, tp(q, A) = tp(P, A), and such that q n ( E ; q) are pairwise non-equivalent. Thus for all i < j < (([A1 + 2)IT1)+ there is gf,, such that hp,,[6f,,; q] =

E Av(P,(I), 6,,,). Thus Y, ( I ) , P,(I) are not equivalent (see Lemma 1.8). Since I is based on A and E; A is the identity we have that each F f ( I ) is also based on A .

Vwcan-l.lw1-n A l c w ~ ( 3 ; a,)* T h for all large ~ h , q n i8 a l m t

l p n [ 6 i , j ; a71- Thus, by the above, v(&,j; 8) E Av(P,(I)y & , j ) 0 T T ( & , j ; 5)

OH. 111, $21 THE FINITE EQUIVALENCE RELATION 97

Thus we have ((I A1 + 2)lTl) + non-equivalent indiscernible sets based on A , in contradiction to Lemma l . l l (4 ) .

LEMMA 2.6: (1)* If p €Sm(B) does not fork over A c B and q z p is a type (in the same wa?$abka) which i8 dm8t over By then q does not fork over A . Also Rm(q, A , No) = Rm(p, A , KO) for all jinite A .

(2)* If p E Sm(B) does not fork over A G B, E ( f ; g) E FEm(B), and q = p u {E(Z;a)} is umaistent, then q does not fork over A . Also R”(q, A , No) = R’”(p, A , No) for all (cjinite) A .

(3) For all A , B, C (with A c B) there is an elementary mapping 4, Dom F = A u C, F 1 A is the identity, such that for all 8 E C, tp(F(8), B) does not fork over A , and stp(8, A ) = stp(F(8), A ) .

Proof. (1) Assume that q does fork over A ; then by Lemma 1.1(6) q has a finite subtype q’ which forks over A. Every formula in q’ is almost over By so by Lemma 2.2( 1) ~ ( 3 ; 6) = A q‘ is almost over B. Thus Q

depends on some E ( f ; ti) €FEm(B). Let realize q. Clearly k(E)[E(Z; a) + Q(Z; 6)], so if tp(f; 6) forks over A so does p u {E(Z; a)}. Thus the firat part of (1) follows from the first part of (2).

The second clause (about Rm(q, A, No)) similarly follows from the proof of Rm[p u {E(Z; a)}, A , No] = Rm(p, A, No).

(2) Notice 8 does not necessarily realize p. Choose a maximal set al, . . , , an such that tp(iZk, B) = tp(8, B) but

k7E(Z;ak), for all 1 s 1 # k I n. (n < w since n(E) < No.) Thus tp@, B) u { - ,E(f ; a”): 1 I k I n} is inconsistent. tp(7i, B) is closed under conjunctions so there is tp(f; 8 ) E tp(7i, B) such that tp(f; 8 ) t

Assume p u { E ( f ; a)} forks over A . Then for all k , p~ {E(Z; ak)} forks over A . So by Lemma 1.1(9) p u {VL,l E ( f ; a“)} forks over A .

p u {E(Z; ti)} is consistent so ( 3 g ) [ ~ ( J ; 8 ) A E(E; g)] ~ p . Thus if 6 realizes p there is 7i* satisfying tp(p; 8) A E(6; I), hence V1=l E(7i*; 8”). Thus also VLs1 E(6; a”). In other wordsp t VzSl E ( f ; Ek), or p t p u {VpPIE(Z;t ik)} . Thus p also forks over A , in contradiction to the hypothesis. The second clause is proved using Claim 11, 1.7 and Exercise 11, 1.1.

VFX1 E(Z; a k ) .

(3) Clear by Corollary 1.3, Theorem 1.4 and (1).

COROLLARY 2.7: If p is alm8t over A then p does not fork over A .

Proof. Let q 2 p be a complete m-type over A u Dom p . q 1 A E @(A) and so does not fork over A by Corollary 1.3, and clearly q 1 A u p is a

(consistent) type almost over A. Thus q 1 A u p does not fork over A by Lemma 2.6( l ) , and so neither does p .

THEOREM 2.8 (The Finite Equivalence Relation Theorem) : Let p, q E S ~ ( ~ M I ) , p # q, A E 1611, p , q do not fork over A , and M is (([A1 + 2)1TI)+-suturated. Then there is E ( f ; j i ) E FEm(A) such that P ( f ) u q(j i ) k 1E(f ; 9).

Prmf. Since p # q and both are complete over 1611 there is c p ( f ; 6), 6 E 1611, such that cp(Z; 6) ~ p , l c p ( f ; 6) E q.

By Conclusion 1 .12 there is a set I = {6*: i < W } c 1611 indiscernible over A and based on A such that 6, = 6. p, q do not fork over A, thus they do not strongly split over A, so c p ( f ; 6,) ~p and lcp(f; 6,) E q for all i.

By Lemma 2.6 (and using its notation) there is n < w such that cpn(P; 6") is almost over A. Let E ( f ; j i ) E FE(A) be such that cp,, depends on 4: (Eji)[E(E; j i ) +. ( c p , , ( f ; 6") = cp,(ji; En))]. Notice c p , , ( f ; 6") E p , lcpn(f; 6") E q. Thus p ( f ) u q(#) t- ( c p , , ( f ; 6") h T c p , , ( j i ; 6")) I- l E ( Z ; j i ) .

COROLLARY 2.9: ( 1 ) For dl and A , stp(a, A ) b s W k r y over A . ( 2 ) If p , q eSm(B) do not fork over A c B, p # q, then there is

E ( f ; j i ) E FEm(A) swh that p(Z) U q(j i ) t- i E ( f ; j i ) .

Remark. This improves Lemma l . l l ( l ) , 2.8.

Proof. ( 1 ) Let p = stp(si, A). p is almost over A, so by Corollary 2.7 p does not fork over A. Now if p were not stationary over A, then p would have two contradictory extensions pl, pa which do not fork over A. Let M be a (([A1 + 2)ITI)+-satwated model such that p1,pa are over [MI. So A c [MI. Let p', p a be extensions of pl, pa, respectively, in Sm( 1611) which do not fork over A. Let E ( f ; j i ) E FEm(A) be such that

pa(y) k l E ( f ; j i ) A E ( f ; a) A E(@; a), a contradiction. (2) Assume that there is no such E. Let r = p ( f ) u q(j i ) u

{ E ( f ; 8): E E FEm(A)}, r is consistent (since FEm(A) is closed under conjunctions). Let f I+ a, j i H 6 realize I'. Thus stp(8, A) = stp(6, A), a realizes p and 6 realizes q. So stp(a, A) is not stationary over A, in contradiction to ( 1 ) .

pl(Z) U pa@) k i E ( f ; j i ) . But p c pl, pa, E ( f ; a) E P , SO that ~ ~ ( 2 ) U

100 QLOBAL THEORY [CH. 111, $ 2

(2) If J M an in$nite ~ndiscernible set based on A , t h there an equivalent indiecern~le set I c 1M1 (I ie also indiscernible Over A) . 8ee De$nition 1.6.

Remark. Notice that when 12'1 is small there is a simple direct proof. The interesting case is when I 2'1 is very large in relation to A.

Prmf. Let p be an m-type almost over A . By Corollary 2.7, p does not fork over A , so there is a complete m-type q z p, over 12111 u Domp which does not fork over A ; so q I E Sm( 1M1). Clearly q 1 1 MI also does not fork over A . Now define a sequence (8,: i < w + w} in M such that a, realizes q 1 ( A u U {a,: j < i}) for all i < w + W. This is possible by saturativity. So by Lemma 2.12 q 1 ( A u {a,: j < w} is stationary over A. By Lemma 1.10 I = {a,: w s i < w + w } is an indiscernible aef over A , whose average is q. And so by Lemma 2.10 we see that &,, realizes p. This proves (1). For part (2), choose distinct 7 i , ~ J , and 6,oIM1 realizing stp(i%,, A ) and 6,,~ lMl such that tp(?ion. . . ha,,, A ) = tp(6,^- - . 6,,, A ) and let I = {b,,: n < a}.

LEMMA 2.14*: If p = tp(iz, A ) ie 8&Li%Onary over B c A , t h p k stp(a, A), and thw, for every 'p which is d w t over A , p k 'p or p t- -,'p.

Proof. Let E E FEm(A) where m = I @ ) . We must show that p t- E(Z; a). If not, then there is 6, not E-equivalent to 3, such that q = p u {E(Z; 6)) is consistent. Then by Lemma 2.6(2) q and p u {&'(it; a)} are two contradictory extensions o f p which do not fork over B, contradiction to the fact that p is stationary over B.

Proof. (1) Let 'p(Z; 6) depend on EtzFE(lMl) and let n(E) = n < W.

Since M is a model there are representativea a,, . . . , E 1M1 of all the equivdence claases of E. We have C ( E ) Vr<,,B(Z; $) and t(VZ)(E(Z; a,) + 'p(Z; 6)) or C(E)(E(Z; a,) + -,'p(Z; 6)) for i < n. With- out loss of generality we can assume c(vz)(E(z; a,) +'p(~; 6)) for

a. m, § 31 THE INSTABILITY SPEOTRUM 101

i < m I n and k(Vz)(E(Z; a,) + +it, 6)) for m 5 i < n. So k'p(Z; 6 ) E V,<,,, E(3; a,) if m > 0 and k+Z; 6) if m = 0.

(2) Let 8 realize p. By Corollary 2.9(1) it is sufficient to prove that p t- stp(a, A), i.e., for all E E FEm(A), p k E(Z; a). So let E E FEm(A). By definition n = n(E) < No, and so k(33,- - .Z,,)(Vg) V!-l E(g; 3,). Now since the parameters of E me from A E 1x1, and M is a model, there are a,, . . . , a,, E ]MI such that C(VQ) VTI1 E(g; a,). Let CE@; a,). Thus p k E(3; a,) and so p t- E(3; a). The second clause follows h m the first clause by Lemma 1.1(8) and the fact that stp(8, lHl) k

tP@i, 1x1,.

EXERCISE 2.1: Is Lemma 2.16(2) still true when T is not stable?

EXERCISE 2.2: Show that i fA G B, p = tp(@, B) does not fork over A then p is stationary over A iff p I- stp(a, A ) iff p t- stp(a, B).

EXERCISE 2.3: The E' we get in 2.4 is the coarsest equivalence relation over A refining E .

EXERCISE 2.4: If I , , I , me (inhib) indiscernible sets b w d on B, and for some a, E I, , stp(a,, B) = stp(ai,, B) then I , , I , are equivalent.

EXERCISE 2.5: If stp(B, A) = stp(6, A), then there is an indiscernible set I based on A such that I u {a}, I u (6) me indiscernible over A.

EXERCISE 2.6: If p does not fork over A, p E Sm(B), stp(a, A) = stp(6, A), Ti, 6 E B then for every 'p, 'p(3; a) EP o 'p(Z; 6) ~ p .

EXERCISE 2.7: If T is stable in A, IAI 5 A, then the number of non- equivalent types almost over A of the form stp(a, A) is 5 A.

EXERCISE 2.8: Show that ifp is stationary over A but p u q forks over A then there is 'p E q such that p u {'p} forks over A.

III.8. The instability spectrum

THEOREM 3.1: The following conditions on K , T , m are equivalent:

that for all i < ~ , p 1 fork over A,. (1) There is an bm?&ng sequence A,, i 5 K , and p E S ~ ( A , J such

( 2 ) Like (1) but now p 1 A, + , ql i t s strongly over A,.

102 QLOBAL THEORY [OH. 111, f 3

(3)A There exists an increasing sequence A,, i I K , a type q E Sm(A,), sets I f = {a!: j < A} E A, + indiscernible over A,, and tp,(B; a&) E q such that j > 0 =r 7tpf(3; 7i;) E q, and there are m, < w m h that {tp,(z; a;): j < A} is m,-i&stent. (So q 1 A , + splits strongly over A, for all i < K . )

(4)A fib ( 3 ) ~ but Where A, = INiI, H1.1 being (IN,I1+-8&urated.

DEFINITION 3.1: K"(T) = the first infinite K for which (1) above fails; K ( T ) = sup,,, K"(T). The first regular K 2 K ( T ) is denoted by K,(T), or m(T) (so it is K ( T ) or K(T)+). For unstable T we stipulate K"(T) = K , ( T ) = 00.

PVOOf Of Theorem 3.1. Trivially (4)A + (3 )A and by definition (3)A 3 (2) (for any A). By Theorem 1.6(1), (2) * (1). So it remains to prove (1) * (4)**

Since p / A , + , forks over A, we have p t A , + l I- vk<, , ,&?8;@j)

where each tpk(B; aL), k < n,, divides over A,. So by Lemma 1.1(3) there m {aL,[: 1 < w} indiscernible over A,, a3, = a;,, and {tpk(B; i&): 1 E o} is mL-inconsistent. Without loss of generality we may wume A d =

u f < d A,, S a limit ordinal (otherwise define A; = A,, for i a succesmr, and A; = u f < d A,; then A; G A d and p r A d + l = p 1 forks Over A d , so it forks over A;, and thus the A;'s satisfy (1)).

and models Mi such that Dom P d = A d ,

Now let B, = U k , l g , [ . Define by induction elementary mappings P,

A,+1 u B,, M, is ( A + IA,l)+-saturated, is 11M,11+-sa,turated, i < j + P, A, = f, A,, i < j * Mi E M,, Range P, E lM,l and for any 7i

For i = 0 take Po as the identity on A,, and take any appropriate

we may define A,,, LJ B, + !Mil u Pf+l(A,+l u B,) with the desired property, and then extend ]Mil u EI,+l(A,+l u B,) to an 11 Mi (1 + -saturated model Mi + 1. This completes the definition. Now choose

= u ( < d M,, Dom P,+ =

u B,, tp(Pf+l(a), IM,I) does not fork over P,(A,).

Mo. For i = 8 let F d = u , < d A,, M6 = u i < d Mi. BY (h'0lhy 1.5

Q ~~"(IM,I)su~hth&tP,(1]) E q.ForeveVi,p 1 I- V k < q &(B;ai)

pc(p A t + l ) v k < n C 't+l(%))* so q v k < m dC@; Pf+l(8k))* q is 80 P,+,(P t vk<,,, CP;(Z; P,+,cm. E', t A,+ , = r A,+, 80

complete so for each i < K there is k(i) < n, such that tpfccr,(l; F, + l(%k(,,))

~ q . { ~ o , , I : 1 < o} is indiscernible over A, so { y i + l ( z k ( f ) , [ ) : 1 < W} is indiscernibleoverP,+l(A,) = EI,(A,), andifSiEB, uAi+l,tp(F,+l(7i), 1Mi1) does not fork over P,(A,). Hence by Corollary 2.11(1) {P,+l(a3,,i,,,): 1 < w} is indiscernible over ]Mi l . Define 6; = F, + l(?&,),t). Since M , + is IIM,(( +-saturated, and IIM,(( + > X we can define 6; E JM, for

OH. 111, Q 31 THE INSTABILITY SPECJTRWM 103

w I; j < A such that {6j: j < A} is indiscernible over IM,I. Now {tpfc,(f; 6j): j E w } is mi,,,-inconsistent. Hence {j < A: tpis(f; 6:) E q} is finite. We may in fact aasume (by omitting if necessary) that only y&,)(f; 6;) E q, and j > 0 implies 7&,)(f; 6;) E q. Let I f = {6$: j < A}. The conclusion holds with this If, &,), M,, q.

COROLLARY 3.2: For all p E S"(B) there is A c B, IAI < K"(T) such that p does not fork Over A .

Proof. By way of contradiction, aasume there is no such A . We shall define A,: i I; K"(T) such that A, c B and 1Ai1 < lil+ + H, for i < K"(T). Take A , = 8, Ad = U,<dA, , AKyT) = B. Assume A, is defined. By rtssumption p forks over A,. By Lemma 1.1(6) p has a finite subtype p' which forks over A,; let A, + = A, u Dom p'. So by Theorem 3.1(1) K"(T) < K"'(T), contradiction.

Proqf. If not, then by Theorem 3.1(1) there are an increasing sequence A,, i I; IT1 +, m < w , and p E B " ( A , ~ , + ) such that for all i < IT1 +

p t A,+l forks over A,. Also, by Theorem 11, 1.2, for every A there is a(A) < !TI+ such that P ( p , A , No) = P [ p r &,),A, No]. The number of possible finited is IT1 so a = supdrute a ( A ) < IT1 +. But p forks over A, and so by Lemma 1.2(2) there is A , such that P ( p , A,, No) < P ( p 1 A,, A,, H,), contradiction.

LEMMA 3.4: If J iS an indi8CemGible set Over A , 6 any sequence, l(6) = m, t k n there iS I c J with I I I < K"( T ) m h that J - I i8 indi8CemGible Over A u 6 u U I .

Proqf. By Corollary 3.2 there is B c A U U J , IBI < K"(T), such that tp(6, A U U J ) does not fork over B. Let I c J, 111 < K"(T), be such that B c A U U I . Clearly J - I is indiscernible over A u U I , and thus by Corollary 2.11(2), 1.6(1) J - I is indiscernible over A u 6 u u 1.

Remurk. If T does not have the independence property (but is not neceswrily stable) a similar conclusion holds, using qnp( T), see Defini- tion 7.3.

104 @LOB& THEOBY [OH. III, 5 3

COROLLARY 3.6: If J i8 an indbcernibk 8d Over A , B i8 any S d , then there i8 I E J such that J - I i8 indiscernible over A V B V U I , and

(1) IIl 5 K ( T ) + IBI. (2) If IBI < Cf(K(T)) then III < K(T) . (The interuting m e is where

I JI is hrge enough in r e W m to I BJ .)

Proof. By way of contradiction msume there are A , m such that IAI I; p < ISm(A)I. By Lemma I, 2.1 we can wsume m = 1. By Corollary 3.2 for every p € # ( A ) there is B, E A, lBpl < ~~(2') such that p does not fork over Bp. The number of possible Bp's is so there is a B for which 1{p E ~ ( A ) : B,, = B}I > p. By Corollary 3.3 K(T) s ITI' and 80 IBI 5 ]TI. Let B E ]MI, 11iK11 = ITI. Then p < I{~EB(A): Bp = B}I i I { ~ E B ( A ) : p does not fork over B}I s l{p E 8( 1M( U A ) : p does not fork over B}I = I{p E 8( lM1): p does not fork over B}I (this by Lemma 2.16(2)) s ISr(lNl)l.

But clearly IS( 1MI)l s 2ITl, contradicting p 2 2ITl. And if T is stable in po and 11M11 = IT1 s pop then IS(lMI)I 5 po contradiofing po s p.

THEOREM 3.7: If p < pXdn then T is nd 8tabk in p.

Proof. Since p < peMn there is K < K(T) such that p" > p. Let K be the minimal one. Thus peE = p. Now since K < K( T), for some m K < ."( 2') 80 by Theorem 3.1, (3), we have an inoreeeing sequence A,, i s K, a type p E Sm(An), sets I' = {q: j < p} indiscernible over A,, I' c A,, md cpi(Z; $) EZ) such that j > 0 =+ yv@; 4) EZ).

Now for r ) E " p define an elementary mapping F,, suoh that (1) Dom F,, = U {$: j < p, i < Z(r))},

(2) 7 = p i Fn extends F,,, (3) if Z(q) = i then for every j:

(4 &-<,>(a = & r < O > ( a l ) ,

(B) q-<,>($) = F V < O > ( a d ) , (C) for everg a # 0,j = &-(o>(G).

The definition prom& by induction on Z(r)). For Z(7) = 0, F,, is the

CH. 111,s 31 THE INSTABILITY SPECTRUM 105

empty mapping, and for Z(7) limit take P, = uy<It,) P,ly. If Z(q) = a + 1, P,ts(o) is any extension of P,ra with the right domain and

Now P, is elementary (and well-defined) since {%:j < p) is indi~cern-

Let B = U {P,(ai): q E "'p, i < l(q), j < p}. Then 1B1 5 p - ~ .

For 7 E "p, P,(p 1 Dom F,) is consistent since P, is elemenfary. Let P,(p 1 Dom P,) s q, ~8"m(B). We shall show that for 7 # p E "p,

Let a be the firat ordinal such that ~ [ a ] # p[a] ; let q 1 a = p 1 a = v.

Assume q (a + 1) = u-(jl), p (a + 1) = v-(j , ) , jl # j,. We may assume without loas of generality that j, # 0.

If also j, # 0, then .Fp@yl) = Pv-(t,)(iZ~l) = li'v-(o)(a~l); P,(a;) =

q a ( t ; F , ( a ~ ) ) E q,. T T a ( t ; BYl) €2) since j, # 0 SO i 'pa(Z; Pp(Zyl)) E q,,,

Similarly if j, = 0, Pp(a:) = F,(ay,) but tpa(t; Pp(iit)) E qp and

Thus IlIm(B)I 2 I{q,: 7 E "p} = pn > p and T is not stable in p.

P,rc-<n[ol) is defined by (3).

ible over A, z Dom Pnrr.

L C P ' P.

Qn # Qo.

Pv-(jl)(iig) = Py-(o)(ay1). Thus Pp(ayl) = P,(G$), ~p.(iE; at) E p SO

thus 1vcr(E; P,(m) E q p , and Q, # Qe.

l T a @ ; P,(a?,)) E ~ l r * Again q n f q p -

COROLLARY 3.8: (1) If po ie th first cardinal 2 IT1 in which T is etable, t h po 5 2ITl, T is stable in p iff p = p, + P<"(~ . In fact po = sup(l8(A)1: A of cardinality 5 IT[}.

( 2 ) T ia awperstable iff K ( T ) = No. ( 3 ) Eithr (i) K"(T) = K ( T ) for every m or (ii) for some m, m < m, iff

K"( T) = K,( T ) if K"( T ) # K ( T ) . In t7re latter m e K,( T ) is singular and K ( T ) = ( K ~ ( T ) ) + .

Remark. See Exercise 3.3.

Proof. (1) Follows from Lemma 3.6 and Theorem 3.7, and the remark to 3.6.

(2) If K ( T ) = No then p = p<'c(T) for all (infinite) p. So T is stable in p for p 2 2ITI, thus T is superstable. On the other hand if K ( T ) > No then for every cardinal N,, cf(N,+,) = w so N>+a, > Ha+,. Thus K,="(,) > N,,,, and T is not stable in Ha+, so T is not superstable.

(3) Clearly K"(T) 5 K"+'(T) (by extending the m-type in Theorem 3.1 to an (m + 1)-type). For every regular K there is a strong limit cardinal A > 2ITI of cofinality K , so A'" = A, A" > A. If K < K(T) then

106 GLOBAL THEORY [OH. III,f) 3

T is not stable in h by Theorem 3.7, and if K 2 ~~(2') then T is stable in h by Lemma 3.6. SO for regular K , K < K(T) iff K < K,(T), SO we finish.

DEFINITION 3.2: I is a maximal indiscernible set over A in M if I is indiscernible over A, I E ]MI, and there is no J 3 I, J E IMI, J indiscernible over A.

DEFINITION 3.3: dim(I, A, M) = min{l JI: J is equivalent to I , and J is a maximal indiscernible set over A in M}. If A = 0 we just write dim(I, M). If for all J as above dim(I, A, M) = I JI, we say that the dimension is true.

Remark. This definition differs technically from Definition 11, 4.5, but no confusion will arise.

LEMMA 3.9: I f I is a d m a l indhcernible set over A in M, then 111 + K(T) = dim(1, A , M ) + ~ ( 5 " ) and ifdim(I, A , M ) 2 ~ ( 5 " ) ~ then the dimension is true.

Proof. Let J be a maximal indiscernible set over A in M, equivalent to I, and for which IJI = dim(I, A, M). Clearly IIl 2 IJI, so that IIl + K(T) 2 IJI + K(T). Thus it is sufficient to prove that III + K(T) I I JI + K(T) or in fact IIl I IJI + K(T).

By way of contradiction assume IIl > I JI + ~(2'). From Corollary 3.6 we have I, E I, !Il I 5 I JI + ~(2") such that I - I, is indiscernible over A u U I , u U J. From the equivalence of I and J we can find an infinite I* such that (I - 11) u I* is indiscernible over A u U J and J u I* is indiscernible over A (take I* = (8,: n < w } where the type 8, realizes over A u u I U u J u U {al: 1 < n} is the average of I over the same set).

Now with the aid of I* we shall show that (I - I,) u J is indiscernible over A. Let a,,. . . , 8, be distinct sequences in (I - 11) u J , E E A , r p ( f , , . . . , 3,; E) a formula. Let 6,, . . . , 6, be any distinct sequences in I*. Define 8; aa follows:

for 8, E J, 6, for 8, E I - I , .

Now Crp[zi1, . . . , 8,; EJ o Crp[8;, . . . , 8:; E l since (I - I,) u I* is indiscernible over A U U J, and hp[8;, . . . , 8:; E l o C&, . . . , 6,; a] since J u I* is indiscernible over A. But then the truth of rp(8,, . . . , 8,; a) is

CH. 111, Q 31 THE INSTABILITY SPECTRUM 107

not dependent on the particular a,. Thus (I - I,) u J is indiscernible over A, in contradiction to J’s maximality in M.

LEMMA 3.10: (1) Let M be an ( K , + K( 2’))-saturated model. Then M is h-saturated iff for every infinite indiscernible I c 1M1, dim(I, M) 2 A.

(2) We can replace the msumption “ (N, + K ( T))-saturated ” by “every type which is almst Over some A c I MI, I A I < K( T ) , is realized in M ” (i.e., by “F~tn-satu*ated”. Bee Definition% I V Y 1.1 and IV, 2.1).

Proof. (1) If M is h-saturated then clearly there is no indiscernible set of power < h which is maximal in M. So one direction is clear. Now assume M is not h-saturated, h > K, + K ( T), and let p E #“(A),

A c IMI, IAI < A, be a type omitted in M. Extend p to p ~ P ( l M l ) and choose B c “1, IBI < K(T), so that q does not fork over B (by Corollary 3.2). By 1.12 and 2.13(2) since M is (N, + K(T))-saturated there is I c ldll which is an infinite indiscernible set such that p = Av(I, IMI). Let J z I be an indiscernible set maximal in M. If dim(I, M) < A, we are through. Otherwise I JI 2 &(I, M) 2 A. By Corollary 3.6 there is J’ c J , I J‘I s lAl + K ( T ) such that J - J’ is indiscernible over A (K(T) < A, IAI < h so I J’I < hand thus J - J ‘ # 0). But then since Av(J, A) = Av(I, A) = p 1 A = p we have that all the sequences in J - J’ realize p, contradiction.

(2) Similar proof.

THEOREM 3.11: If { d f ( } { < h iS an ~nCreaeing 8- Of h-8&U&ed

modele, and K ( T ) Cf 6, then = ui<6 Mi i8 h-8dUTded.

Proof. Clearly if h s cf(6) then M is h-saturated. So we may assume h > cf(S), and in partioular h > K(T). First assume cf S > No so we can use 3.10(1). Let I c IiKl be an infinite indiscernible set. Let B c lH), IBI < K(T), be such that Av(I, ]MI) does not fork over B. Since K( T ) s of S and cf 6 is regular, there isj < 6 such that B E 1M,1. By Corollary 2.13(2) there is J E lM,1 equivalent to I and since M, is h-saturated there is J’ z J , J ‘ c IM,I, indiscernible and IJ’I 2 A. Thus by Lemma 3.9 dim(I,M) = dim(J’,M) 2 h and by Lemma 3.10(1) M is h-saturated. If cf S = 8, we can use 3.10(2) instead of 3.10(1) (by 2.13(1)) and easily reprove a suitable version of 2.13(2) (by Exercise 2.4).

THEOREM 3.12: If T is A-stable then T h a h-saturated nzodel of 7 A.

108 GLOBAL THEORY [OH. 111, 8 4

Proof. We define an increasing sequence {&f,};s), such that IIM,II = A, M,, = U, < M, and every p E B( IM, I) is realized in M, + It is easy to see that M,, is cf(b)-saturated. Thus if A is regular then MA is A- saturated.

If A is &gulax then A = x;<,,f(A) A,, cf(A) < A, < A,’for all i < j c

cf(A) then by Theorem 3.11 we

If ~(2’) > cf(A) then A<dT) 2 Aof(”) > A. So by Theorem 3.7 T is not

cf(h). SO for all i < cf(A) MA = Uj<AMj+A:; cf(j + A,+) > A, SO

M,+,: is A,-eaturated. Now if K(T) have that MA is A,-saturated, for all i. Thus M A is A-&urated.

stable in A, contradiction.

EXERCIBE 3.1: In Lemma 3.4 prove that we have 111 < ~ ~ ( 2 ” ) .

EXERCISE 3.2: Give an example in which K(T) is singular.

EXERCISE 3.3: Prove ~ ~ ( 2 ’ ) = ~(2’). [Hint: Suppose A, (i < K ) is increasing, tp(an6, A,+l) forks over A, for i < K. By 4.16 tp(iZ, A, +1) forks over A, or tp( 6, A, + u a) forks over A, u a.3

EXERCIBE 3.4: Prove that in Lemma 3.4, when J is infbite, there is a minimal I = I , among the possible I’s, i.e., I , is contained in every possible I .

EXERCISE 3.5: Prove the paxallel of 3.11 for A-aomp&ctnesa.

III.4. Further properties of forking

THEOREM 4.1 : Let p E Bm( B), A E By then p doe0 not fork Over A iff for allJinite A , Rm(p, A , No) = Rm(p t A , A , KO).

The analogous theorem for stationary types is:

THEOREM 4.2: Let p E Bm(B), A E By then (1) p M etatimry Over A i+@ for all Jinite A , P ( p , A , KO) =

P ( p r A , A, KO) and Mlt(p, A, No) = 1. If p M etationary wer A and the rn-type q 2 p , then q doe% not fork Over A iff for all Jinite A and all A, 2 s A s KO, R”(p, A , A) = Rm(q, A , A).

(2) p it3 e t a t i m r y over A iff p is etation4zry Over B and &oes not fork over A .

( 3 ) In (1) we can restrict ourselves to all large enough A .

OH. m, 8 41 FURTHER PROPERTIES OF FORKING 109

Proof of Theorem 4.1. The direction e follows from 1.2(2). Now wume p does not fork over A. Let 8 realizep. Let q = stp(8, A), r = stp(8, B). By 2.6(1) (and since q t -p t A, r t -p) , for every finite d, R"(r, d, No) = Rm(p u r , d , No) = Rm(p,d, No); Rm(q,d, 80) = W q U P t A , d , NO) = R"(p A , d, No). So it is sufficient to prove Lemma 4.3.

LEMMA 4.3: Let q = stp(8, A), p a type realized by 8 which does not fork Over A , and q E p . Then for everyfinite A , 2 5 A s No,

(1) Rrn(q,d, 4 = Rrn(p,d, 4. (2) Mlt(q, A, A) = Mt(p, 4 A).

Proof. First we prove (1) for A < KO. Let p be over B, where A G B. By 11, 1.1 Rm(q,d, A) 2 Rm(p ,d , A). By 11, 2.13 and 11, 2.2

Rm(q, d, A) = n < w for some n. By 11, 2.9(2) there are cpf;j(E A ) , iZ$j for q E ">A, v E "<A, i < A, j < A, such that for q E "A

q u {cp:i@; q,{)t: l < n, i < A, j < A, i # j;

i = q[l] and t = 0 or j = q[l] and t = 1)

is consistent; let a,, realize this set. Define 6"Jq E "A) 6:' (q E ">A, i < A, j < A) by 2.6(3) such that (i) stp(b,, A) = stp(7i,, A), q E "A

(ii) tp*(u {b,,: q E "A} u U {&j: q E n > A , i < A, j < A}, B) does not fork over A, and extend the corresponding type of the

By 2.9(1) 6,, (7 E "A) realizes p . Clearly hpnll[6n; 6;i,lat when 1 < n, 9 E "A, i # j < A; and i = q[l], t = 0 or j = q[Z], t = 1. So by 11, 2.9(2) (now using the other direction) Rm(p,d, A) 2 n = Rm(q,d, A ) . So we prove the equality.

The proof of (1) for A = KO, and of (2) is similar using suitable parts of 11, 2.9.

Now we return to

atp(6f;j, _A) = stp(7it;', A), q E "'A, i < A, j < A.

Proof of Themem 4.2. (1) Suppose p is stationary over A. So it does not fork over A, hence by 4.1 for every finite d, Rm(p, A , No) =

Let M be a (I BI + + I TI +)-saturated model, B c "1. Let r E rSm( 1M1) p E T, T does not fork over A, and let 8 realize r . By 2.14 p k q =deI

stp(si, A), hence by 4.3 Mlt(p,d, A) = Mlt(q,d, A ) = Mlt(r,d, A) for

R"(p r A , 4 No).


every finite A, and 2 I A I 8,. But by 11, 1.10(1) Mlt(r, A, A) = 1 so Mlt(p,A, A) = 1 and we finish.

Now suppose p is not stationary over A. If i t forks over A, then by 4.1 for some finite A, Rm(p,A, 8,) # Rm(p 1 A , A, 8,). So assume p does not fork over A. By 2.9(1), if p = tp(8,, B), q = stp(8,, A) then p If q. So for some E E FEm(A) p u {-,E(Z; a,)} is consistent; let si, rertlizeit.By2.6(1)Rm[p u {E(Z;aI) } ,A, KO] = Rm(p,A, N,)forZ = 1, 2 (where A = {A')). But E(Z; 8,) k -,E(Z; a,). Hence Mlt(p, A, KO) 2: 2. The second sentence is emy, so we have finished the proof of 4.2(1).

(2) The proof is immediate by part (1). From 4.1 we see

COROLLARY 4.4: Let A E B E C and p E Sm(C). p does not fork over A i$p does not fork over B and p 1 B does not fork over A .

THEOREM 4.6: Letp be an m-type over A , q a A-m-type, A not necessarily finite, swh t?ut p U q (b coneistent and) forb over A . Then for A 1 8,: * ( p , A , A) < 00 * Rm(p u q, A , A ) < Rm(p, A , A).

Remark. (1) This improves 1.2. Note that in 1.2(4) we have some unspecified A,, and in 1.2(3) we have a formula which divides.

(2) In the come of proving this theorem we shall show:

(*) If p forks over A, A c B, then there is an elementary mapping F, F 1 A = the identity and P ( p ) forks over B.

Proof. Let M be a p+-saturated model (p = IAI + A + I TI) such that A c [MI and q is a type over [MI. Assume that a = Rm(p,A, A) = Rm(p u q, A , A) < 00 and we will get a contradiction.

Since A 2 No, by 11, 1.6 and 11, 1.1 there is q' E S T ( ~ M ( ) such that Rm(p u q', A , A) = a. By the definition of rank there axe .c A such q'. Let them be {q,: j < j , < A} and let 8, realize p u q,. Choose A, c 1M1, IA,I 5 IT1 such that tp(a,, [MI) does not fork over A, and let B = U,<fo A, u A. Then I BI 5 p.

By 11, 1.8(2) we may wume without loss of generality that d is closed under conjunction and negation and by 11, 1.1 that p and q are closed under conjunction. So there is $(Z; a) ~ p ( 3 E A) and d(Z; 6) E q such that #(Z; a) A d(Z; 6) forks over A. So there are {EI , , : i < w}l<Io<a, such that { E I , t : i < o} is indiscernible over A, (~~(3; E I , , ) : i < o} is m,-

CR. 111, $41 FURTHEBPROPERTIESOFFORKINQ 1 1 1

contradictory, $ A 8 t Vz vZ(Z; E l , , ) . Since M is a model we can choose (by 2.6(3)) 6', E;,, E IMI, I < Z,, i < w, such that

stp,(6 u U {E l , , : I < I , , i < w}, A )

= stp*(6' u (J {Ei,,: I? < z,, i < w}, A )

and tp*(6' u U {Ei,,: I < I , , i < w}, B) does not fork over A. Thus also

and {pz(5; Ei,,): i < w} is m,-contradictory, I < I,. But stp(E,,,, A ) = stp(E,,,, A), I < I , , i < o (by the indiscernibility of {Ez,,: i < w } over A) so by 2.9(1) tp(Ei,,, B) = tp(E;,,, B), I < Z,, i < w. Hence $(if; a) A O(Z; 6') forks over B. By 11, 1.6 we may extend p 1 = pu { $ ( ~ ; a ) A 8(Z; 6')}toatypepuq',q'ES~(IMl)suchthatRm(puq',d,A) = u (because Rm(pl,A, A) = a by Exercise 11, 1.1).

Thus there is a j < j, such that O(2; 6') E q, (since 8(Z; 6') is a A- formula). Since @, realizes q, u p and tp(Z,, IM[) does not fork over B, the formula $(Z; a) A B(3; 6') does not fork over B. Contradiction.

$(z; a) A e(z; 6') k vz vl(z; E:,,), = ~ m [ p u {$(z; a) A e(z; 6')}, d, A]

EXERCISE 4.1: Show that in 4.6 the condition A 2 KO cannot be omitted.

Remark. See Exercise 4.18.

Hint: Let Af = (B, R) be a finite model, R a two place relation, such that for any a, b E B there is an automorphism f of M, f ( a ) = b. Let us define a model N:

IN1 = w x B x{O, l},

RN = {<<% a, 1)s (n, b, 1)): M t= R[a, bl} BN = {((n, a, 0), (m, b, 1)): n = m or a # b}

PN = {(n, a, 1): n < w , a E A}.

Let T = Th(N); it is KO-stable; let A = 0, p = {P(z ) } , q = ((0, a, 0)Sz: a E B}, d = (8, It}.

COROLLARY 4.6: Let p = stp(3, A), A 2 KO, Z(a) = m and d is not neceesarily finite.

(1) There do not exist two contradictory A-m-types q,, qz such that

(2) If Rm(p, A , A) < co then Mlta(p, A , A) = 1. (3) If Q E S ~ ( ~ M I ) , Rm(q,d, A ) < 00 then Mlta(q,d, A) = 1 (see

R"(p,d, A) = R y p u q1, A , A) = R"(p u qa, A , A) < 00.

2.16(2)).


LEMMA 4.7: Let q = stp(& A ) , l(a) = m , p be a A-m-type such that p u q does not fork over A (A not necessarily Jinite). Then for A 2 KO, R"(q, A, A) = a < 00 Rm(g U p , A , A) = a.

Proof. Let B 2 A u ?j u Dom p. Choose r E Sm( B) extending p u q such that r does not fork over A. Then r k q u p t- q and thus R"(r, A, A) 4 R"(p U q, A, A) 4 Rm(q, A , A) = a. On the other hand by 11, 1.6 there is p' E S t ( B ) , such that Rm(q u p', A , A) = Rm(q, A , A). By 4.6 q u p' does not fork over A and thus by 2.9(1) p' c_ r hence p E p'.

LEMMA 4.8: (1) If p does not fork over A then p is "de$nuble almost over A". This meuna thad for all 'p there is a formula t,hcp(Z; Ecp) which U almost over A, swh thad p t 'p is de$nuble by t,hcp(z; E,,,); i.e., ~ ( z ; a) E p * I=+&; Ecp] and -,'p(E; a) ~p

(2) If A c B E C, p E Sm(C), p 1 B is stationary over A andp does not fork over A, then p is deJintabk over B.

k + J a ; E e l .

Proof. (1) Immediate by 4.3 (for A = 2) and 11, 2.12(3). (2) As of (l), noticing that by 2.14, if ti realizes p , p t B k stp(zi, A).

LEMMA 4.9: Letp ES"((M(),A$nite. Foreueytypeq €Srn(B) extending p , which does nd fork over IMI, q t A = Avd(l, B) for some in$nite A - i d i s m i b k set I E IMI.

Proof. By Lemma 11,2.20 there is a finite A, such that for all 'p(Z; 8) E A, a, and any infinite A,-indiscernible set J, I'p(J, a)] < No or 17'p(J, ti)(

Using Theorem 11, 2.17 we can find a hi& A* such that if pk = tpd.(7ik, Ak), Ak = A u u {al: I < k}, and for all 'p E A* and k < w

is a 4,-indiscernible set over A, i f p k satisfies the condition from 11, 2.17. suoh that p 1 'p is (+@, A)-dehable, for every

'p € A * ('pa from 11, 2.12) and define tik (k < w ) accordingly. As in 2.13 the result follows (by defining {a,: w s i < w + w } and Exercise 11, 2.9).

< No.

Rm(pk I'p,'p, 2) = *(PO 'p, ' p a 2, (where l @ k ) = m), then { a k : k < w)

Choose a finite A c

EXERCISE 4.2: Show that I does not depend on q but only on p and A.

UH. 111, 8 41 FURTHER PROPERTIES OF FORKING 113

COROLLARY 4.10: (1) The formula ~(3; a) does not fork over IN1 iff there is 6 E

(2)* If there is 6 E A m h that k q [ 6 ; a] then ~(3; a) does not fork over A . m h that C Q [ ~ ; a].

Proof. (1) Assume ~ ( 3 ; a) does not fork over IMI. Then there is q E Sm(lN1 u a) which does not fork over JNl, ~(3; a) E q (by Theorem 1.4). Let p = q 1 1Y1. By the previous lemma there is I c IYI such that q 1 Q = Av,(l, u a). Thus ~(3; a) is realized in N. This proves one direction, and the other follows from (2).

(2) If ~(3; a) forks over A then ~(3; a) k V,<,, ~ ~ ( 3 ; a,) where ~ ~ ( 2 ; a,) divides over A. But if ~ ( 3 ; a) is realized in A then there is i such that ~ ~ ( 3 ; a,) is realized in A. By the definition of "divides" there are n, c w and sequences a{, j c w, such that tp(a{, A) = tp(a,, A) and for every i c n, (~'(3; a{): j c w} is n,-contradictory. But the sequence in A which realizes Q,(P; a,) realizes also all the ~ ~ ( 3 ; a{), contradiction. An alternative proof of Corollary 4.10. Choose q E S"'(liKl) such that q u { ~ ( 3 ; a)} does not fork over 1Nl, and n(tp) c w such that for any indiscernible set I , lrp(I;a)l < n(9) or ) 7 ~ ( I , s i ) l c n(cp). Let I = {6,,: n c w} be an indiscernible set based on q, so ~(3; a) E Av(I; T i ) hence

LEMMA 4.11 : If p E ST( / M I ) ( A is not necessarily finite) is definable almost over A where A G lYl (see 4.8(1)) then p doef, not fork over A .

Proof. Without loss of generality Y is IAI +-saturated, since we may extend the given N to a IAI +-saturated model and then extend p to a complete A-m-type over the new model which is still definable almost

114 QLOBALTHEORY [OH. 111, 8 4

over A. The implication about forking will then hold for p. Also we may assume that A is closed under conjunction and negation. By way of contradiction assume p forks over A. Thus by Lemma 1.1( 6) we get a ~(3; Z) ~p which forks over A, and by definition there is #@; ii) almost over A which defines p t tp. Without loss of generality ii E 1M1. So if Z' E lMl and stp(Z', A) = stp(Z, A) then C#[Z; a] = #[Z'; ii] and thus

Now define sequences En E lMl, n < w such that tp(Zn, A u U,<, a,) does not fork over A, and each En realizes stp(Z, A). By Lemma'l.lO(1) {En: n < w } is an indiscernible set bawd on A. Clearly C#[Z,; ii]; thus

[by l.l(9)] it forks over A. This contradicts 2.6 (because of 2.7).

v(3; a') E p .

~ p @ ; an) E P ; thus for every n, 4n = V w s a n - l , l r u l - n A 1 . W ~(3; a,) EZJ and

EXERCISE 4.3: (1) Find an example of p E S,"(B), p is definable over A c B, but p forks over A.

(2) In 4.11 let q be an m-type over A, p u q consistent; show that p u q does not fork over A.

COROLLARY 4.12: Let p ES"(IMI), A c ]MI, p does not fork over A iffpi8deJinablealmostoverAiffforall~, Rm(p,q , No) = Rm(p A , v , KO).

Proof. By 4.8, 4.11 and 4.1.

THEOREM 4.13 (The symmetry theorem): If A c C , B c C, tp(ii, C) does not fork over A , tp(6, C ) does not fork over B then tp(ii, C u 6) forks over A iff tp(6, C u a) forks Over B.

Proof. Suppose not; so by the symmetry of the hypothesis on ii, 6; we can assume tp(ii, C u 6) forks over A, but tp(6, C u 7i) does not fork over B. Hence there is Z E C and a formula ~(3; 6, Z) which forks over A, and Cv[ii, 6, Z]. Now define by induction on n c w sequences a,, 6, such that

(i) stp(ii,,C) = stp(ii,C) (ii) stp(6,, C ) = stp(6, C) (iii) p , = tp(ii,, si u 6 u C u U {ai: i c n} u U,{6,: i c n}) does not

fork over A. (iv) qn = tp(En, 7i u 6 u C u U {7ii: i 5 n} u U {6{: i c n}) does not

fork over B (this is possible by 2.6(3)). Now t+p[ii, 6,,, Z] [aa by 2.9(1) stp(6, A) is stationary over A, we have

assumed tp(6, C U ii) does not fork over B, and by (iv), (ii)] hence

OH. 111, 8 41 FURTHER PROPERTIES OF FORKING 115

hp[Z,, 6,, 81 for k I n [q,, does not fork over B, hence, by 4.8(1) is definable almost over B; but by (i) stp(a,, C) = stp(7i, C)].

On the other hand if k > n, as q(E; 6,E) forks over A, by (ii) also q(3; 6,, E) forks over A, but by (iii) p , does not fork over A, hence

So bq&, 6,, E l o k I n, hence by 11, 2.13(5) T is unstable. Contra- -lq(f; En, a) Ep,. so C - l C p [ i z , , 6,, E l .

diction.

THEOREM 4.14: If A c B, p = tp(Tt-6, B) does not fork over A , then q = tp(& B u 6) does not fork over A u 6.

Procf. Suppose the hypothesis holds, but the conclusion fails. So for some 5 E B, !=tp[Z, 6, but v(Z; 6, E) forks over A u 6. Now define by induction on n sequences En, a,, such that

(i) stp(Zi,, A u 6) E stp(zi, A u 6). (ii) stp(En, A) = stp(E, A). (iii) pn = tp(a,,, A u 7i u 6 U E U U {tii: i < n} u U {E i : i < n}) doe8

not fork over A u 6. (iv) r , = tp(E,,, A U a u 6 U E u U {a,: i I n} u U (5,: i < n}) does

not fork over A (this is possible by 2.6(3)). As E E B, tp(a-6, A U E) does not fork over A ; hence by the previous

theorem 4.13 (and by 1.3), tp(E, A U @ u 6) does not fork over A, hence by (ii), (iv) and 2.9(1), tp(E,, A u 7i u 6) = tp(E, A u si u 6), hence

But by (i) stp(zi,, A u 6) = stp(Z, A u 6), so by (iv), ~(a,,, 6; Z) = q@, 6; 3) E r,, when n I k. Combining we get Cq,P,, 6, a,] when n s k.

Suppose now n > k, by (ii), (iv) tp(EnE, A ) = tp(6-& A) hence q(3; 6, a,) forks over A , hence by (iii) -,q(Z; 6 , E J ~ p , ; hence b+Z,, 6, E,]. Combining we get kq[a,,, 6,4] iff n I k, so by 11, 2.13(5) T is unstable, contradiction.

!=tpFi, 6, znI*

LEMMA 4.15: (1) If tp(Z, B) does not fork over A E B, and tp(6, B u a) does not fork Over A u a, then tp(a-6, B ) does not fork over A .

(2) If thejrst two types are stutimry over A , the third is also. ( 3 ) In ( 1 ) we can use an infinite a.

Proof. (1) Let {gi: i < i,} be a maximal sequence with the property that for all i # j, stp(&,, A u 7i) f stp(6,, A u a). By Lemma 2.2(2) i, is certainly s ~ ~ ~ I + I ~ I . Now by Lemma 2.6(3) there is an elementary mapping F with Dom F = A u 3 u U (6,: i < i,} such that F 1 A is

116 QLOBAL THEORY [CH. 111, § 4

the identity, and for all E E Dom P, tp(P(E), B) does not fork over A, and stp(5, A ) = stp(P(Z), A).

Thus, in particular, stp(8, A ) = stp(F(Si), A) and tp(7i, B), tp(P(a), B) ES~(')(B) do not fork over A. But by Corollary 2.9(1) stp(G, A) is stationary over A, and thus tp(a, B) = tp(F(Ti), B).

Thus we can find an elementary mapping G with Dom G = B u Range F such that G B is the identity and C(P(7i)) = 3. Now let bf = G(P(gi)) . By the maximality of the 6, there is no 6' such that stp(6', A u a) $ stp(67, A u a) for all i < i,. Thus there is an i < i,, such that stp(6, A u a) = stp(67, A u a) (6 is from the statement of the lemma).

Now tp(67X, B) does not fork over A since an@' E G(Range P). Thus tp(6f, B u T i ) does not fork over A u a, by Theorem 4.14. But we have also assumed that tp(6, B u 7i) does not fork over A u a, so by Corollary 2.9(2) tp(67, B u a) = tp(b, B u a). Thus tp(a-6, B) = tp(7in6:, B), and tp(Si-6, B) does not fork over A.

-

(2) Immediate by 2.14. (3) Same proof as (1).

EXERCISE 4.4: Prove that if p # q E Sm( B) p A = q 1 A = r and Rm(p, L, A) = Rm(q, L, A) = Rm(r, L, A) < co, then there is E E FEm(A) such that p(Z) U q(#) k ,E(Z; 5).

EXERCISE 4.5: Prove that if p €Sm(B), A c B and Rm(p 1 A, L, A) = Rm(p, L, A) < 00 then p is definable almost over A.

EXERCISE 4.6: If in addition to the assumptions of Exercise 4.5

Mlt(p A, L, A) = 1, then p is definable over A.

EXERCISE 4.7; !Cry to generalize Exercise 4.4 and Exercise 4.5 to other infinite A's, where Sy(B) would replace Sm( B).

LEMMA 4.16: Let p E Sm( I M I), w k r e A c IM) and M is I A1 + -saturated.

A is stationary over A andp does not fork over A ] iffp does not (1) p f w k g over A iff p strongly splits over A. (2) [ p

split over A .

Proof. ( I ) One direction is just Theorem 1.6(1), and the other direction is left as an exercise.

(2) One direction is Lemma 1.9, and the other direction is left as an exercise.

OH. 111, 8 41 FURTHER PROPERTIES OF FORKING 117

LEMMA 4.17: Let I, J be inJinite indiscernible sets. (1) If Av(I, U I ) = Av( J , U I) and Av(I, U J) = Av( J, U J) t h

I and J are equivalent. (2) If for all distinct a,, . . . , a,, E I and distinct 6,, . . . , 6,, E J

tp(a,^ - - . a,,) = tp(6,^. 'Ên), then Av(I, U I ) = Av( J, U I) =- I, J are equivalent.

(3) AV(I, U I) is St&WMW'Y Over U I . (4) Av(I, U I,) is stationary Over U I , for some I , E I , 1111 <

min{N,, K(T)}, and Av(I, U I ) does not fork over U I , .

Proof. Let M be a (IIl+ + IJI+)-saturated model such that U I u U J E [MI. Let p = Av(I, [MI) , q = Av(J, [MI) . Clearly p and q do not split over U I and U J respectively. So by Lemma 4.16, p U I is stationary over U I and p does not fork over U I; a similar claim holds for q. This proves (3) .

Now from Av(I, U I) = Av(J, U I) we get that p U I = q t U I and thus (by Theorem 4.2(1)) for all finite A, Rm(p,A, 2) = Rm(p 1 U I , A , 2) = Rm(q 1 U I , A , 2) 2 Rm(q, A , 2 ) . If we assume that tp(a,^- - -â,,) = tp(6,". - .Ên) as in (2), equality holds; thus p = q so (2) holds. Also in case (l), by the symmetry in the assumptions, again equality holds, thus (1) holds.

Let us prove (4). If I, s I, 11,1 = No then by (3) Av(I,, U I,) = Av(I, U I,) stationary over U I,, and as Av(I, 1M1) does not fork over U I, (as it does not split over U I,) Av(I, U I) does not fork over U I,. If KO < K(T), we finish. If K(T) = No, for some finite I, E I, Av(I, U I) does not fork over U I,. Choose a E I - I,; let I, = I, u {a}. As Av(I, U I,) k stp(8, U I,) we finish.

DEFINITION 4.1 : An m-type p is stationary if for every A, Dom p E A, there is a (unique) type q €&"(A) such that

(i) P c q. (ii) For every finite A Rm(p, A , 2) = P ( q , A , 2 ) .

Remark. The uniqueness follows by 11, 1.4(1).

DEFINITION 4.2: (1) The stationary m-types p , , pa are called parallel if for every A, Dom p1 u Dom pa E A, the unique types q,, q2 from the previous definition are equal.

(2) We call p a stationarization of q if q is stationary p parallel to q OT q is complete over some A, and for some E realizing q, p is parallel to stp(E, A). A stationarization of I, over A is any stationarization q E Sm(A) of p.

118 QLOBALTHEORY [CH. 111, 5 4

Remark. In (2), in any case p is stationary, and the two cams are wnsistent .

LEMMA 4.18: (1) Let p sSm(B), t h n p is stationary over B iiff p is

(2)* In Dejinition 4.2(1) it suflces to say that this ?wLolds f o r some A. (3)* Parallelism is an equivalence relation. (4)* If p , q are stationary and f o r every jinite A , Rm(p, A , 2) =

(6)* If p is a stationary m-type, Domp E A, there is a unkpe

8kl4%Onary.

Rm(p u q, A , 2) = Rm(q, A , 2) then p , q are parallel.

q E &'(A) parallel to p .

Proof. ( 1 ) By 4.2. (2) Immediate. (3) Immediate, by the definition and (2). (4) Immediate. (6) Immediate by the definitions.

DEFINITION 4.3: An (infinite) indiscernible set I is based on a s t a t k r y type p if Av(I, U I ) is parallel to p.

DEFINITION 4.4: (1) A set I of sequences is independent over (A, B) where A E B if for every B E I, tp(B, B u U ( I - a)) does not fork over A. When A = B we write A instead of (A, A).

(2) We say I is independent over p, if p E Sm(A), I is independent over A, and each a E I realizes p .

LEMMA 4.19: (1) If I is independent over p , p E Sm(A), p is stationary, then I is an indiscernible set over A based on p .

(2) If I is an indiscernible set over A, based, on p €Sm(A), then I is independent over A and over p . If C E A, p does not fork over C, then I is independent over (C, A).

(3) If I is independent over (A, B), J independent over (A , B u U I ) then I u J is independent over (A, B).

(4) If I is independent over (A , B), A E C E B, E B, J E I , then I - J is independent over (C, B, u U J ) . (6) If I = {ai: i < a}, tp(B,, B U U {B,: j < i}) do@ not fork over

A E B then I is independent over (A, B).

CH. I I I ,§ 41 FURTHER PROPERTIES OF FORKINQ 119

Proof. (1) By l.lO(1). (2) E a y by 1.7(2). (3), (6) Immediate, by 4.13, 4.14 and 4.16. (4) Immediate.

DEFINITION 4.6: Let A c B, J a set of sequences of length m. (1) If p ES"(A), dim(p, B, J) is min{)Il: I c J, I a maximal inde-

pendent set over (A, B) of sequences realizing p}. (2) If p is a stationary type, and there is q ES*(B) parallel to it,

dim(p, B, J) = min(lI1: I a maximal set indiscernible over B, of sequences in J realizing q}. See Lemma 4.20. (3) The dimensions in (l), (2) are called true if for every such

maximal I, IIl is equal to the dimension. (4) If J = "A, we write A instead of J. If B = Domp, we omit it.

LEMMA 4.20: (1) Definitions 4.5(1) and 4.5(2) are Consistent, i.e., when both apply the resulting dimemione are equal.

(2) If I is based on p ( p stationary, I an (infinite) indiscernible set) then dim(I, A , M ) = dim(p, A , M).

Proof. Immediate (use 4.19, see Definition 3.3).

THEOREM 4.21: (1) If I is idpendent over ( A , B), K 2 K(T) , cfK > ICl, then for some J c I , IJI < K , I - J is independent over ( A , B u C u J ) . Moreover there is a minimal such J .

(2) If dim(p, A, J) is 2 K(T) then it is true; and if the dimemion is not true, for every suitable maximal J (from Definition 4.5) I JI < K,(T).

Proof. Similar to 3.6, 3.9, Exercise 3.4.

EXERCISE 4.8: Show that for some unstable T , there is p ~S"(141), A c ICl, such that p does not split over A but is not definable over A.

EXERCISE 4.9: Show no stable T satisfies Exercise 4.8.

EXERCISE 4.10: Let A c B, 7i algebraic over A. Then tp(8, B) does not fork over A (see Definition 6.1).

CLAIM 4.22: (1) Let A c B E C be such that for all ~ E C there is ZEC such that tp(Z, B) does not fork over A and tp(Z, A ) = tp(6, A). If ii is any sequence such that tp(8, A ) is stationary and tp(8, C ) does not fork over A then: tp(8, B) k tp(8, C) iff tp(7i, A) t- tp(7i, C).

120 GLOBALTHEORY [m. 111, § 4

( 2 ) We can waive tire statiOnarity of tp(8, A), if tp(E, A) is stationary for every E E C ; and then:

stp(i3, B) I- tp@, C) if tp(& A) F tp(8, C). Proof. ( 1 ) -= is immediate.

=- Let q(Z ; 6) E tp(& C), 6 E C; we shall show that tp(8, A) I- q ( Z ; 6). Let a be aa above. By Lemma 1.9 tp(8,C) does not split over A, and q(Z; 6) E tp(8, C), so q(3; a) E tp(8, C). Since tp(8, B) t- tp(8, C) we have tp(8, B) k q(Z ; a). Thus there is a E B and #(Z; a) such that k#[8; 83 and I -#(Z;a)+q(Z;5). Clearly tp(8, A u z) does not fork over A so (by 4.13) tp(& A u a) does not fork over A. So as before, we can define {a,,: n < w } to be an indiscernible set over A u Z based on A such that stp(a,,, A) = stp(a, A). By Lemma 2.6 there is n for which

#n = V 4) wS2n-1 few

Iwl =n

is almost over A. Of course, #(Z; a,,) I- q(Z ; a) for all n and so #,, I- q ( f ; a). So by Lemma 2.14, since #,, is dmost over A, tp(8, A) I-#,, (or tp(& A) I- -I#~, let q E Sm(C u Un<@ a,,) be a stationmization of tp(8, C), so #(Z, a,,) E q hence #,, E q, hence tp(8, A) u {#,,} is consistent; contradiction). Thus tp(8, A) I- q(Z; a); but tp@, A) = tp(6, A) 80

@(a, A) I- q ( E ; 6). (2) Similar proof.

EXERCISE 4.11: Suppose A is finite, ~ E S ~ ( B ) and rl, i < A, are A-m-types such that the types p u r, are rn,-contradictory, rn, < w ,

i < A, and IBI + IT1 < A. (1) Show that for some i, p u ri forks over B. ( 2 ) Show that if 1231 + I TIK < A we can relax the finiteness condition

on d to [dl < K. In fact, A 2 No, A > 2141 suffioe.

QUESTION 4.12: Generalize the properties of the rank D such aa symmetry, monotonicity, etc.) to the c&88 where T is unstable but D(p, L, a) < a for every type p.

THEOREM 4.23:Suppose A > No is regular and (Vp < A)(pCKcT) < A). If IIl L A > IAI then there are J , I , c I , IJI = A, II,l < K(T), such that J i s independent over A U U I , (hence if stp(8, A u U I , ) , 8 E J , is $xed, J i8 indbcernible over A u u I,,).

Remark. We give this theorem partly to show an alternative proof of Theorem I, 2.8: the existence of large indiscernible sets.

CH. 111, Q 41 FURTHER PROPERTIES OF FORKING 121

Proof. Let a,, i < A, be distinct members of I , and let A , = A u U,<, a,. For each i let B, c A,, 1B,1 < K(T) be such that tp(a,, A,) does not fork over B, and let f(i) = min{j: B, c A,} so cf 6 2 K(T) =- f(i) < i.

As (Vp c A)(p<n(T) < A), A > K,(T) hence by 1.3(3) of the Appendix, So = (6 < A: cf 6 2 ~ ( 2 ’ ) ) is stationary. Hence by 1.3(1) of the Appen- dix, there is a stationary set S, E So on which f is constant, i(0) say. As

by 1.3(1) of the Appendix there are a stationary set S, c S, and B c A,,,, such that i ES, 5 B, = B. Let I , c I be minimal such that B c A u U I , and let J = {a,: i E S,}. Clearly the conclusion holds.

EXERCISE 4.13: Given an increasing sequence of sets A, (a < <), for every [ we can find elementary mappings F,, q E c>[, such that:

(i) q + v implies F, extends F,. (ii) Dom P, = Al(,). (iii) tp,(Range F,, U {Range F,: Y E [>t, not q + v}) does not fork

If tp,(A,, U,<, A,) is stationary, this construction is unique and for any 8, E A,+, , we call {FJa, ) : q E ,+l<, i < [}an indiscernible tree, as it satisfies Definition VII, 2.4, and VII, 3.1.

over Ua < I(,) Range Fnlw

EXERCISE 4.14: Suppose K < K(T) and q~,, a < K , are as in Theorem 3.1. Find sequences a,,, q E ” A such that:

(i) {a,: q E “‘A) is an indiscernible tree. (ii) hp,[E,; ?iv]ii(v *TI) where q E “A, v E + A. (iii) {rp,(Z; a,,-( ,)): i < A} is m,-contradictory for v E ,A, some m, < w.

EXERCISE 4.15: Show that if T is stable then we can take n = 1 in Definition 1.4 of forking. Hence a formula divides iff it forks.

EXERCISE 4.16: Show that in 4.23, if we weaken the conclusion “lIol < x(T)” to “(Iol < A ” we can weaken the hypothesis on A to “ A > K,( T) is regular”.

EXERCISE 4.17: (1) Suppose M is K-compact, p an m-type, 1p1 < K.

Then p does not fork over [MI iff p is finitely satisfiable in M iff p is realized in M.

(2) Suppose M is K-SatUrated, p €Srn(A), K,(T) + [ A [ + I K. Then p does not fork over [HI iff p is finitely satisfiable in M iff p is realized in M.

122 QLOBALTHEORY [ax. III,§ 6

EXERCIBE 4.18: Show that in 4.5 we can omit A 2 No if we add A = clz(A) or p is stationary over A. [Hint: We can aasume that q is finite, so in either case that q = {q (Z ; a)}, so we can msume p is finite. Let M be a ( A + IT1 + IAl)+-saturated model, A u si s 1611. Let for (a < A) be elementary maps, fa 1 A = the id., fo 1 1M( = the id., tp*(f,(lMI), U B , , f s ( l M l ) ) does not fork over A. Then note:

(A) q, = { ‘ p ( f ; f , ( s i ) ) u ( 4 = j ) : a < A} are pairwise explicitly contradictory, and of equal rank, hence Rm(p, A, A) > Rm(p u qo, A , A). .

(B) No element of 1M1 realizes ‘p(Z;f,(si)) A A p (a > 0) (by 4.10, Exercise 4.19), hence Rm(p u q, A , A) = Rm(p u qo, A , A ) (by Exercise 4.20).]

EXERCISE 4.19: Suppose A E B, C, tp,(C, B) does not fork over A, and q is a type over B forking over A. Then q forks over C. [Hint: Otherwise let tp(& B u C) extend q and not fork over C. By the Symmetry Theorem 4.13 tp*(B, C u a) does not fork over C. But (by symmetry) also tp*(B, C) does not fork over A , hence by transitivity (4.4) t p , ( B , C ~ d ) does not fork over A, so by monotonicity tp*(B, A u 2) does not fork over A. So by symmetry, tp(& B) does not fork over A, but q c tp(& B) and q forks over A, contradiction.]

EXERCISE 4.20: Suppose p is an m-type over l M ) , p 2 1131’ + A, M is p-saturated, p ~ { q ( Z ; s i ) } is not realized in ill; and pl = p u {-,c@; a)}. Then P ( p , A, A) = P ( p l , A, A). [Hint: Trivially P ( p , A , A) 2 Rm(p l ,A , A), so we prove by induction on a that R”’(p,A, A) 2 a =- Rm(p l ,A , A) 2 a. The point is that the types “witnessing” for the rank can be chosen over !MI and if r is an m-type over lM(, Irl < p, then p u r is consistent iff p1 u r is consistent. Use Exercise 4.17(1).] In fact, M p+-compact is sufficient.

111.5. The first stability cardinal I

From now on, until after Lemma 6.11 in this section let B c IMI, p €Sm(B), and B = {q ~S~(lill1): p c q and q does not fork over B}. By Lemma 2.16(2) every q E B is stationary over By and by Corollary 2.9(1) for every si realizing p there is a unique q,, E B such that q,, k stp(si, B) so if si realizes q E B then q,, = q.

CH. 111, 8 61 THE FIRST STABILITY CARDINAL 123

LEMMA 5.1 : Choose a jixed a* which reulim p . Then there are A* and Er(3; #) E FE”(B) for i < A* m h that:

(1) p(Z) u {Er(Z; a*): i < A*} is stationary over B. ( 2 ) For all j < A*, p(Z) U {Ef(Z; a*): i < j} U {-$?(Z; a*)} i s ~ 0 1 ~ -

eistent.

Prooj. By Corollary 2.9(1) p(Z) u {E(Z; a*): E E FE*(B)} is stationary over B. So there is a set Qil of minimal power A* such that pgl = p(3) U {E(Z;B*):EE @,}isstationaryover B.Let Qi, = {I#,@;@: i < A*}. Let U be the set of all j < A* such that p(Z) u {E,(Z; a*): i < j} u {-,E,(Z; a*)} is consistent, and let Qi = {E,(Z; j j ) : i E u).

We claim that p z I- pgl. We shall prove this by proving by induction on i that p$ F E,(Z; a*). Assume this is true for i < j. Now if j E U the result is immediate. If j 4 U then by the induction hypothesis p: I-p(Z) u {E,(Z; a*): i < j} and aa j 4 U p(Z) u {E@; a*): i < j} I- E,(Z; a*), 80 pg F E,(Z; a*). This proves the claim, and in particular p z is stationary over B. Thus lQil = A*. Sincep(3) u {E,(Z; a*): i < j, i E U} U {-,E,(Z; a*)} is consistent for each j E U, by changing the indices we get exactly (2).

LEMMA 6.2: (1) For all a,, i < A*, if p‘ = p(Z) u {Er(Z; a,): i < A*} is &tent (A* and E: are from the previ0u-s lemma), then p’ W 8tatimry over B .

(2) For all ai, i < A*, if p’ = p ( ~ ) u { E ~ ( z ; a,): i < j } is coneistent then pf u {-,I#?@; a,)} is codstent.

Proof. (1) Let a’ realize p‘ and set r = p(Z) u {Ef(Z; a’): i < A*}. Now by Lemma 5.1 (1) p(Z) u {Ef(Z; a*) : i < A*} is stationary over B,

and a’ realizes the same type over B ~ J J Z*(p); so r is also stationary over B. It is not hard to see that p‘ I- r I- p’; thus p’ is also stationary over B.

( 2 ) Now let a‘ realizep’ and take r = p(Z) u {Er(Z; a’): i < j}. Since Z* and a’ realize the same type over B, r u {,E,*(Z; a’)} is consistent (by Lemma 5.1(2)), and of course r u {E?(Z; a’)} is consistent. As above p’ I- r I- pj , so that p’ u {E?(Z; a’)} and$ u {-,E;(Z; a‘)} are consistent. If t=E?[aj; a‘] then p’ u { 4 ’ ? ( 2 ; a’)} I-pj u {-,E;(Z; a,)} and if C-,E,*[aj; a‘] then pj u {E,*(Z; a’)} I- p’ u {,E;(Z; a,)}. In any caae (2) is proved.

Remark. This shows that A* is in fact not dependent on the choice of a* as assumed in Lemma 5.1. Clearly it does not depend on M either, 80

124 GLOBAL THEORY [a. III,§ 6

DEFINITION 6.1: A*(p) is the A* defined in 6.1.

Proof. (1) For all q E U ~ < ~ * ~ + ~ ~ define a,, by induction on I(q)

C7E:[+<,,; a,,-<1>] for i = Z(7). Assuming p , is defined for I(q) = i , choose such that p , u {E:(Z; 7i,-<o))} is consistent. By Lemma 6.2(2),p, U {--,EF(Z; iZ,-<o))} is also oonsistent, so any sequence realizing it is a suitable a,,-<l),

For all 7 E "2 pn is stationary over B, and so p , has a, unique stationarization in 9. Thus 19'1 2 2"'.

On the other hand let a:, 2 < n(@) be representatives E lHl of the different equivalence classesof E:, and l' = {B?(% a:): i < A*, I < n(E:)}. Then by Lemma 6.2(1) q1 n r # qn n r for q1 # qa E 9 So it is clear that 19'1 I 2Irl = 2"' for No I A* and 191 5 2Ir1 < No for A* < No.

(2) C = U (8:: i < A*, 2 < n(E,*)} is a set of power A* for A* 2 No and is finite if A* is finite. It is readily seen that C satisfies the requirements stated in the lemma.

such that p , = p u {El * - (z; a,lcr+l)): - i < I(q)} will be consistent and

DEFINITION 6.2: Let Eo EFE"(B), @ E FE"(B). ECN(Eo, @) is the maximal n such that for some E realizing p , r(@, E, n) = UfSl p(Zf) u {E(zf; a): E E 0, 1 s i 5 n} u {-,EO(Zf; z,): 1 5 i < j 5 n} is consistent. (ECN stands for class equivalence number.)

Remark. Clearly ECN(Eo, @) I n(Eo).

LEMMA 6.4*: (1) In the above &$nit& ''80me 15'' may be replaced by any E". (2) If Q1 E @a then ECN(Eo, @a) 5 ECN(Eo, Q1).

4'

Proof. (1) is true because p is a complete type over B so for every E, 8' realizing p there is an elementary mapping P such that F 1 B is the identity and P(E) = 8'. (2) is immediate.

DEFINITION 6.3: Let {EO}, @, 'YE FE"(B). Then Eo depends on @ (mod Y) if ECN(Eo, 'Y) > ECN(Eo, @ u Y).

CH. 111, 5 61 THE FIRST STABILITY CARDINAL 125

DEFINITION 6.4: Let l7, @, Y E FEm(B). l7 is independent over Qi (mod Y ) iffor all E E 27, E does not depend on @ u (27 - {E)) (mod Y).

LEMMA 5.6*: (1) For every 0, E there is a flnite Y E Qi such tluct ECN(E, @) = ECN(E, Y).

on 27 (mod Y). (2) If E depend% on @ (mod Y) then for someflnite l7 E @, E depends

Proof. (1) Let n = ECN(E, @). Then F(@, 8, n + 1) is inconsistent. Let F' be a finite inconsistent subset and let Y be the set of E E FEm( B ) appearing in I", I" E F(Y, 8, n + 1 ) so ECN(E, Y ) I n. But by Lemma 6.4(2) ECN(E, u') 2 n, so we have equality.

(2) Immediate by part ( 1).

LEIKMA 6.6*: If El depends on @ u {Ea} (mod Y), but El doe% not depend on @ (mod Y), then Ea depencls on Qi u {El} (mod Y) and even on {El} (mod Y u @).

Proof. Let a realize p and denote pg = p u {E(Z; a): E E l7}. Let a:, . . . , ail (a!, . . . , a:,) be representatives of the equivalence classes of El (Ea) which contain sequences realizing p.

Let S f = { k : 1 I k I nl, p& u { E l @ ; a;)} is consistent}. Since El depends on @ u {Ea} (mod Y ) but not on @ (mod Y ) there is k E S1 such that p&y u {Ea(Z; a), El(%; ah)} is inconsistent. Let a* realize

u { E ~ ( z ; a;)}. Thus for E E @ u Y, I. E[a*; a] A E1[a*; ail. Also for some m 5 na k Ea[a; a%] so pgUy u {Ea(Z; a;)} is consistent

(it is realized by 3). But p&,u~Bi) U {Ea(Z; a:)} is inconsistent as every sequence which realizes it also realizes p&, u {Ea(Z; a), El(%; a;)}.

Thus ECN(Ea, Qi u Y U {E l } ) < ECN(Ea, Qi U Y ) I ECN(Ea, Y) , and the conclusion follows.

LEMMA 6.7*: If for all i < a E' does not depend on Qi u {Ej: j < i) (mod Y) , then {EL: i < a} is independent over Qi (mod Y) .

Proof. By Lemma 5.6 it suffices to prove this for finite a. For a = 0 or a = 1 it is trivial. Now suppose the claim is true for a but fails for a + 1. For some i < a E* depends on l7 = @ u { E f : j < a + 1 , j # i} (mod Y) , but EL does not depend on l7 - {E"} (mod Y) . So by Lemma 6.6, with E' for E l , E" for Ea, l7 - {E"} for @, and Y for Y, we get a contradiction.

126 QLOBAL THEORY [CH. 111, f 6

LEMMA 6.8*: Let l7 be independent over @ (mod Y). Then:

independent over @ u {E} (mod l7* u Y).

that n - n1 is independent over @ U

(1) for every E there is a Jinite II* c l7u @ a c h that l7 - 17* is

(2) For every Q1 there is l7, E nu@, < 1a11+ + No, such (mod U n1).

Proof. (1) By Lemma 6.6(1) there is a finite l7* G l7 u @ such that ECN(E, Y U @ u n) = ECN(E, Y u n*). Clearly by 6.4(2) l7 - l7* is independent over @ (mod Y u n*). Suppose that - 17* is not independent over @ u {-E> (mod Y u n*). Then for some E* E l7 - l7*, E* depends on @ u {E} u (l7 - 17* - (E*}) (mod Y u n*). But E* does not depend on @ u (l7 - l7* - {E*}) (mod Y u l7*) (by 5.4(2): the monotonicity properties of dependency). So by Lemma 6.6 E depends on @ u (I7 - I!*) (mod Y u n*), hence ECN(E, Y u l7*) > ECN(Z, 17* U Y U @ u (l7 - l7*)), contradiction to the definition of l7*. Now the claim follows by Lemma 6.4(2).

(2) Similar.

LEMMA 6.9*: If No < A I A* (see Lemma 5.1) and A is regular, then there i s a J i n i t e Y c { E : : i < A } a n d S s A , l S l = A , s u c h t h a t l 7 = { E : : i ~ S } is independent over the empty set (mod Y) and ECN(Er, Y u (l7 - {E:})) 2 Bfori€S.

Proof. Let l7, = {E:: i < j}; by Lemma 6.1 ECN(E,*, nj) 2 2. If 6 is a limit ordinal less than A then by Lemma 6.6(1) there is f ( 6 ) < 6 such

By the regularity of A and 1.3( 1) of the Appendix there is i, < A such that So = (6 < A : f (6) = i,} is of cardinality A. For each 6 E So there is

As I17i,l < A, IS,l = A, A is regular, and l7*J has c A finite subsets, for some finite l7, S = (6: f (6) = i,, n6 = l7} is of cardinality A. The result now follows by 6.7.

that ECN(E,*, n d ) = ECN(E,*, nf(d)).

afinite ITd E l7,, such that ECN(E,*, n d ) = ECN(E,*, n,,) = ECN(E,*, nd).

LEMMA 6.10: Let A, I A* be singular. Then there is Y G {Er: i c A,}, IYI = cf A,, and S E A,, IS1 = A,, such that {E:: i ES} is independent over the empty set (mod Y ) and ECN(E:, Y) 2 2 fbr i ES.

Proof. Let A, = ~ o < c r < c f h o Aa where Aa is regular and for 0 A, > cf Ao. Let l7, be defined aa in the previous lemma.

CH. 111, 6 61 THE FfR8T STABILITY CARDINAL 127

Now define by induction finite Y, E nAa+, and S, c lSal = such that {Ef : i ELI,} is independent over UB<a yfl U nAa

(mod Fa), and without loss of generality i E 889 j E Say /3 < a, == i < j. The proof is as in Lemma 6.9. We define by induction on n < w ,

V; c 8, for every a < cf A,, such that: (1) VZ = s,. (2) v;+1 c Vm (I) IS, - V ; + y s cfh,. (3) {E{: i E V;+l} is independent over

(by 6.8(2)). Let V , = fin<, V;, then IS, - Val I cf A, and {El: i E V,, a < cf A,} is independent over

0 mod u Y,u U { E , : ~ES, - V,, a < cf A,}].

LEMMA 6.1 1 * : Lei l7 be independent Over @ (mod Y) and ECN(E, Y) 2 2 for E E l7. Then for every nl E l7,

Fnt = p(P) U p(@ U {E(P; 8): E E @ U Y U nl}

[ a<cfho

U {TE(E; 8) : E E l7 - nl}

is cm&tent.

Proof. By the compactness theorem it suffices to prove for finite l7. We shall prove it by induction on Il7 - nll. If - nll = 0 there is nothing to prove. So assume it is proven for n, and let Il7 - rill = n + 1 and E* E l7 - nl. Thus Fn,u{E., is consistent; so let Z H a, ij H 6 realize it: so CE*[8; 61. Now we know that

ECN(E*, Y U @ U (l7 - {I#*))) = ECN(E*, Y ) 2 2 .

so r = p ( z ) u {E(z; 6): E ( Z ; 8) E Y u 0 u (n - {E*))} u { 7 ~ * ( ~ ; 6)) is consistent. Let E H E realize it. Then P H 8, jj i+ Z realize Fnt.

QUESTION 5.1: There is a countable C E IMI such that for q # r E 9' q 1 C # r 1 C (assume M is IBI +-saturated). See Lemma 6.3.

Now we abandon our fixed By M y p .

DEFINITION 5.6: D,(T) = LIm(0), D ( T ) = U, D,(T).

LEMMA 5.12*: There is a countable A such that 1Sm(A)I 2 ID(T)I for 8 m e m < w.

128 GLOBAL THEORY [CH. 111, 8 6

Remurk. (1) We can take m = 1 unless I (D( T)I is singular of cofinality > w. (2) If cflD(T)I > KO we can choose an empty A.

Proof. Let p = ID(T)I. For all m < w and A 5 IDrn(T)I, A regular, there is a set AA,, of power < m such that lS1(AA,,,)1 2 A (this by Lemma I, 2.1).

Case 0. If p 5 KO there is nothing to prove.

Case 1. If p > No is regulax then p = IU,,,D,(T)I 5 2 ID,,,(T)I, so there exists m for which p = ID,(T)I and thus lS1(Afi,,,)1 2 p and lAfi,rnl < m < No*

Case 2. If p > No is singular of cofinality > No, then again there is m for which IDrn(T)I = p and so ISm(0)l 2 p.

Case 3. If p > No is singular of cofinality No, let p = Zn<@pn and A = Urn,, Afi+,m. Then IAI s K O and IS(A)I 2 ID(T)I.

LEMMA 6.13: Bwppme IS(A)I 2 A > IAl<n(T), A regular. Then one of the following holck:

(1) ForsomeB, 1BI < K(T), IS(B)I 2 handhence ID(T)I'"'n 2 A.

Proof. For every p €# (A) there is B, c A, [BPI < K(T) , such that p does not fork over B,. IS(A)I 2 A > IAI<"(" so for some Bo l{p E S ( A ) : B, = Bo}I 2 A. Now if IS(BO)l 2 A then (1) holds. Other- wise IS(Bo)l < A. Thus for some po, 9' = { p ES(A): p 1 Bo = pot B, = Bo) is of cardinality > A . If A*(po) 5 KO (from 6.1) then by Lemma 6.3, 2w0 2 2A*(P0) 2 19'1 2 A. Thus (2) holds.

If A*(po) > No, by Lemmas 6.9, 6.10, 6.11, ISa(Bo)l 2 2A*(po) 2 I{p E S ( A ) : p Bo = po, B, = Bo}I 2 A. So for some B', IS(B')I 2 A and IB'I = 1BI + 1 < K(T).

(2) A I 2%

THEOREM 6.14*: If X = ID(T)l < IT1 then L has a sublangwcge L* of power A such that for every predicate R of L there is a formula (in fact a predicate) R* of L* such that (VZ) (R(Z) = R * ( Z ) ) E T . (We say in this w e that T is a dejinitionul extension of T n L*.)

Proof. For all m and all p # q E D"(T) choose a formula v = rp,,q(Z) such that tp ~ p , l p E q. Let Qrn = {v,,&Z): p # q E Dm( T)}, and let Y,

m. ID,§ 61 TEE FIR8T STABILITY UARDINAL 129

be the closure of Qm under the connectives. Claim: For every formula $ ( E ) ( I ( % ) = m) there is an equivalent formula in Ym.

Prwf. By the definition of the ( P ~ , ~ , the set r = {'p(Z) = ' p ( i j ) : 'p E Gm} u {$@) = +(@} is inconsistent.

Thus there is a finite @ c Qm such that {'p(if) E 'p ( i j ) : ip E @} u {$(it) E +(jj)} is inconsistent. Let @ = { 'pl: I < n}. Thus for every 7 E "2 there is t(7) such that ~ ~ ( Z ) n [ ~ l k I@)~(*). Thus

$(z) E V A ' p l ( ~ ) n [ l I , nEna l<n

ttn) = 0

This proves the above claim. Now we define L*. For every m and ' p ( E ) E Ym, if there is a predicate

R such that (VZ)('p(Z) = R(Z)) E T let R, be any such R. If there is no such R, let R, be an arbitrary predicate. Then L* = {R,: 'p E Urn Ym} is the required sublanguage.

Remark. Since lD(T)J is always 2 KO we can assume by the above theorem that ID( T)I 2 I TI.

Now we come to the main theorem of this section.

THEOREM 5.15: Let A designate an in$nite cardinal. Then every stable theory T eatie@9 one of the following:

(1) T ie stable in A iff h = ID(T)I + hex(*). (2) T is etuble in X iff A = 2"o + Acre(*); and also ID(T)I = \TI

< 2%

Proof. If T is stable in A then lSm(0)I I A and so ID(T)I I A. Also by Theorem 3.7, A = If there is a countable A such that [#(A)[ 2 2No then we also get A = 2Ko + A<rc'T).

Now assume that (1) does not hold. Then there is A such that T is not stable in A but still A = ID(T)I + A<k(T). Now A+ is regular and there is A such that IS(A)I 2 A+ > A 2 IAI. Thus the assumptions of Lemma 5.13 hold for A+ in place of A, but 5.13(1) does not hold (otherwise ID(T)I<"'*) 2 A + ; by our hypothesis ID(T)I I A and thus > A, a contradiction). Thus 5.13(2) holds, i.e., 2No 2 A + . In other words A < 2Ho; so ID(T)I < 2No and thus IT1 = ID(T)I by Lemma 11, 3.15. Of course there is a countable A for which IS(A)I 2 2% (by Theorem 11, 3.2).

Thus T is not stable in any power < 2% But by our hypothesis if A 2 2n0, A = ACE(*) then T is stable in A. Thus (2) holds.

so A = ID(T)1 +

130 OLOBALTHEORY [CH. 111, Q 6

COROLLARY 6.16: If T is superstable then I D(T)I or ID(T)( + 2 b is thefivst power in which T i s stable and it i a I8(A)J for some countable A.

LEMMA 6.17: If T is countable and superstable, but not stable in So, then there is m such that IDm(T)I 2 No. (Thus T is not N,-categorical by I X , 1.6.)

Proof. Similar to the proof of Lemma 6.13.

III.6. Imaginary elements

DEFINITION 6.1 : (1) The sequence 6 is defined by the formula I@; a) if ~(6; a) = {6}. It is defined by the type p if 6 is the unique sequence which realizes p . It is definable over A if tp(6, A) defines it.

(2) The formula v(Z; a) is algebraic if v(Q; 3) is finite. The type p is algebraic if it is realized by finitely many sequences only. The sequence 6 is algebraic over A if tp(6, A) is algebraic.

(3) The definable closure of A, dcl A is {b: b definable over A}. (4) The algebraic closure of A, rtcl A is {b: b algebraic over A}. If

(6) When dcl A = acl A, cl A will be their common value. A = acl A we say A is algebraically closed.

LEMMA 6.1 : (1) The type p defines a sequence iff for some finite q c p , A q defines that sequence.

(2) The type p is algebraic i;fs for some Jinite q c_ p , A q is algebraic (moreover, we can choose a q such that p and q are realized by exactly the same sequences).

Proof. Immediate.

LEMMA 6.2: (1) A c dclA c acl A. (2) When A G By dcl A c dcl B and acl A c acl B. (3) If B = dclA tiLen B = dcl B. (4) If B = acl A then B = acl B. (6) 6 &a definable [azgebraic] over A if 6 E dcl A [6 E acl A].

Proof. Easy, e.g., (4) If c E acl B then for some b,, . . . , bk E By n < w and formula 9, b(3 '"z)~(z, b,, . . . , bk) A ~ [ c , b,, . . . , b,]. AS b, E B =

OH. 111, 8 61 IMdGlINARY ELEMENT8 131

acl A there are #,[b,; a,]. Define

E A, n(i) < w and #' such that k(3s%t$4f(z; $) A

k

O(z; * . * Y 7ik) = * * Y k ) [ A #i(Yi; %) A (3 '"z)Cp(z~ y1, - * * 9 yk) 1 - 1

A d z , Yl,. * * Y n)] ;

so l=O[c; a,,. . .,ak] (where yf H bi). But the number of possible (y,, . . . , y k ) is I nf=, n(i), and for each such (y,, . . . , yk) we can find I n 2's. Hence I O(C; a,, . . . , ak)l 5 n n:= , n(i) < KO.

LEMMA 6.3 : (1) If p E Sm(A) then p 7uz.~ a unique exteneion q E Sm(dcl A) , in factp = q and every formula Over dcl A is equivalent to a form& over A.

(2) If a E acl A, then v(Z; 7i) is almost Over A. HoreOver if O(Z; 6 ) is almost Over acl A then it is almost Over A.

( 3 ) If q E Sm(acl A), B E A , then q forks Over B i f l q / A forks over B. (4) stp(a, A) = stp(a, acl A), tp@, acl A) k tp(7i, A). ( 5 ) Let T be stable, I an (injinite) indiscernible set based on A. Then I

is trivial (all of I's elements are e q d ) iff Av(I, A ) W algebraic.

Proof. (1) is trivial, and (3), (4) follow from (2) by 2.6(1) (and 1.1(8) of course). So let us prove (2); so by 6.2(5) msume #@;a) is algebraic, Z E A, k#[iz; Z]. Define E(Z; jj; a) = (VZ)[#(Z; Z) --f cp(Z; 2 ) = p(g; Z)]. Clearly E = E(Z; 9) is an equivalence relation, and if I#(Cr; E)I = n, it has I 2" equivalence classes. So E E FE(A). As !$[a; Z], E(Z; 8) --+

q(Z; 2) = c p ( j j ; a), i.e., v(Z; 7i) is almost over A. The assertion about O(Z; 6) follows by 2.4.

Now ( 5 ) is easy.

Now we define the model C q .

DEFINITION 6.2: (1) The universe of C q is {a/E: m < w , aim16QI, E = E(Z, jj) E L is an equivalence relation (without parameters)} (we mean that ZJE, = aa/Ea iff El = E,, l=El(Zl,B2)). We identify a/= with a.

(2) The relations of QeQ will be equality and (i) P, = {a /E: 6s 161),

(ii) a function P, from P, onto P, defined by P,@) = 8/E (so F,

(iii) for every formula p(Z) E L R, = {(a,/=, . . .): Crp[a,, . . .I}. is a partial function, i.e., a relation),

132 QLOBAL THEORY [a. 111, 8 6

(3) The language of Cq is Leq, and its theory Teq. (4) 6:a wi l l denote a submodel of Cq whose universe is P, u ITs, u u Pgn for some P, and some n < w . So V$ is not uniquely defined.

LEMMA 6.4: For every f m & c p ( f ) in Lea there is an equivaknt form& which ie a Boolean cmnbinath of formzclae of the following forma:

(i) a pmpoaitiOrca2 txn&znt (true or f a k e ) , (ii) x = y ,

(iv) AT- 1 pEI(xi:I) -+ W i , - - - 9 &)[AT- 1 FE,(&) = xt -+ W L * - - 3 &)I- Proof, Left as an exercise for the reader (equivalenae is relative to T).

(%) p E ( Z ) ,

CONCLUSION 6.5: Qbq w not saturated; but if we a&d to it llCll mure elenzents, wit- extending the r e l a t h (80 a 1 2 a ~ element 4 us P p ) we get a eaturated model.

DEFINITION 6.3: (1) If p is a set of m-formulas in Q, then pel =

(2) If M < Q we let Meq be the submodel of Cq, whose universe is

(3) In attributing properties we shall not distinguish strictly between

{B&; T i ) : c p ( f ; a) €I)} u {P=(xt): i < I ( f ) } .

{a/,?#: a E IiKI}. See Lemma 6.6.

iK and MeQ, p and pm (this is justified by 6.6).

LEMMA 6.6: (1) p is &tent iff pw is Coneietent. If p is an m-type in W, p k A,<,, P-(x,) then for some m-type q in 6, pq = p .

(2) When M < Q, MeQ < C a .

(3) For every W < Cq, there is a un&w Jf < Q such that Meq = H'.

Proqf. Easy.

DEFINITION 6.4: (1) The kind 0, of Zi = (ao, . . . , is the unique formula O(xo, . . . , xrn- 1) = A,<,,, PEI(xt) such that CQ C

(2) By a ( < Ko)-type in Cq we shall mean a type p such that 0, E p for any 7i realizing p.

LEMMA 6.7: (1) For h 2 ILI, T is stable in h iff Teq is stable in A. Also K(F) = .(TOq).

( 2 ) If p is a type in Q, A c 161 t h : p forb Over A (in 6) iff pea forb over A (in Cq) and p is statbnary iff peq i s etatbnary.

( 3 ) If 7i E Q, A c 1Q1 then [tp(7i, A, Q)rq = tp@, A, Cq).

(4) If B c A, a E acl B, then tp(a, A ) not fork over B.

OH. 111, 5 61 IMAGINARY ELIEMENTS 133

( 6 ) If B c A, 6 E acl(B u a), 6 # acl A, and tp(6, A) f o r b Over B, then tp@, A) f o r h over B (hot?& in Q and in P a , in particular for b =

FE(W

Proof. (1) By 6.6; and the second phrme by (2), (6). (2) By 6.4.

(3) Easy. (4) mvial. (6) Easy by (4) and 4.16. Following Definition 6.4(2), the theory we have developed for Q bide for

Pa. We shall we this freely.

LEMMA 6.8: For any equivalence relation in Qeq, E = E(Z,g) = E(Z,jj; E ) ( l (Z ) = I @ ) = m) between 8eQuences of k h d 8 (80 E(Z, g) k 8(Z) A tJ(g)), we can c h e a relation P,, and partial function F , (both definable by fmulu8 in Leq with E a8 parameter; but the f m u l u 8 them- selves do not depend on E ) which satisfy:

FE is a function from sequences of length m of kind 8 to P,, Buch that t e [ q A epa] , implies FB(E1) = F&) 0 ~E[E, , cia].

Proof. Technical, so we leave it to the reader.

Remark. We shall not distinguish strictly between the FE, P, from 6.8 and those from Definition 6.2. So we write ii/E instead of YE(@.

DEFINITION 6.6: For a formula ~(3; jj) let EJjj, 3) = (VZ)[cp(Z; 8) E ~(3; Z)] (in this is done for fixed kinds of j j ,5 . ) Clearly E, is an equivalence relation, and ?(Z; a) is equivalent to a formula over 7i/E,.

LEMMA 6.9: I n gW, for algebraically ched A: (1) Every p E Sm(A) is stationary over A (for stable T). (2) Every formula cp(Z; 3) which is almost over A , is equivalent to a

formula Over A.

Procf. (1) By 2.9(1), (1) follows from (2). (2) As cp(Z; a) is almost over A, it depends on some E E FEm(A). Let

b,, . . . , 6, be representatives of the equivalence classes of E. Clearly for ell i FE(6,) is algebraic over A [as P E ( P E ( 6 , ) ) and I-P,(s) = VYI x = p E ( 6 i ) ] , So FE(6,) E A . As a(Z; a) depends on E , for some w, I-cp(Z; a) = V,Ew E(Z; 6,). But kE(Z; 6,) = [FE(6,) = FE(Z)] hence kcp(Z; 7i) = [V,Ew FE(Z) = FE(6,)]. The second formula is over A, 80 we f i s h .

-

134 GLOBAL THEORY [OH. 111, f 6

THEOREM 6.10 (The Canonicity Theorem for Types): Let T be stable. In CQ, for e v e y statiomy m-typep there is a 8et A = Cb(p) (the canonical h e of p ) and a s t a t k r y type q E P ( A ) , q = Ctp(p) ( = tlre canonical type of p ) euch t W :

(1) p and q are pralkl. (2) Any automorlphh F of Cq t a b p to a parallel type iff F f A =

( 3 ) Tor any a E C q , a E A iff every autommphism. F of Cq which take8

(4) There is B E A , I BI < K(T) Over which no type parallel to p forb. (6 ) If a type in Sm(C) is parallel to p and CEO@ not fork over B E C then

(6) A E dcl(Dom p) .

identity.

p to a parallel type necessarily satisfie F(a) = a.

A E ad B.

Proqf. Let r E Sm(QBO) be parallel to p. By 11, 2.2 for every 'p(Z; 8) there is a formula &(g; 5,) which defines r f 9, i.e.,

C+,[Z; E,] o 'p(3; 3) E r .

A = dcl{e,/E*.: 9 E Lea}, Let

q = r f A . Clearly r is definable over A , hence does not fork over A by 4.11. So

q is stationary and q, r are parallel, hence p , q are parallel (this is (1)). Clearly an automorphism F of Cq takes p to a parallel type iff

F(r) = r if€ for every 'p, k+&; a,) = +,(g; F(E,)) iff for every 'p, E,IE,. = F(E,J/E*e ifF F f A = identity (this is (2)). If a $ A then as A = dcl A there is a' # a, tp(a, A ) = tp(a', A ) hence there is an automorphism P of QBO, P r A = identity and P(a) = a'; so P(r) = r, F(a) # a' (this is the missing part in (3)). F'rom (3) it is clear that A does not depend on the particular choice of the $I,: Also A depends only on r , i.e., on the equivalence class of p under parallelism. The same is true for q, hence the notation A = Cb(p), q = Ctp(p) is justified.

As we could have chosen E, E Domp (by 11, 2.12) and as E,/E*. E

dcl(5,) {E,/E**: p€Leq} E dcl(Domp). Hence by 6.2(2), (3) A E dcl(Dom p), so (6) holds. As for (4), for some B s A, 1B1 < K ( T ) , q does not fork over B, but

r does not fork over A, hence by 4.4 r does not fork over B. As any type parallel to p is c r also (4) holds.

So we &118 left with (6). By 4.4; r does not fork over B, hence is definable almost over B, hence by 6.9(2) definable over ad B, so we could have chosen Ee E acl B hence A E acl B.

OH. 111, Q 61 IMAQINABY ELEMENTS 135

CONCLUSION 6.11 : When T ie etuble, for eue y etutionary type I, in Q there is a p r d W type q, euch t7td for every atdmnorphh F of B:

F ( p ) i e p r d k d t o p iff F(q) = q.

LEMMA 6.12: In P, if I is an in&iecernib& eet baaed on the etcrtiOnary type p (i.e., Av(I, U I ) , and p are p r d W ) then I ie i n d h i b l e over

Cb(p).

Proof. Any permutation of I is induced by some automorphism F of V q which maps Av(I, U I ) onto itself, hence p to a parallel type hence F 1 Cb(p) = identity. So the result is clear by 6.10(2).

The following theorem shows that if T is stable but K( T ) is large, then T contains an underlying structure of equivalenoe relations. This corroboratea our intuition that the natural example is in fact quite general.

THEOREM 6.13: In Q, if No < K < K(T) < 00, K ie regular, then there are m < w and fomnukza cpt(E, #) E L i < K , I @ ) =. I ( @ ) = m euch t h d :

(i) cpt(3, #) &fine0 an equivalence relation. (ii) For every h there are an (7 E "A) such t7td ktp,[a,,, a,] i f lv 1 (i + 1) =

rl t (i + 1).

Proof, Let M, (i I; K ) be an increaaing sequence of models, 42, < 6,

is 11M,11 +-saturated. (This is by 3.1, 4.16(1).) Choose a, (n < o) such that tp(a,,, lM,l u Ute,, a,) extends p and

does not fork over 1M,I. Hence I = {a,,: n < o} is indiscernible over 1M,1 and baaed on it.

Now work in CQ. Clearly I is baaed on p hence Cb(p) E. acl(U I ) . On the other hand Cb(p) c INEqI but for i < A, Cb(p) $ lM:aI (all by 6.10). Hence we can define, by induction on i, an increasing sequence a(i) < K (i < K ) and c, E Cb(p), c, E lM&l, c, E 142:?,+l)l. (We use the regularity of K. ) So for every i c, E acl(U I ) hence for some m(i), k(i) , $,

and

(see Exercise 6.1) end w.1.o.g. $,(lQml; a,, . . ., a,,,(,)) n 1M,1 c lMacL+l)l. Clearly if tp(cl, U I ) = tp(c2, U I ) then cl E Cb(p) o c2 E Cb(g)

hence Ili<lQeql; a,, . . . , awd c IJf2,+1)1.

= UL<C Mi, m d p E 4 1 M " I ) * P t lM,+ll forks over 1MLI and M:+,

b$:[c,, a,, * * * 9 a,:)], W s'c")z)$:(~, a,, - * 9 am(,))

$&; am * - * 9 am(,)) 1 ~P(c:, U I )


As K > KO we can assume m(i) = m, k(i) = k, 1, = l,(i) =

I#,( lpq1 ; a09 . ., am)l, 1% = ldi) = l#;(Ma(f); . + 3 am)l (for eve^ i). Let

Trivially cp, ie an equivalence relation and as #,(ICaI ; a,, . . . , am) G 1 M;?, + , ) I , $ 1 ME?,)] we can find 7i, satisfying (ii) (as in Theorem 3.1). By 6.4 we can go back to cE.

EXERCISE 6.1: Prove that if 6 E acl A, then for some Si E A and tp, cp(3; Si) = tp(6, A).

EXERCISE 6.2: Suppose c, E acl(A u 8) for some Si but c, + acl(A u (c,: j < i)), for i < K. Prove K < K(T).

EXERCISE 6.3: Prove that in 6.13 the restriction on K cannot in general be weakened.

QUESTION 6.4: Can we in 6.13 replace "K > H, regular" by "of K > Ho"?

CLAIM 6.14: Let T be stable. (1) (in CQ) If i3 E ad(A u B), tp,(A, B u C) does not fork over B, then

tp(a, A u acl B) k tp(a, A u B u C).

(2) 8-e B E C and (i) stp(6, A u B) I- tp(6, A u C), (ii) tp,(6 u A, 0) does not fork over B,

(iii) tp,(6 u A, B) i s stationary, then tp(6, A u B) k tp(6, A u C).

Proof. (1) As tp,(A, B u C) does not fork over B, by 6.7(5), tp,(A u a, B u C) does not fork over B; so by 6.3(3) tp,(A u 5, acl B u C) does not fork over B, hence over acl B. So tp,(C, acl B u A u i3) does not fork over acl B. But, by 6.9 tp,(C, acl B) is stationary hence by Exercise 2.2 and Lemma 6.3(4),

tp,(C, acl B u A) t- tp,(C, acl(B u A)) t- tp*(C, ac1 B u A u i3)

hence tp(a, A u ac1 B) k tp@, A u ac1 B u C) (see IV, Ax(V.1) for F", IV, 2.8).

OR. I n , 8 71 INSTABILITY 137

(2) We work in Qeq, 80 by 0.3(4), Exercise 2.2, tp(6, acl(A u B)) k stp(6, A u B) so by (i) (a) tp(6, acl(A u B)) k tp(6, A u C). By 6.14(1) and (ii)foreveryFEacl(A U B), tp(c,A U aclB) I- tp(6,A U C), hence: (8) tp,(acl(A U B), A U acl B) 1 tp,(acl(A u B), A u C). Now (a) and (8) imply: ( y ) tp(6, A u acl B ) t- tp(6, A u C). [If 6, realizes tp(6, A u acl B) let f, be an elementary mapping, f, r (A u acl B) is the identity, f(6) = 6,. Now let f extend f,, Dom f = 6 u acl(A u B), hence Range f = 6, u wl(A u B). Let g be the identity mapping on C. Clearly (f (A u acl B)) u g is an elementary mapping. Hence by @) (f acl(A u B)) u g is an elementary mapping, hence by (a) f u g is an elementmy mapping, 80 6, realizes tp(6, A u C).] As tp(6 u A, B) is stationary, by 6.3(4) tpJ6 u A, B) 1 tp( 6 u A, acl B)

hence (8) tp(6, A u B) k tp(6, A u acl B). By ( y ) and (8) tp(6, A u B) k tp(6, A u C), which is our desired conclusion.

III.7. Instability

DEFINITION 7.1 : (1) Let K s d ( T ) be the first infinite cardinal K for which T doea not have K independent orders by m-formulas: i.e., there are no formulas ~ ~ ( 3 ; &) (i < K ) l(3) = m and 4 (j < w) such that for every r) E

{Q'(z; $)U'"["r":j < w, i < K }

is consistent. (2) = Kkl(T).

Remark. The expression "ird" is an abbreviation for "independent orders".

Proof. Immediate.

THEOREM 7.2: P ( h ) 2 sup{(pX)+: p < Ded A, K < K&(T)} (we m- 8ume A 2 Kird(T), of course) (see DeJinition I I , 4.4).


Proof. Let p < Ded A, and Z an order 111 = p, with a dense suborder J, 131 = A. Let K < t&(T) and let tpr (i < K ) be the formulaa from Defini- tion 7.1. By compactneae define (8 E J ) so that for every r) E "I,

is consistent. Let A = U {g: i < K , 8 E J}, and p,, E q,, E Sm(A). Then IAI s h and

THEOREM 7.3: Led A and A be infinite set8 such that: (i) J8r(A)I 2 A > 214 + IAI, where h i e regu2ar. (ii) J8Tl(A)J < h for uZZA, E A , !All < Id!. (C) There i8 no S c &'(A) such that IS( 2 h and I{p tp: p s

IAI for every 'p E A . Thf?7& < K z d ( T ) .

Proof. Let A = {tp,(Z; 8): i < ldl}, and A, = {tp,: j < i}. We d e h e by induction on a a function F, from Q ( A ) to subseta of I A I :

In the definition a appears only once: in (Vfl < a); hence for y < a

PAP) 2 PAP). Now for some p E SZ(A), Inn<(u P,(p)l = Id I (in fact this holds for

a's much higher than w). For suppose there is no such p. As ISZ(A)I 2 A, h > 214, h regular, for some w, E ldl, n < w , S , = { p E S ~ ( A ) : P,(p) = w,foreveryn < w}hascardinality Th.Letd, = { t p , : i E ~ , < , w , } , so by assumption lA*I < ( A 1, hence by (ii) lST*(A)l < A. As h is regular, for some po E ST.(A),

has c d i n d i t y z h . Now for every t; < 141, 1{p 1 tp,: p ESa}l 5 IAI. We prove this by

induction on t;; if t; E nneop w,, p ES,, then tp, ELI* and p 'pt =

CH. III, 5 71 INSTABILITY 139

po rcpt. If for some n [$w, then by the definition of F,, and as (Vp, q E B,)(VI < w ) [ F I ( p ) = FI(q)] we get (lettinggA. = { p A*: p EBS}):

I{P r 'Pr: P E&}l

s 1 u { p vc: p E Sa and the number of q vr, for q E 8 2 , A * S A (

' lA*I<&

q Id* = p rd* i s s1~1}1

I 2 2 1{p 1 tpr: p ES,, p d* = r and the number of A * E A ( TESd.

lA*I <no q r ~ r , f o r q ~ & , q rd* = r ia ~1.41}1

I Idl.lSd*l.I4 I I4 as

IBAOI s n 18@,1 5 IAI'"'' = IAl cpiEA*

by the induction hypothesis, and Id1 < IAI as

2141 < h I ISd"(A)I 5 214+14.

So by S, we get a contradiction to hypothesis (iii) of the theorem. Hence for somep* €&(A), IwI = Id1 where w = on<, Fn(p*) .

By compactness it suffices to prove that for any i(0) < i(1) < i(2) < .. . < i ( n - 1) < ldI , i ( I )Ew,Z<nandk<othereare~~((I<n, j< k) such that for any 7 E "k, {tprcI,(P; Bj)ift*f112j): 2 < n, j < k} is consistent.

Denote d1 = {'pica,: u < 2). Now we define by induction on I I n for every 7 E I ( IA1+ ) a type

q,, E ST(A) such that (i) i ( O ) , . . . , i(n - 1) E P,+ -l(q,,); (ii) when v is an initial segment of 7, qv r At(") G q,,; (iii) i # j => qV-<() dl # qv-<,) d1 when I = I(v) + 1.

We let qo = p * ; and if q,, is defined, I(q) = I we can define q,,-<r)(i < IAI +) by the definition of B'n+l-I.

Let r,, = qn tdl('J)+l. For every q ~ " - l ( l A ] + ) , by Exercise I, 2.7(1) there are q,il (j < k) such that for every I < k there is an a such that rn u - aj,v *-l ) u ( j 2 1 ) : j < k} G Q , , - ( ~ ) . By 2.6 of the Appendix (the "pigeonhole" principle for n-trees) and renaming we can assume Ti?,;' = a3-l for every 7 E "-l(IAl+) (note that the number of possible %s q,iS[j .c k, 7 E IAI +)I, and continuing by downward induction we define all the izi (I < n , j < k) and clearly they satisfy our demands, hence we finish.

-n-1-. . . - 3 - 1 u ~ - ~ , , , is sIAI < IAI+). Now we choose similarly

140 OLOBALTHEORY [CH. 111, 8 7

CONCLUSION 7.4: If for 8ome A , IS(A)I 2 A, A regular, A > 2ITI + JAJITI and for p. < Ded,JAJ, peK < A then ~i, .d(T) > K .

Proof. By 7.1 we can assume T does not have the independence property Choose a d with minimal cardinality for which ISd(A)I 2 A. By 11, 4.10(2) when Idl! < K , IS,,(A)I 5 naed, ISo(A)I < A hence 2 K .

So our conclusion follows by 7.3 as (i), (ii) hold by the choice of d and the existence of S as mentioned in (iii) contradicts (AIITI < A.

THEOREM 7.6: Suppose T does not have the independence property, A c B, p ES"(A). Then there is C E B, ICl I IT1 and q ES"(B), 8wh that p c q, and q does not split over A u C .

Proof. For every C E B let

r(C) = {rp(Z; a) = v(Z; 6 ) : a, 6 E B, tp@, A u C) = tp(6, A u C)}.

It is clear that tp(E, B) does not split over A u C iff i3 realizes r(C). So it suffices to prove that for some C, C E B, ICl 5 IT1 and p u r(C) is consistent. Suppose there is no such C, and define by induction on a < IT!' sets C,E B, IC,l I IT]. Let C, = 0, and C, = Ut<aC,. Suppose C, is defined; so p u r(C,) is inconsistent, hence there are n(a) < W , af, 6f E: B and formulas qf for 1 < n(a) such that

(i) tp(af, A u C,) = tp(6f, A u CJ, (ii) p I- VI<,,(,) F~(z; sip) = l~f (~ ; 6:). Now let C,,, = U {a?-@: 1 < %(a)} u C,. So C, is defined for every

As the number of possible (n(a), . . . , vf, . . is s I TI, there are ci < IT!'.

n < w and v1(Z < n) such that

l{a: a < lTl+, %(a) = n, vf = v1}1 = ITI'.

So assume that n(k) = n, q$ = tpf for every k < w (by renaming). Let A = {vI: 1 < n}, and define p,, ES"(A u CI(,,J for q E to>2 by induction on l(q). Let p0 = p, and if p,, is defined 1(q) = k, let p,,-(,) be any extension of it in Sm(A u Ck+l)' So by (ii) for some 1 < n, yI(Z; a:) = +pI(Z; 6:) ~ p , , - ( , ) . Let ~ ~ ( 3 ; ~ p , , - ( , ) , then p,, u {TrpI(Z; a:)t} is consistent (by (i)) hence there is ES"(A U Ck+l), p,, _c p,,-(l), lvI(Z; ~ p , , - ( ~ ) . Now for every k < w let Ak = U { E { : j < k, I < n}; for q # v e k 2 , by the definition (p,, Id) rAk # ( p v Id) A,, hence IflT(Ak)I L 2k, and lAkl 5 ksm,, where m, = ZI<,, @!). so by 11, 2.1

OH. 1x1, 8 71 INSTABILITY 141

and 11, 4.11 some q(3; j i ) E A has the independence property, a contradiction.

DEFINITION 7.2: (1) Kedt(T) is the first regular cardinal K such that: there are no formulas q,(3; 8,) ( l ( Z ) = my i < K , i successor) numbers n, < w and sequences sin, r) E Iczw, for which

(i) p,, = {q@; a,,l,): i su.ccessor, i < K } is consistent for r) E Icw,

(ii) {q@, a < w, r) E "'w, i = l(7) + 1) is n,-contradictory, i.e., for w c w, IwI = n,, C7(3P)[l\, , , q , ( Z ,

(2) K r O d t ( T ) = e d t ( T ) .

Remark. The expression "cdt " stands for "contradictory types".

Proof. (1) Let 'p(z; j i ) have the strict order property. Choose q, =

~ ( z ; j i ~ ) A lq(z;jids ni = 2. (2) Because we can choose all the (p,% equal. (3) By adding dummy variables. (4) By 11, 3.6. (5) Clearly Krgdt(T) 5 K,(T); the converse holds by (4) and by 6.13.

THEOREM 7.7: The following conditions on T , m < w and a regular cardinal K are equivalent.

(1) K < Kedt(T) .

(2) K < K&(T), and moreover, in DeJinition 7.2(1) we can have n, = 2 for every i.

(3) For every A, A'" = A, there is a set A, IAI = A, and a set S of m-types over A which are contradictory in pairs, IS1 = A", where p E S *

(4) There is a set A , a set S of m-types over A and a cardinal x such that: (i) 181 > IAI'" + 21Tl+X, (ii) no sequence realizee > x types from 8, (iii) for every p E 8, 1231 5 x.

14 = K .

142 GLOBAL THEORY [m. m, § 7

Proof. ( 2 ) s- (1) Immediate. (1) s- (4), ( 2 ) =. ( 3 ) Easy, by compactness. (3) s- (4) Choose in (3) a A which is a strong limit cardinal > 21Tl+",

which hes oofinality K. Then the A and S from (3) will satisfy (4), for

(4) =. ( 2 ) Let w.1.o.g. IS1 = A + , where A = IAl'" + 21Tl+X, andB = {pi: i < A+}. We can mume the types in S am pairwise contradictory, for define by induction ui E A+ aa follows: u( is a maximal subset of A+ , for whioh j < i * zc, n uj = 0, and U {pa: a E u,} is consistent. By (4)(ii) luil 5 x ; so letting pi = U {pa: a E u,} we get what we want. So now let p , = {q',@;g): a < x}. As the number of sequences

(9;: a < x ) is 5 ITIx 5 2ITI+X I A we can assume tp; = rpa; and similarly that tp*(A') is fixed where A' = ua For i < A+ we define finite subseta u', of x by induction on a. If u; is defined for fl < a and there ia a hite u E x and there am distinct [(n) < A+(n < a) such that qQ) = 4 for j E UBea u;, $((A) = uj; and the formulaa Arm cp,(Z; a:((A)) am pairwise contradictory then we let u; = u. The first a for whioh we cannot define uI, will be denoted by a(i) . Clearly a(i) < x ; and it suffices to prove that for some f < A + , a([) 2 K. (Then the rpl of Definition 7.2 will be A {tpj:j E w f } remembering that tp,(A,) is fixed.)

So suppose for every f < A + , a(6) < K, and we shall get a contradiction. As there are xen I 2% < A possible sequences (&: a < a( i ) ) (as IAI'" I A < IS1 5 lAlx, K 5 x ) for some S1 s 8, ISll = A + , and p , E S ~ implies a(;) = a*, and u', = w, for a < a*. As IAI'" I, A, there is Sa s 8, suoh that IS21 = A + , and pi E S ~ implies % = aj for

Now let S, = {[(i): i < A+}. For i < j < A+, pcCO U pCw is contra- didory (we nofed it at the beginning of the proof), so there is a finite subset h( i , j ) of x suoh that

X = K.

j E U a < A

is contradictory. But A+ 2 (2X)+, hence by 2.6 of the Appendix for some S3 E Say IS3] 3: x , h has a fixed value, u. This shows that for any pi E fI3 we could have defined u;. = u; a contradiction.

THEOREM 7.8: T h e k a themy T , without the Btrict oraZerpoperty, for whkh K&(T) = a.

CH. 111, Q 71 INSTABILITY 143

Proof. The set of axioms of T will be as follows:

alaeees (1) E is an equivalence relation with infinitely many equivalence

W w E 4 ( V q ) ( z E y = YE%), ( V q z ) ( d y A yEz + xi%),

@x0, . . ., z,,) A lz i~z, for every n > 0. i < j s n

(2) For two distinct B-equivalence classes x, /E, x2/E, the family of sets

{z: z E q , zRy} for y E x,/E

forms a partition of x l /E

(VX?.l)(ZBY -+ -BY),

( V q ) ( i ~ E y -+ ~ Z ( Z E Z A ~ R z ) ,

(vZyZ)(Z&/ A ZR2 A 2 E y j - y = 2).

(3) These partitions are independent, moreover, for m s n < w,

(vzi, . * * , Xn)(vyl, - - 8 yn)(vz)[ (i\ T z ~ y i A i\ l z ~ z i 1-1 1 1 1

A - @ I E Y k i s i < k r n

A A 1x1 = x k ) Iernsk

z*Ez A A z*Ry, k a s n

m - 1

A A x~Rz* 1 - 1

A 1-m 7 x l R z * ) ] .

We leave to the reader to prove that T admits elimination of quantifiers is complete and satisfies our requirements.

THEOREM 7.9: There ie a theory T , and a type p ES(O), Such that for every cardinal A there M a set A , IAI = A, which sa.tisjEes: for every q E S(A) , p E q, and B E A , IRI < IAI, q aplita strongly over R.

144 GLOBALTHEORY [CH. 111, f 7

Remark. This shows the limitations on generalizing 1.3 to unstable theories.

Proof. A set of axioms for T will be:

has exactly two elements (1) E is an equivalence relation, on {z: P(z)}; each equivalence c l w

(Vz)[zEz = P(z)],

(vq)[zEY = yEz1,

(Vz)[P(s) + (31YM # Y A XjJY)l.

(Vz?dCP(z) v lP(Y) ---+ lZRY1,

(Vzyz)[~Ey A yEz --f zEz],

(2) When -4'(z), P(y) then R chooses an element from y/B,

(vZy)[iP(S) A P(y) + (3!Z)bEZ A ZRZ]]. (3) The choices in (2) are independent, i.e., for every w( l ) , . . . , w(n)

E (1,. . ., n}

It is emy to see that T has elimination of quantifiers, and when M is a model of T, A = P ( M ) , p = {lP(z)} satisfies our demands.

Proof. (1) We can use 7.4 and the independence proofs of set theory or prove like 11, 4.16.

CH. 111, $71 INSTABILITY 145

(2) As K&t( T) s Kcdt ( T), it suffices to prove by induction on m that Kr,”at(T) s K&t(T). For m = 1 it is trivial, so suppose we have proved it for m and K < KrtL l( T) is regular; we shall prove K < Kr,’dt( T). So there are formulas tp,,(y, Z; 2,) (Z(2) = m) (0 < a < K , a successor) and a,, ( r ) E K A ) as in Definition 7.2; and n, = 2, and {a,,: r ) e K’A} is an indiscernible tree (see Definitions VII, 2.4, VII, 3.1, VII, 3.2, Theorems VII, 3.6, VII, 3.6) and A > I TI is a strong limit cardinal of cofinality K .

Choose a maximal set w c K such that Fw is consistent, where rw = {tpa(yn, 2; ii,,,,,) 0 < a < K a successor, r) E %, and y + 1 # w 3

r)[y] = 0) (note that if w(i)(i < 6) is increasing, then F,,,, is increasing and Fv,w(,) is consistent iff U, F,,,, is). If IwI = K clearly {tp,,(y; Z, 2,,): a E w } proves our assertion. Otherwise let Po < K , (VaEw)(a < Po), and for every r) E KA let (8 = (0, . . . ), < ,,,,)

F“ = {tpa(yv-,,, 2; a(,,-,,)lu): 0 < a < K , a successor, v E flu,

[Y + 1 # w m d y < &I 5 4yl = O},

rn = . . . ) A p : p G Fn is finite}.

Now no sequence Z realizes > 2IT1 of the types 9. [Otherwise by 2.8 of the Appendix there are q, (2 < w) and y < K such that r), y = qo 1 y , and the q,[y] are distinct. Then w u {y + 1) contradicts the maximality of w.]

DEFINITION 7.3: (1) &,(T) is the first cardinal K , such that there are no formulas tp,(Z;&) (i < K ) (Z(2) = m) and natural numbers n,, and sequences iii(a < w) such that

(i) for any r ) E IEw, {tp,(f; iikLt1): i < K } is consistent, (ii) for all i < K {tp,( i t ; a:): a < w} is n,-contradictory.

(3) K r E p ( T) [Krinp( T)] is the first regular cardinal 2 K E ~ ( T) [Kin,( T)]. (2) Kinp(T) = SUP, .Ep(T)*

Remark. The notation “inp ” stands for “independent partitions ”.

DEFINITION 7.4: (1) Krrd( T) is the first cardinal K such that there are no formulas tp,(Z; &) (i < K ) (Z(x) = m) and sequences iii(a < w) such that

(i) for r ) E Icw {tpl(2; i i i) if(n[f12u): i < K , a < w } is consistent, (ii) for i < K , a < w , k(V2)[tpi(2; iii) +- tpl(%; a:,,)]. (2) Ksrd(T) = Krrd(T)*

Remark. The notation “srd ” means “strict independent order ”.

146 QLOBALTHEORY [CH. I n , 5 7

THEOREM 7.11: 8-e Kcat(T) = 00, K~,, (T) < 00 then there ie a fomzukc (~(2; #) and 8equenCes a,, (q E ,>w) such that (i) (~(2; a,,,,,): n < w }

i8 d S t e n t for each q E Ow (ii) if 7, v E are 4-~nconzparuble, then k(3z)[q(z; 4) A p(z;

Proof. By the assumption K&t(T) = 00, the proof of 7.6(2), 7.10(2), and 7.7(2), there are a formula cp(z; fj) and sequences a,, (7 E ,'w) such that

for k # I < w , 7 E ">o. By Definitions VII, 2.4, VII, 3.1 and Theorems VII, 3.6, VII, 3.6, we can assume

0) depends on the lexicographic order among the q,'s, and Z(qf), h(qf, 7,) = max{Z: qr I = qj t I). By Ramsey's theorem, the fact that we can lake any subset of the levels and compactness we can assume

(a) {~ (z ; a,,,,,): n < w } is consistent for each 7 E ,w,

W 3 z ) [ d z , %-u)) A ~ ( 2 ,

( y ) {an: q E O > W } is an indiscernible tree, i.e., tp(Znln. .

( 8 ) in (y ) only the order relations among the Z(qf), h(qf, 7,) matter. For 0 , . . . , 0 (k times) we write 0,. Now {~ (z ; licI,ol+l)): 2 < w } cannot

be consistent (as then {{(~(z;7i<~,~,,~>): k < w}: 2 < w } proves K,,,(T) = 00). Hence there is a maximal ra for which

{dz ; a < t , o l + 1 ) ) : < n} u { d z ; a<n,,,>): k > n}

is consistent. Hence there is m < w such that

{(P(z; ~ )}

u { d ~ ; a < n + l , o k > ) : n + 2 5 k < n + m + 2) is incorisistent. Now define, for every q E O>w

h(q) = (n, O n + l , q [OI, Om, ~ [ l ] , Om, - - - 9 7[J(7) - 1],Om>, -* - v*@; an 1 - A (P("; ~ < l , o ~ + ~ > ) A A ~(2; ah<q)-ol)-

I < n l c m

It is easy to check that: (1) For every q E ,'w, {v*(z; CZJ: 2 < w } is consistent (by the choice

of n). (2) If q, V E ' O ' W , l(q) < Z(v), q < I r v (lexicographic order) but not

q Q v , then Cl(3z)[q*(x; a:) A 'p*(z; $71 (remember that the indiscernibility applies to levels too, and the definitions of m and n).

So the 'p*(z; a:) satisfy "half" of our demand (ii) (and (i), of course). By compactness we can define (q E @>w) which satisfy (a), @), (y) , (6) and when q, v E @>a, Z(q) < Z(v), v < I Z q but not q 4 v then C1(3z)[v*(z; at) A v*(z; a:)] (by reversing order).

147 CH. 111, 8 71 INSTABILITY

Now repeat the process and we get tp** as required.

THEOREM 7.12: If T does not have the independence property, t k n

K,",d(T) = Kh(T)*

Proof. By definition, always KTrd(T) 5 K Z d ( T ) , so it suffices to prove K < K!&(T) assuming K < K$(T), and let tpi(Z; $) (i < K) exemplify the latter. We define by induction on 5 formdm #& gf) (i < 5 ) such that, letting

(*) (i)

there are LZ; (a < w , i < K) such that: for every q E ICw, UieK pici, is consistent where

p i = {fit(%; ak)uUra): a < w},

(ii) t,hf(%; G) I- #@; for i < 5, (iii) Clearly (*),, holds, and (*)r suffice and (*)a follows from (*)E ( 5 < 8). So suppose (*ky and we shall prove (*)t+l. As T does not have the independence property, for some n < w and finite w c n, and q E 5,

(J picf, u {vf(it; G ) u ( a e w ) : a < n}

% is inconsistent. So as in the proof of 11, 4.7(2), there are u, v c n, k c n such that (k + 1) E U - v, k E V - u, u - {k + 1) = w - {k}, and for every q E Kw

{?I!! : a < w } is an indiscernible sequence over U, # UB < a$.

u {tpf(Z; L Z ~ ) ~ ( ~ E V ) : a < n} f < K f +c

is consistent but for some p E ICw,

u pLCa u {vi(z; ak)if(aeu): a < w } l < K 1 # c

is inconsistent and we can replace u, < K , , + pLril by a finite subset q. We continue as in 11-4.7(2), conjuncting t,h with A q. We let t,hE = t,hy and

is satisfied because by the indiscernibility assumption for any k < w , we can assume

q G {tpi(z, GL)u(D(')"a): i # 5 , u < w ; but not Z(i) s a < I ( $ ) + k}.

148 QLOBAL THEORY [m. m, § 7

EXERCISE 7.1: Prove that K&,(T) 5 K'"(T), K&,(T) s e d t ( T ) . Show that even for stable T equality need not hold.

EXERCISE 7.2: Prove that when T does not have the independence property, em 2 e -a (T) .

EXERCISE 7.3: Suppose T does not have the independence property; and there are K cp,, $a, i < K , a < w such that (where Z(Z) = m) for any r) E =w, {cp,(Z; G)K(a*"[fl): i < K , a < w } is consistent. Prove K < K&,(T).

EXERCISE 7.4: Prove that in Defhition 7.3 we can w.1.o.g. take n, = 2.

QUESTION 7.5: Prove that q,,(T) = KE,(T) = KLJT).

QUESTION 7.6: IS Km(A) = K1(A)?

EXERCISE 7.7: Suppose A E By p €Sm(A). T does not have the independence property. Then there are q E Sm(B), q 2 p and C G By ICl < K&,(T) such that: if {a,,: n < w } is an indiscernible sequence over A u C, and a,, E By then cp(3; a,) E q e cp(3; a1) E q.

Remurk. Instead of "T does not have the independence property" we can add "where {cp(Z; a,,) = +Z; n < w } is k-contradictory for some k < w y y .

EXERCISE 7.8: Prove Theorem 3.8 by the,method of the proof of 7.3 (changing the definition of Pa).

QUESTION 7.9: Investigate K F ~ ( T ) . If T does not have the independence property is K ~ ~ ( T ) = K ~ ~ ( T ) , at least when both are regular cardinals?

PROBLEM 7.10: Characterize the possible functions R(A), and K(A, K ) = sup{ISI +: S is a set of 1-types, contradictory in pairs; p ES == 1p1 < K ; and the types in S are over some A, IAI 5 A}. (Possibility: Maybe the function A(A,@) should be used, where @ = (w,: i < K ) , w, a subset of {j: j < i} and k(A, a) = sup{IZ n +: Z E rc2A; Z closed under initial segments; and for i < K ,

I{7 w,: r) E Z } ~ I A} (remember r) is a function with domain {j:j < i}).}

CH. 111, f 71 INSTABILITY 149

EXERCISE 7.11: Suppose in P, in acl(A v a) there is I independent over A, assuming T is stable (see Definition 4.4). Prove K ~ ~ ~ ( T ) > IIl.

EXERCISE 7.12: There is a theory T without the strict order property, Kcdt(T) = 00, K ~ ~ ( T ) < 00. Hint: b t T C O r M b t Of

(1) P, & form a partition. (Vx)(P(z) = 4J(x)) . ( 2 ) R is a symmetric relation on &, E on P x &: Vz(lzRx),

P Y ) W Y = Y W , (VW)(ZRY --+a4 A & ( Y h (VW(z E Y --+ P(4 A

Q(Y) ) - ( 3 ) xRy implies 2, y are disjoint: (Vqz ) ( sRy A z E x -+ lz E y ) . (4) (Vxyz)(xBy A &(z) -+ xRz v yRz). ( 6 ) There is an x satisfying a quantifier free formula, except when

(1)-(4) prevent it, e.g.,

QUESTION 7.13: Can we in 7.10(1) replace K r by K ?

DEFINITION 7.6: Let K2ct(T) be the first cardinal K for which we cannot find an m-formula cp,(Z; ti,) (a < K ) and sequences a, (q E ='u) such that:

(i) For every q E {va(Z; a,,,): 0 < a < K , a a successor} is consistent.

(ii) If TI%, V E ~ , a,/3 < K are successors, and q, v are 4- incomparable then t7(3Z)(v,(Z; 8,) A v8(Z; By)).

We let ~ ~ ~ ~ ( t ) = sup,,, Krct(T), KeCt(T) [ K ~ ~ ~ ~ ( T ) ] be the first regular oardinal 2 K,",t( T ) [KSct( T ) ] .

QUESTION 7.14: Investigate .Fct( T ) , e.g., is Kict( T ) = KsCt( T ) is KrCdt(T) = -iIlp(T) + Kract(T)?

EXERCISE 7.15: Show that if Kctd(T) < 00, then 1.3 holds.

CEUTEB IV

PRIME MODELS

IV.0. Introduction

‘We give here an axiomatic way to construct prime models (Sections 1 and 3) and prove that some sakisfy the axioms (Section 2) and then give more informetion (mainly characterization (Section 4)) under stability assumptions. We first examine here the most common case.

So let T be a countable theory. We call M prime = (FMo-prime) over A if A c 1M1 and whenever A c INI, there is an elementmy mapping f: M + N, f A = the identity. So, in a sense, a prime model “contains” only what every model should contain. It is easy to see that the sequences algebraic over A should be in the prime model, and no other elements should be in it. But, aa we are considering elementary mappings, it is natural to look at the complete types over A which sequences from M realize. Clearly if M is prime over A, p E Sm(A) is realized in M ifF it is realized in every model N, A c 1471. Which me those types? Clearly. they include all the isolated types (p is isolated if for some tp(Z; a) ~ p , tp(Z; a) k p ) , and we prove (6.3) that for countable A the converse holds too. On the other hand ifa E A, C(3z)tp(z, a), there should be an element c realizing tp@, 7i) in M. So we prove (6.10) that a necessary condition for the existence of prime models over every A, is that for any A, E A, and consistent tp(z, a) there is an isolated q, tp(z, a) E q E &A). On the other hand this is sufficient, for then we can define inductively cI (i c ao) such that tp(c,, A u {c,: j < i}) is isolated, and A u {q: i < ao} is the universe of a model M (such a model is called primary and the sequence a construction). If A c INI, we can easily define f : M + N, by defining f(ci) inductively. Naturally after proving existence, we want to prove uniqueness. For the primary model it is not too difficult ; but for the prime model it fails in general (see [Sh 79al).

Here we come again to stability. If T is N,-stable, we can prove that over every A a prime model exists (by 2.16(4), 2.17, 3.12) and charac-

150

OH. Ivy 5 01 INTRODUCTION 151

terize the prime model by cclocal ” conditions, and prove its uniqueness ( M is prime over A iff A s [MI, M is atomic over A (i.e., for each E E ]MI, tp(E, A) is isolated) and there is no uncountable set z M indiscernible over A). For superstable T, we cannot prove the existence of prime models, but if one always exists we can characterize it similarly (see 4.18). For stable T we cannot prove existence nor (natural) characterization, but when prime models always exist (or even if over A there is a primary model) the prime model is unique (6.6).

Those uniqueness theorems reprove the uniqueness of the algebraic closure of a field (which is a trivial application as this was the classical example model theorists had in mind), but also prove the uniqueness of the differential closure of a differential field (for characteristic zero we have an KO-stable theory, for other characteristics we have a stable theory). However the algebraic closure has an additional property: minimdity. We call M minimal over A if A c 1M1 and A c IN1 G implies N = M . In general the prime model need not be minimal (e.g., when T is the theory of infinite models, with the equality sign only). However for KO-stable T, a minimal model N over A is necessarily prime over A; and a prime model over A is minimal iff in M there is no infinite indiscernible set over A.

Now a natural generalization of “prime model” is “model prime among the A-saturated or A-compact models ”, with suitable generalizations of isolated types. However for stable T, the notion E-saturated ( =every type almost over a set of cardinality < A is realized) is better than A-saturated. More exactly: for A > KO we get the same saturation, but the definition of isolated satisfies more axioms, insuring existence when A 2 K( T) and helping to prove uniqueness. A fourth example is the following construction, analogous to the construction of primary models: if T is stable and countable, for every A there is My A G ]MI, such that for every Z E IMI, Q E L there is #(it; 6) E tp(Z, A), #(E; 6) 1 tp,(E, A ) (this is helpful for two cardinal theorems for stable theories, see V, Section 6).

Trying to develop each of these cases separately will cause much repetition, so some general setting is needed. We suggest here one, letting F be the set of pairs ( p , B) where p E S ~ ( A ) , B E A, p is "isolated" over B in some proper sense (the main example was F& where FA = { ( p , B): there is q E p 1 B, IpI + IBI < A, q t-p}). So an F-construction over A is (A, (Zf: i c a), (B, : i < a)), where (tp(Zi, A u {E,:j < i}), Bf) E F; and we define naturally F-saturated, -primary, -atomic.

In Section 1 we list the axioms, and give some basic definitions and

152 PRIME MODELS [a. w, § 1

trivial lemmas. We do not claim this set to be dehitive, it is enough for us if it is quite useful; and it covers more cases than it was originally tailored for. More exactly, though we have “ p is isolated over By’ in mind, what we get is applicable to “p is definable over B ”. So in Section 2 we prove that certain F’s satisfy most axioms (sometimes under additional hypotheses). In addition to the four previous cmes, we deal with F{ = {(p, B): p does not fork over B} for stable T used in 6.15 and VIII, 2.7. In general, we prove that existence axioms hold using various stability assumptions. We sum up our results in a table.

In Section 3, we investigate the conclusions we can draw from the axioms, mentioning each time exactly on which axioms we rely (this is important aa some of the examples fail to satisfy some of the axioms). We shall mention here only the uniqueness of the F-primary model and its F-atomicity.

In Section 4 we concentrate on stable theories and prove characterization and uniqueness theorems. The way is to find some properties of prime models, proved via their being included in primary models; and show that any two models having these properties, are isomorphic, e.g., for Fff-primeness we prove it when cf h r: K( T); partitioning it to two cases (cf h > KO, cf h = KO) covered by two %heoreme with quite different proofs. (For cf h = KO, K( T) = No, so we oan use induction on ranks; for cf h > KO we use a different argument which fails when cf h = KO.) An important technical lemma is that if p E #“(A) is FA- isolated T stable, p does not fork over B G A then p B is Fi-isolated too. The case h 2 K( 2’) > of h remains obscure (we do not know even if Fi-primary models are Fi-atomic). The last section, 6, deals with various theorems. The most important theorem is the uniqueness theorem. In the characterization theorems we demand mainly cf h 2 K(T), in 6.6 for regular h this can be replaced by A+ 2 K( T). We use F{ to construct models with an absolute indiscernible set (if A 2 p, T stable in p, then there is I G !MI, 111 = 11M11 = A, I absolutely indiscernible) and investigate the possibilities of omitting types (e.g., if I TI = h is regular, p , E D(T) are not Fi-isolated, (i < i(0) < 2A) then T has a model omitting every pi (see 6.17)).

IV.1. The set of axioms

We give here a list of axioms for the concept of primeness, and mention some obvious connections. We will not msume all the axioms hold

OH. I v , 8 13 THE SET OF AXIOMS 153

all the time, but specify each time the axioms used in proving a certain daim .

F will denote a family of pairs (p, A) where p ES"(A') (m < w ) for some A', A E A'. We assume that F is closed under changing the names of the variables and their order, that is, if p ES"(B), A E B, and q = {v(yo, . . . , ym- 1; a): P)(z~(~) , . . . , T i ) E p } (u a permutation of (0, . . . , m - 1)) then (p, A) E F o (q, A) E F. We let A(F) [&(F)] be the first [regular] cardinal A (if it exists), such that (p, A) E F =- IAI < A. For any A let FA = {(p, A ) : IAI < A, ( p , A) E F}. We call a type p F-isolated over A if (p, A) E F; p is F-isolated if there is A such that

Let p E F(B) mean (p, B) E F. In order to get an intuitive feeling for the following axioms think of (p, B) E F as meaning that the type p is isolated (in'the usual sense) over B.

( p , A ) E F.

The A x i m Ismnorphiern. Ax(1): F is mapped onto itself by any automorphism of Q, i.e., for any pair (p, A) and automorphism F of (E, (p, A) E F e (P(P) , W)) E F.

Trivially Ieolated Type0

Ax(11.1): If si E B E A, IBI < A(F), p = tp(Z, A), then (p, B) E F. Ax(II.2): If B G A, Ti is definable over B, IBI < A(F), p = tp(si, A), then (p, B) E F. Ax(II.3): If si is algebraic over A , p = tp(Ti, A), then (p, B) E F for some W t e B c A . Ax(II.4): If E E B C A, q(Z; 8) E p = tp(G, A), IBI < A(F), then ( p , B) E F.

Monotonicity

Ax(III.1): If B c A E Dom p, (p, B) E F, then p A E F(B). Ax(III.2): If B E A E Domp, IAI < A(F), (p, B) E F then (p, A) EF.

Ax(1V): If Q = tp(si-6, A), p = tp(si, A), B c A and (q, B) EF then

Notation. If p = tp*(C, A), B c A, then we say (p, B) E F when for every E E C, tp(8, A) E F(B). We say tp*(C, A) is F-isolated if tp(E, A) is F-isolated for every E E C. Note that when Ax(1V) holds, tp*(E, A) E F(B) iff tp(E, A) E F(B).

( p , B ) E F.

154 P-1 MODELS [OH. Iv, 5 1

8y?ndry and Transitivity

Ax(V.1): If q = tp(6-6, A), p = tp(6, A u 6) and B 5 A, and (q, B) E F then (p, B u 6) E F. Ax(V.2): I f ICl < h(F), q = tp*(6 u C, A), p = tp(6, A u C), B c A and (q, B) E F then (p, B u C) E F. Ax(V1): If By C c A, tp(6, A u a) E F(C) and tp(a, A) E F(B) then tp(a, A u 6) E F(B). Ax(VI1): If B E A, tp(6, A u C) E F(B u C), and tp*(C, A) E F(B), then tp& u C, A) E F(B).

Continwit y

Ax(VII1): If A, (i < 8 ) is increasing with i, ~ES~(U,<, ,A, ) and p 1 A, E F(B) for i < 6, then p E F(B). Ax(1X): If A,, B, (i < 6) are increasing with i, 6 < cf A(F), p~ sm(utcd and p 1 E F(Bf), then ( p , ul<d B,) F*

Eietence

Ax(X.1): If 6 E A, k(3z)(p(z; a), then for some pair ( p , B) E F, B c A,

Ax(X.2): I f A, c A, for i < j 5 a, 6 E A,, k(3z)(p(z; a), then for some p ES(A,) , (p(z; 6) ~ p , and p r A, is F-isolated for every i s a. Ax(XI.1): If (p, B) E F, p E @(A), A c C, then there are B c C, and q E Sm(C) such that p E q, (9, B') E F. Ax(XI.2): If A, E A, for i < j 5 a and p E S ~ ( A , ) is F-isolated, then for some q EB"(A,) p c q, and q r A, is F-isolated for every i I a.

(p(z; 6) E p E S(A).

Generalized Transitivity

Ax(XI1): If C,, Ca E B E A, p E Srn(A), ( p then (p, C,) E F.

B, C,) E F, and ( p , C,) E F,

Remurk. We make no use of Ax(XI1) and use Ax(X.2) and (XI.2) rarely.

LEMMA 1.1: (1) Ax(II.4) implies Ax(II.3) and Ax(II.2); Ax(II.2) implies Ax(11.1). If A(F) = No, Ax(X.1) impliefl Ax(II.3).

(2) Ax(V.2) and (IV) imply Ax(V.1). (3) Ax(II.1) and Ax(VI1) imply Ax(1V). (4) If aX(V.l), (IX) Itold, then Ax(V.2) lsoIde for ICl < cf h(F) 80 it

holds when h(F) is regular.

m. IV, 8 13 THE SET OF AXIOMS 155

( 6 ) Suppose Ax(X.1) h0ld.a f o r jnite A. Then Ax(XI.l) implies

( 6 ) Ax(X.2) implies Ax(X.1); and Ax(XI.2) implie8 Ax(XI.1). (7) If Ax(1X) hot%, then Ax(XI.l) implies that Ax(XI.2) h o b f o r

Ax(X.l), and Ax(XI.2) implies Ax(X.2).

a < cf h(F).

Remark. Except for trivial cases Ax(VI1) implies Ax(IV) (as we may take C G B assuming Ax(II.1)).

Proof. (3) Apply Ax(VI1) with B, A, an6,0 for B, A , a, C.

DEFINITION 1.1: (1) A set A is called ( F , p)-saturated if for every (p, B) E F , Dom p c A, lDom pI < p, p is realized by some sequence from A.

(2) p ( F ) is the first cardinal p such that for every h 2 p and A ; A is (F, p)-saturated iff A is ( F , A) saturated (if there is no such cardinal, p ( F ) = co O f course). (3) A is F-semi-8c~tUTatd if it is ( F , lAl)-saturated. (4) A is F-saturated if it is ( F , IAI +)-saturated (or equivalently,

( F , p(F))-saturated; or (F, p)-saturated for every p).

LEMMA 1.2: (1) Assume Ax(1V) and (V.1). A is (F,p)-saturate& if wknpES(A'),A' E A,IA'I < p , B s A',@, B)~F,pierecclizedinA.

(2) Assume Ax(VIII), (111.1). ( p , B) E Fi,fforevery$niteC G Dom p ,

(3 ) When p < h I 00, ( F , &saturation impliea ( F , p)-ttaturati~n. (4) When p 2 p ( F ) , ( F , p)-saturation ie equivalent to F-saturation. ( 6 ) Every F-saturated set is F-semi-saturated. ( 6 ) Assume Axiom (X.1). Then every ( F , X,)-saturated set is a model.

P t ( B u C) E F(B).

Proof. Immediate. The difference between (1) and the definition is that here p is a one-type, and not an arbitrary m-type. This is where Ax(1V) comes in. (2) is proved by induction on lDom pI .

DEFINITION 1.2: (1) An F-mtruction is a triple

.d = (A , {a,: i < a}, (B,: i < a))

such that pi(&) = tp(a,, U {a,: j < i} U A ) E F(B,). (2) U, = U,( .d) = {j < i: (B, n a,) - ( A U U {ZB: f l < j}) # 0).

156 PRIME MODELS [CH. Ivy 5 1

A set U E u is close& (i.e., &-closed) if (Vj E U)(U, E V ) . U: = U~(d)isthesmallestclosedsetcontainingUf-B,* = B,u(B,uZ,: j c U,"}. A set B of elements is closed if for some closed U c u, UjSu 7i;, E B E UjSu a, u A , and j E U implies B, E B.

( 3 ) F o r U c ( j : j < u } l e t d ( U ) = U ( S i , : i ~ U } u A a n d D o m d = d(4.

(4) When it is clear what d is, we omit it. We denote 4 0 ) by A ; we also sometimes ignore (B,: i < u) in

defining d. (5) Instead of a, we can use any well ordered set I, and in this case

al(1) ( = after last) will be an imaginary element of I which comes after all its (real) elements. In this caae d ( s ) will mean &((t: t < 8, t E I)); so Dom d = d(al(1)).

DEFINITION 1.3: C, is F-constructible over A , if for some F-construction d, d ( 0 ) = A,, Dom d = Co.

DEFINITION 1.4: (1) If C is F-constructible over A , and C is F-saturated [(F, p)-saturated] then we say C is F-primary [(F, p)-primary] over A.

(2) C is F-primitive [(F, p)-primitive] over A if A E C and for every C', such that A E C' and C' is F-saturated [(F, p)-saturated] there is an elementary mapping f, from C into C', where f 1 A is the identity.

(3) C is F-prime over A if it is F-primitive over A and F-saturated, C is (F, p)-prime over A if it is (F, p)-primitive over A and (F, p)- saturated.

(4) If in any of the above C is the universe of a model M, we say M is an F-primary [ - . -1 model. The same holds for Definition 1.1.

LEMMA 1.3: Assume Ax(IV), (V.1). (1) If C is F-constructible over A, then C has a F-construction over A in

(2) If for j < a, A j + l is F-mtrmtible [pimitive] over A,, and for limit 6 I u, A , = ul<, A,, then A, is F-constructible [primitive] over A,.

(3) If for 0 < j I u, A , is F-prime [primary] over u, < , A,, then A, is F-prime [primary] over A,.

(4) A is F-primitive over A, and F-constructible over A. If A is (F, p)- saturated, it is also (F, p)-primary and (F, &prime over A.

which l ( 4 ) = 1.

Proof. Immediate.

a. IVY 5 21 EXAMPLES OF F 157

LEMMA 1.4: (1) If A(F) is regular, then in De$ndtdon 1.2 A(F) > IB:I + IUfl. So if in addition Ax(III.2) 7rOl&B; then w.1.o.g. we can a88ume

U: = U,, B: = B,. (2) Asclume Ax(III.l). If B ie F-mtmt ib l e over A , C E B, then there

ie C' c By C c c' euck that A V c' is F-conetructible over A , and

l q < ICl' + w 9 .

DEFINITION 1.6: We say A is F-atomic over B if B c A, and for every a E A, tp(8, B) is F-isolated.

QUESTION 1.1: Are there any more connections between the axioms ?

DEFINIRON 1.6: For a class K of models, we say "M is prime over A for K (or among the members of 9) if M E K, and A c N E K implies there is an elementary embedding of bl into N, f A = the identity; let K-prime stand for "prime among the tc-compact models of T. Similarly for "primitive for K".

IV.2. Examples of F's

Here we present some examples of F's, and show which axioms they satisfy. For F = Fy,, we get the usual concept of primeness.

DERTNIITON 2.1 : (1) Let p be an m-type, or even a type with infinitely many free variables.

(i) p is a E"t,-tYpe if 1p1 < A. (ii) p is a PA-type if lDomp1 < A. (iii) p is a Fi-type ifp is almost over some B of cardinality < A. (2) For 2 E (8, t } = { ( p , B): for some A, my q; B c A , p ES"'(A),

I BI < A, q an Fg-type over By q E p , q k p ] . FR is defined similarly, but with q almost over B.

(3) M is Fg-compact if every c-type over ldll is realized in M.

Remark. For understanding Ff, let us for a moment work in aeq, and define card* A = min{l BI : B E acl A = acl B}. Then Fi is defined as F; replacing IAI c A by card* A c A.

DEFINITION 2.2: Fj; = {(p, B): for some A, m , p E S ~ ( A ) , B G A , IBI < A and p does not fork over B}.

DEFINITION 2.3: F i = { ( p , B): for every 'p there is q* E p I By Iqo(pI A,

158 PRIME MODELS [OH. Ivy 8 2

such that qq t -p I+ cp =def{tp(Z;a): p(Z ;a )~p} and IBI 5 h + IT]; c f h > IT1 * IBI < A}

Note that p 1 +p differs from p p in that the latter also contains formulas of the form +Z; a).

In the above deftnitions the letters t , a, a, f, and 1 stand for type, set, almost, forking, and local, respectively. To be precise we should say “Ff in CE” or write Fg[Q] for all X E

{a,f , 1, a, t } , that is the definitions depend on the model Q. We use the square brackets to differentiate this from F:(IQI) = {p: (p, IQI) E F ~ } . We use Ff only when T is assumed to be stable.

We use the following lemma freely in dealing with Ft.

LEMMA 2.1: Let B G A , p = tp(Z, A). Then (4) * (1) o (2) o (3). If T is stable all the conditions are equivalent.

(1) ( p , B) EF:, i.e., there is q E p , q t-p, q almost Over B. (2) For a m type q almat Over B, q t- p . (3) StP@, B ) t - P . (4) stp(Z, B) k stp(Z, A).

Proof. Trivially (1) =r (2). By theremark to Definition III,2.1(2) =- (3). Suppose T is stable, (3) holds and F’ realizes stp(F,B), by 111, 2.6(1) stp(F,B) does not fork over B, so by (3) (and 111, 1.18) also p does not fork over B, but F’ realizes p (by (3)); hence, by 11, 2.6(1) stp(?,A) does not fork over B, so by 111, 2.9 stp(F,A) = stp(C’,A). By the choice of F’, (4) holds. Hence by 111, 2.9 stp(F,A) = stp(F’,A); so (4) holds. Letusassurne(4)or(3)andweprove(l).Letq = {F(Z, G):rp(Z, a) ~ p ,

p(Z, a) is almost over B}. It suffices to prove q t-p. For any O(Z; a) ~ p , by (4) or (3) there is E E FEm(B) such that E(Z, a) t- O(Z; a) (remember that by 111, 2.2(1) FEm(B) is closed under conjunctions). Let #(Z) =

(Vjj)[E(Z, fj) --f O(g; a)]. Clearly the parameters of # are from A ; also E(Z; 2) t- #(Z) E #(a) hence # is almost over B. As E(Z, Z) k O(5; a), C#[Z]. As #(Z) is over A, #(Z) ~p and as #(Z) is almost over By #(Z) E q. From #’a definition it follows that #(Z) lo(?.; a) (as E(3, Z)) hence q I- O(3; a). So q t p and we finish.

In the following lemma we indicate many trivial facts concerning the Fg, their relation to previous notions, and the connections among the various Fg.

m. IV, 5 21 EXAMPLES OF F 159

LEMMA 2.2: (1) A is F&-sdurded if it is the universe of a model; if Ax(X.1) hot%, ale0 the converse is true.

(2) We get equivalent definitim of Fa,, FL, if we assume q consists of one formula and in all me8 (in Definition 2.1(2), 2.2 and 2.3) we can a%&um thd q i8 c h e d under conjunction8 (for FE, by 111, 2.2(1)).

(3) For x = t , 8, a, f, h(Fg) = A, and h(Fi) = A + + IT1 + when of h I ITI, and h(Fi) = h othemuiee.

A+ + ]TI+ and when of h > IT[, p ( F i ) s h (for F! w e theproof of (7); for FL we 111, 1.3,III, 1.4). (Bee Exercise 2.2.)

8, a, [t], x &J Fz-8durated ifl [v] &f is Ff-compact.

(4) AqJ s A, A%) p(F3 + XI, p ( F 0 = a, p(Fi) 5

(6) A is F ~ - 8 d U T d e d if for 80112e h - m w t Af, lMl = A. FW x =

(6) A i8 F1,-8dUrded iff for some h-SdUrded Af, 1M1 = A. (7) FOP h > No, A is Fi-saturated iff fOr 80me h-8dUrded Af, lMl = A. (8) If h > I TI, Ff, = q, and if cf h > I TI , Ff, = FA. If h > I TI, p is

(9) If A s K , x = 8, t , a, f, 1 then Ff E F. F,0-iSO&C?d iff it i8 pA- lkOla ted .

(10) Fi 2 FA E Fi c Fi E FL (the last in~luSim for 8tUbk T, Of

course).

A s lal, p E Bm(A, (E), B c A).

l+mn&wtion in QBq. Hence, for A, C c 161:

in geq, also for ecl C,

F;-saturated.

(11) For x = t , 8, a , f , 1, ( p , B) E Ff in (E iff (p", B ) E Fg in CQ (where

(12) For x = t , 8, a, f, 1, if a? is an E-comtructim in Q, t h it is an

(i) If C ie e-constructible (primitive) over A in Q, then this hold% also

(ii) For x = s, a , if C is F;-saturated then in aeq acl(C) is

(iii) 8imilarly for Ff-primary (prime). (iv) If A E 6 and in aeq acl(A) is Fi-saturated and A = acl A n 6,

then in 6 , A is F:-saturated (z = t , s , a). (v) If B E A E a, A is F;-atomic over B in 6 iff this holds in

tieq, i f this holds when we replace A andlor B by their algebraic closure. (vi) Let x = s,a. If B E A E 6, and in 6"" A is F:-primitive over

B, then in a A is F,"-primitive over B. (vii) If B E M E 6, and in 6"" M"q is F:-prime over B, then in 6

M is F;-prime over B. (viii) A E 6 A = acl A in a. Then A is Fi-compact i# acl A (in 6"")

is Fi-compact.

8tuble 2"). ( B y 111, 2.14.) (13) Ifp€E"'(A),pEF,O(B),I, rBiss ta t ionary thenp€F~(B) (for

Proof. (7) One direction is trivial. Now assume M is A-saturated, p is almost over B c 1M1, IBI c A. We must show that p is realized in M . Let a realize p. Define IN1 (i 5 w ) such that at realizes tp@, B u U,<# a,). Then Bo realizes p . (The proof is like the proof for stable T, in III,2.13( 1). For y E p let E be the equivalence relation over B that p depends on. Now since E has finitely many classes there is i such that Z, and are in the same class. Thus at satisfies B(Z; 7i) and so does ao).

(12) Notice that all the F’s satisfy Ax(II.3), hence acl(A) is F- constructible over A.

Remark. Notice that although F&-saturation and F;,-saturation are the same, not necessarily F;, = F:, (see Exercise 2.2(4)).

Now we check the axioms one by one. In first reading you can omit F;, F:.

LEMMA 2.3: A d m (I) i8 8di8jitd by Ff for x = t , 8 , a, f, 2.

Proof. Immediate.

LEBfMA 2.4: (1) A & m (1I.j) j = 1, 2, 3,4 are 8db@d by x = 8, t , a, f, 1.

for

(2) Moreover, if a ie algebraic over B, B c A, IBI < h thm tP@, A ) E W B ) .

(2) By 2.1. Proof. (1) Immediate.

LEMMA 2.6: Adom (111.1) i8 8diSF by q, fOr 5 = t , 8, a , f , 1.

Proof. For x = t , 8,2 this iS immediate. For x = a this follows from 2.1(3), and for x = f this is immediate (see 111, lJ(6)).

LEMMA 2.6: AXim (111.2) i8 Sat&@& by Pf, fOr X = t , 8 , a, f, 1.

Proof. Immediate.

Proof. So let p = p(B) = tp(7i, A), q = q(3, g) = tp(7in6, A), B c A, and (q, B ) E R, and we should prove (p, B) E Ff. For x = f this holds

CH. Ivy 8 21 EXAMPLES OF F 161

by 111, l.l(f3). If x = t , 8, there is an Fz-type r = r(3, fj) over By r I- q. By 2.2(2) we can m8ume r is closed under conjunctions. So r' = { (3g) (p(B, fj; 8): (p(% fj; 8) E r} is an Fi-type over By lr'l = Irl and r' I- p . Hence (p, B) E FZ. The cases x = a, 1 are treated similarly (for x = a notice 111, 2 4 1)).

LEMMA 2.8: (1) Axiom (V.1) ie eatisfied by F;, for z = 8, t , a, f, 1. (2) Axiom (V.2) is satisfied by Fz for x = 8, a, f; and also for x = t ,

when A is regular. If we assume cf A > ICl it holds for x = t , 8, a, f, 1. (See Exercise 2.3,4.)

Proof. (1) Follows from (2), by 1.1(2) and 2.7. (2) Weassumep = tp(i3, A u C), q = tp,(si U C,A) , B G A , ICl < A

and (q, B) E F;. We must prove ( p , B u C ) E FZ. If x = f this follows from 111, 1.1(5), 4.14. If x = s,t[x = a]

then clearly there is a type r [almost] over B, such that r I- q, r = U { r F : B ~ a U 0, where rcF tp(c,,A) is an F$-type.

Let r' = {cp(E; E , 6): cp(B,,, Z6; 6) E r, 6 E B, 8 E C).

Now r' is a type [almost] over B U C , and r' I- p. If x = 8, a ( B u C( < h hence r' is an Fi-type, so we finish. If x = t , ICl < cf A, then lr'l = 1.1 < A and we finish. So let x = I , ICl < cf A, and let (p = cp(Z; Z).

Assume Z = (zo, . . . , zk - ,), then there are formulas (pl = (pl(Z; Zll; El,,) for 1 < n( = 2k) such that for all partitions {i,, . . . , i,} u {j,, . . . , j t } = k there is I < n such that ( p l ( f ; zil, . . . , z,,; z,,, . . . , z,J = cp(Z; zo, . . . , zk-,).

By assumption for every 1 < n and Z1 E C there is a type r&) c tp(anEl, B) such that r&) k tp(SinEl, A) t + pl.

Define r@,) = {8(z; E,, E ~ ) : 8(g8, B ~ , , z2) E r1(El)},

r = U {r;(8,): El EC, 1 < n}.

It is easy to check that Irl < A, r I- p I+ (p, so we finish the last case x = 1.

LEMMA 2.9: Axiom (VI ) is satisjied by F$ for x = t , s, f, if h is regular or c f h > I TI, also by Fi, and if T is stable, by Ff too.

Procf. We should prove that if B,C E A, (ql,C) EF; where q1 = tp(6, A U a), and (p, B) E F: where p = tp(8, A) then (p,, B) E F;, wherep, = tp(zi, A u 6). Let q = q1 r A .

162 PRIME MODBL8 [OH. Ivy 8 2

For x = f this is 111, 4.13. For x = s, t , we first claim that if Ti, realizes p , 6, realizes q, then

8, 6, realizes tp(an6, A). Because tp(Zl, A ) = tp(8, A) for some elementary mapping f, f 1 A = the identity and f (a,) = a; so w.1.o.g. Ti = 6,. AS C E A , (q, C) E F;, tp(6, A) I- tp(6, A u a), SO tp(iZ-6, A) = tp(Tin6,, A), so the claim is proved. Hence if 8, realizes p necesaarily tp(aln6, A ) = tp(a-6, A), SO tp(iZl, A u 6) = tp(& A u 6) = p,. SO p I- p,. But for some F;-type p' over B, p' E p , p' I- p , hence p' I- p1 hence ( p , , B) E q.

For 1: = a the proof is similar. We first show that if a,, 6, realize stp@, B), stp(6, C) respectively then tp(Ti-6, A) = tp(Siln6,, A). Hence stp(Z, B) I- tp(Ti, A U 6) so we finish by 2.1.

We are left with x = 1. Let ~ ( 5 3 ~ ; Z) be a formula. As in the proof of 2.8 we can find n < w and, for 1 < n, 'pI = cpI(Za; ?&, a,) such that for every E E A u 6, there are I < n and E' E 8, 8' E A such that hp(Za; 8) = (pI(Za, 6, 8'). As (ql,C) E F ~ there are types p i c q1 YC, p i = qi(zh),

{0&3, $): i < (p:(} be closed under conjunctions (by 2.2(2)) where i$ E C. For every 1 < n, i < lp:l let

p1 1' $1, [dl < where $1 = #I(%; zap 21) = FI(Za; 536, zI)* Let (I\ =

0&(Za, z<,i, 2,) = (vZ6)[ef,1(z5~ Z f , J -+ d z a o , QI. As (p, B) ~Ffh, there are types pf,l = pf,1(3a) c p 1 B such that

pf,r k p f + O&; and let p* = Uf,I pr,[ . So as in the case x = 2 we1assume hisregulasorcfh > ITI,wehaveIp*l < h;andofcoursep*sp r B . Now it is easy to check p* I- p1 r' v.

THEOREM 2.10: A d m (VII) is satisfied by Fg when 2 = s or x = a, f, T is stable, or x = t , I , h is regular.

Prmf. Let B c A , p = tp(si, A U C), p = tp*(C, A), r = tp& u C, A) and (p, B u C) E FRY (p, B) E F;. We must prove B) E F; for 8 E C where = tp(a-8, A).

CLAIM 2.11: For sets A , C, suppose for every 8 E C pi I- qe = tp(8, A), and p' I- p = tp(& A u C), p' is over A u C, (p ' , pi are closed under conjunctions). Define

?-& = {(3%d)[#(3a, 5, %d; 6') A t)(?t&, 4, E2)]:

El E y a E cy #(za; z y a, 6,) Ep', 0@Cy Ed; 6') E pied}.

Then ri,c I- ra,e = tp(ii-E, A) .

163 OH. IVY 8 21 EXdlldPLJES OB F

Proof of 2.11. Let 'p = 'p(Zay Z,, 6,) E T,,,, and we shall prove r;,E I- (p,

and tGs will suffice. By the hypothesis p' I-p and t'p[7i, 8,6,] hence p' I- 'p(Zdd, a, 6,) ~ p . Hence (as p' is closed under conjunctions) there exists # = 2, a, 6,) ~ p ' (we add dummy variables if necessary to make room for 2) such that a E C, 6, E A and #(Zay E, a, 6,) I- (p(ZaY E , 60). Hence b * [ Z , a, 6,] where 'p*(Za, 4, 6,) = (E6)[#(Za, Z,, Zd, 6,) +

I- 'p*(ga, Za; 6,) hence for some (p(Zad, $, 6011.

Hence (p*(Zay %a; 6,) E qa-d, SO

It is easy to check that @ a ~ %; 68) E d - d , F(Ed)(vZd)[O(Z& Z d ; 63) + (p*(Ze~ Z ~ Y 6 a ) I a

(33d)[$@tw %Y 4, 61) A @ a ~ Zdp 6311 I- CP(%F,, %Y 6 0 )

and that the antecedent of this implication belongs to &. So we finish.

Continuation of the poqf of 2.10. Clearly if p', qi are over B u C, B respectively (and w.1.o.g. C n A E A), then T $ , ~ is over B ; so the caae x = s is immediate. If x = t , then I C 1 < A as (p, B U C) E Fi and there are p', qi satisfying 2.11 such that 1 p' 1, I As h is regular in that case it follows that IrL,,l < h so we finish. The C= 2 = f follows by 111, 4.16. So now we deal with the m e 2 = I ; again we assume h is regular.

Let 2 E C, and 'p(Za, Za; E ) be a formula. By the hypothesis there is p' c p closed under conjunctions, lp'1 < A, p' is over B u C and p' I- p + 'p. Let p' = {#l(Z,,, E, a(i), 6:): i < a < A}, and 'pr(Zay Zd(:); 6*, 6:) = (VZ,J[#1(Z4y Z,, Zao, b:) -+ 'p(Z4, ZEy 6*)], for a(;) E C, 6: E B, 6* E A . By the hypothesis (q,-dcf), B) E Fi so for some &acn c qpdct, 1 By

< A.

IQ&l < A d-aco I- Qa-aco t' 9:. Let q;-&l) = {4.,(5,%(1); @*,I: j < .(i) < A}

(again wume it is closed under conjunctions), 6;,, E B. Let

e&(Z8~ za; gtj) = (3Zd(l))[#l(% %Y ZrZ(f)> 6:) A ei,j(3t~ Zd(f); 6:,j)1.

Let r& = {O&(z8, Z,; 6f.J : j c a(i) , i c a}. It is easy to check r;,, E r,,a is over B and of cardinality < A . It suffices to show T L , ~ k r5, I+ Q.

For let 'p(?i?,,, Za; 6*) E r,,,, (8* E A ) so k'p[iZ, E, 6*]. Hence 'p(Z,,, 2, b*) ~ p , hence for some i < a, #&,, E, a(i), 6:) I- 'p(Zay 8, 6*) hence

b(fia)[${(Z8Y q) 'p@aY E Y 6*)].

SO t'p:[~, a(i), 6*, 671, hence 'pr(~~, it&:); 6*, 6:) E q ~ - d ( { ) hence for

164 PXME MODELS [CH. Iv, 6 2

some j < a(i) 6:,,) I- &Zd, Zd(,); 6*, 6:). So by the definition of d

~f , j (% gd(f); @,j) A #f(% %Y zd(O; ' qP(zdY %; 6*) hence

(3zd(f>)t#f(Zd, % zd(f;, 6:) A ef,j(Zdt %d(f), 6f,j)1 I- dzdY 6*)1 hence

so we finish also the case x = 1.

d&(za, zd; 67,J I- d z d , zd; g*)

We are left with the case x = a. First we prove

LEMMA 2.12: Let T be atable (1) If B E A, tp(Ti, A) ~ q ( B ) , E ~ a c l l ( B U Ti)thtp(Ti^c', A)EF$(B). (2) Let B c A, tp(Ti, A) E Fa@) iff tp(Ti, ad A) E F"(clc1 B).

Proof. (1) Let O(z; 8, 6) be algebraic, 6 E B and C@, 8,6]. By 2.4( 1) for I$,, we can assume s(z; a, 6) I- tp@, B u a). Let e ( l q a, 6) = { E o , . . ., B,,}, Z0 = 5. By 111, 2.6(3), (1) there are Ti', 5; such that stp(TinZon - . . Z,,, B) = stp(Si' ELn. . .^$, B) and stp(Ti'nZLn- - - ^-' cn, A ) does not fork over B. So stp(Ti', A) = stp(8, A), (by 111, 2.6 and 111, 2.9) hence we can msume Ti' = Ti. So (EL,. . ., 5;) is a permutation of (EOy . . . , En), hence we ctln wume Ei = $. So tp(Si Z,, A) does not fork over B.

By 2.1 it suffices to prove stp(8"Zo, B) I- tp(Ti-Zo, A). If this is not true there is Ti* 8*, stp(Si*^Z*, B) = stp(Ti Zoy B), tp(Ti*^Z*, A) # tp(Ti E,, A). So necessarily tp(Si*Ê*, A) forks over B. Again we can assume Ti* = Ti, hence for some i E* = Ef, contradiction.

(2) Immediate, by 2.1 part (4), and 111, 6.3(4).

Conclusion of thproof of 2.10. We have to prove it for x = a. By 2.2(11), it suffices to prove it in P.

Let C' = acl(B u C), so by 2.12(1) tp*(C', A) EF$(B) and by 2.12(2) tp(Si, A u C') E F"(B U C') = F"(C'). stp*(C', B) I- tp(C', A). Now use 2.11 with p' = tp*(Ti, B u C'), qi a type almost over B, which is over A, and such that qi I- tp(E, A) (qi exists by 2.1). This proves the assertion.

LEMMA 2.13: A d m (VIII) i8 8athfied by

SO tp@, cl) I- tp(Ti, A u C'),

for x = t, 8, a, f.

Proof of 2.13. Let A, (i < 6) be increasing, p E~~(U,, , Af), pf = p 1 A, E F$(B). We must prove ( p , B) E F;. For x = f, it follows by 111, 1 .l( 7).

OH. Iv, 9 21 EXAMPLES OF F 165

For x = 8, t let q, be an Fg-type over By q, E p,, q, t p,. I fx = s ,p rBisanFi-type,p tBtq , tp ,hencep tBtU, , ,p , = p . If x = t , qo is an Fi-type, qo t po; and clearly q, G po, hencep, t q, t p,,

For x = a, by 2.1, if li realizes p, stp(8, B) t p, hence stp(li, B) I- p hence !lo I- Po Uf < d Pr = P.

hence (p, B) E F:.

LEMMA 2.14: Adom (IX) is satisfied by F: for x = t , 8, U, f; for 1, if 6 < Cf A, it8 ConclUsiOn hkh.

Proof. Let 6 < cf A,. A,, B, are increasing with i, p ES"'(U,<~ A,), and

For x = f, if p forks over U,<, B,, some finite p' c p forks over U,<, B,, so for somej < 6 p' is over A,, so by 111, 1.1(8) p A, forks over B,, contradiction.

For x = 8, t[a] there are Fg-types q, [almost] over B,, q, c p , q, I- p 1 A,. Then U, <, q, proves our conclusion. For x = I the proof is similar.

( p I A,, B,) E F$. w e must prove (PY Ul<d B,) E E.

DEFINITION 2.4: (1) Al(T) is the first cardinal A such that there are no A, (i < A) and p ES"'(U,<~ A,) such that i < j =-A, E A,, and p t A, + splits over A, (if there is such A, and a0 otherwise).

(2) A,(T) is the first cardinal A such that there are no A, and p,, E #"'(A,,) for q E h22 such that

(i) Pr-<o> z P,,-<1>3 A,-<o> = A,,,> for rl E (ii) if q is an initial segment of Y then p,, c p v . (3) A3(T) is the first cardinal A such that there are no m-types p,,,

(i) p,,-<o> v P,,-<~> is inconsistent for q E h>2, (ii) if q is an initial segment of Y then p,, c pv .

q E h > 2 such that

DEFINITION 2.6: (1) A1(T) [Aa(T)] i s the first cardinal A such that for every B c A, p ES"'(B) there is C c A, ICl < h and q ES*(A), such that p E q, and q does not split over B u C [and q

(2) A3(T) is the first cardinal A such that for every m-type p over A, there is an m-type q over A, 1q1 < A, and p U q t p' E #"'(A) for some p'.

( B U C ) t q].

THEOREM 2.16: (1) K ( T ) 5 Al(T) I Aa(T) < h,(T). (2) T is unstable in every p < 2"scT). (3) If T is stable then A,( T) 5 I TI + .

PILflldE MODBLS [a. IV, 8 2 166

(4) h3(T) = KO iff T is totally trammndental (see Exercise 2.23). (5) h’(T) 2 P ( T ) 2 h3(T).

Remark. If 2ITI* > 2ITl, then (2) implies (3) trivially.

Proof. (1) If T is stable K(T) 5 X,(T) by 111, 3.1, as strong splitting implies splitting. If T is unstable, it is an easy exercise. W e have, in fact, proved that h,(T) 2 ha( T) in I, 2.7. Now ha( T) 5 h3( T) is trivial.

(2) Let p < 2<WT); for some h < h3(T), p < 2A; taking the first such A, we have 2 < A 2 p < 2”. Hence there are p,,, q E h>2, as in Definition 2.4(3). For every q ~ ” > 2 choose finite B, so that (P, , -<~> 1 B,,) u (p,,n<1) 1 B,,) is inconsistent, and let B = (Jve*>S B,,. For Y E ”2, pv = U,<,, pvr, is consistent, so for some qv E Sm(B), pv u qv is consistent. Clearly Y # q == qv # q,,, hence ISm(B)I 1 2” > p 2

(3) Suppose T ie stable, &(T) > lTl+, so for h = lTl+, there am I), (q E h>2) aa in Definition 2.4(3). Let {A,: i < ITI} be a list of all finite sets of formulas in L. We define by induction on i 5 I TI sequences 7, E a(i) = 9,. For i = 0, or i limit ohoose in any acceptable way. For i + 1 choose q,+’ = q , h ~ ~ A ’ 2 so that P ( p , , , + l , d , , 2) is minimal. For Y = qTI, pvn<o), pVecl, contradict our construction and 11, 1.4(1).

(4) This is a restatement of 11, 3.1 and 3.2. (6) Trivial.

2<” 1 pi.

a(i) < IT1 + and i < j * a(i) < a( j ) , qi

THEOREM 2.16: (1) #(T) 5 h,(T) for 1 = 1, 2, 3. (2) Swppae A, E A, for i < j 5 a and p is an m-type over A,: (i) There ie q ES~(A,), p E q and B C A,, IBI < h,(T) 8wh t7mt

P ” q t @ n A d t-q IA,. (ii) There is qESm(A,) p E q and r E q, Irl < h3(T) swh t h t

p u r i F q t A , w h e r e r ; = r l a , . (iii) There are q ES~(A,) , p E q and B c A,, IBI < K(T) , and

c c q t B, Irl < K(T), m h that p u (r A,) hm no extdm in Sm(A,) which forks over B n A,.

(iv) There ia qE@(A,), p c q and B E A,, IBI < hl(T) m h that: p u q 1 ( B n A,) hm no extewion in P ( A , ) which qli ts over B n A,; for euch i.

Proof. (1) For 1 = 1 it is easy, and for 1 = 2, 3 it follows from (2) for a = 0.

OH. Iv , 8 21 IOXAMPLEB OF F 167

(2) As the proofs are similar, we prove only (ii). Suppose the conolusion fails, and we get a contradiction. We define for every sequence q of ones and zeroes, Z(q) < A3(T) by induction on I(q), cm ordinal P(q) 5 a and an m-type q, over ABtn,, such that Iq,l < Il(q)l + + KO and p u q, is consietent, and for i < p(q), p U (q, A,) has a unique extension in Bm(A,), and v 4 q =+ q, s q,.

For the empty q, q, = 0, B(q) = 0 and when l(q) is a limit ordinal 8 let p;, = U,<a q ~ l , ~ B(q) = U t e * 8(9 t i). If 9,s B(T) are defined, let B a be the first ordinal such that p u q, has at least two extensions in Bm(An). Such B exists, as we assume that the conclusion of (2)(ii) fails, and /3 r B(q) by the induction hypothesis. So for some 8, E An and cp,, p u q, u {cp,,$; 8Jt} is oonsistent for t E (0, 1). Define q,-(I, = q, u {cp,(z; a,)'), /3(qn(l)) = 8. Clearly all the induction hypotheses are satisfied. So the qqYs, 7 E (J (02: a < &(T)), contradict the definition of A,(T).

LEMMA 2.17: (1) If A r A3(T), t 7 m ediofi a&ma (X.l) and (XI.l), and if A r &(T), then eat ieh &o azbm (X.2) and (XI.2).

(2) If A r Aa(T), t h PA eatis@ axioms (X.l) and (XI.l) and if A r Aa(T), then PA satiojk4 ale0 &om9 (X.1) and (XI.2).

Proof. Immediate, by 2.16(2)(i), (ii) (for A r A,(T)) and Definition 2.6 (when A 2 A1(T)).

LEMMA 2.18: (1) Adoma (X.l), (XI.l), (X.2) and (XI.2) are eatkclie& by F(,,PAifA > I T I , T ~ z e m T G e t a b k i n 8 0 m e p < 2A.

(2) Adoma (X.l), (XI.l), (X.2) and (XI.2) are satiejkd by if A r K(T).

(3) A d m (X.l), (XI. I), (X.2) and (XI.2) are eatiefied by F{ if T io etable.

(4) Ax(X.1) ie eatisjied by Fi if T ie stable, A r (TI (e.g., by every countable stable T).

(6) A d m (X.l), (XI. l), (X.2) an& (XI.2) are eatiefid by FW, if T k totally tranecendmtal.

Proof. Part (1) follows by 2.15(2)(3), 2.16(1) and 2.17(1), (2). Part (2) follows by 2.16(2)(iii); part (3) by 111, 1.4. Part (6) follows by 2.15(4), 2.16 and 2.17(1). So we are left with (4).

Let p be an m-type over A, and {cp,(P; g,): i < ITI) a list of the formulas in L. Define by induotion on i < I TI formulaa $,(1; a,) (a, E A)

168 PRIME MODELS [OH. Iv , 5 2

such that p , = p u {$,(3; Z,): j < i} is consistent. If we have defined $,@, a,) for i < a < I TI, clearly p , is consistent also when a is a limit ordinal. Choose $,@; a,) so that Bm(p,+,, tp,, 2) is minimal. It is easy to check that for some q a ~ S ; ( A ) , P ~ + ~ I - ~ ~ . Hence plT, has a unique extension in Sm(A): U,< ITI q,. So Ax(X.1) (and a little more) holds.

LEMMA 2.19: When x = t , 8, I , f, F = Fsf, F 8ati8&? Ax(XI1). Also for x = U, T 8table thb holds.

Proof. For x = t , 8, for some Fg-type q E p C,, q I-p 1 B (as (P t B, Ci) E Fg) hence q I- P t Ca; but as ( p , Ca) E R, p 1 Ca 1 p , hence q b p , SO (p, Ca) ~Fsf. For x = u the proof is similar using 2.1(4). For x = f Clearly p does not fork over B (as C, E B, (p, 0 2 ) E Pi) and p t B does not fork over C,, hence by the transitivity of forking (111, 4.4) p does not fork over C,. We are left with x = I , so for every 'p, for some q E p 1 Ca, 191 < A, Q I-p 1' tp, it is also clew that p t C, kp IC,, hence p 1 C, 1 q, so for every $ E q there is a finite re E p 1 C,, re 1 $. Now r = u {rs: 1,4 E q} E p C,, and r I- q I- p 1 + 'p, so (p, C,) E Fi.

DEEWITION 2.8: Ff: = {(p, A): p E#"(B), where m < w , A E B, IAI < h and p does not split over A}.

LEMMA 2.20: (1) Ff: sat&$es Axbna (I), (11.1, 2, 3, 4), (111.1, 2), (IV), (VII), (VIII) urvd (IX).

(2) = A, p(FR) = a). (3) If h 2 hl(!P), then Ff: 8d i8 f i8 &onZ (X.l) , and if h 2 &(T) then

Ff satisfies axioms (X.2), (XLl), (XII.2) and every p € S " ( A ) is Ff- isolated.

(4) If T doee not have the independence property t7m P ( T ) 5 I TI + . Proof. (l), (2) are immediate, and (4) is a restatement of 111, 7.6. (3) is similar to 2.16.

DEFINITION 2.7: Fg = {(a, B): psSm(A) for some m < w, B E A , IBI < A, and there is q E p t B, 1q1 < X such that q has no extension in Bm(A) which forks over B}.

Remark. We can wume q is closed under conjunctions.

LEMMA 2.21: (1) 8di8fies uxbm8 (I), (11.1, 2, 3, 4), (111.1, 2), (IV), (V.1) an& (IX).

(2) If i8 regdur F{ 8di83U d80 (v.2).

CH. Iv , f 21 EXAMPLES OF F 169

I II.1 II.2 II. 3

II.4

III. 1 III.2 Iv v. 1 v.2 vl VII VIII Ix x . 1

x .2

XI. 1 XI.2 w

+ + + T stable + + + + + + + T attable + + + + + + + T stable + + + + + + + T St8ble + + + Skolem

+ + + T St8ble + + + Skolem

+ + + T stable + + + + + + + T stable + + + + + + + T 8t8bh + + + + + + + T a b l e + + + + Aregulfbr + + T stable x + Am& x + + T stable T stable A regular x X X

Areguler + Tatable TstableAmgular + ? + + + + T Stable x + ? + + + + T stable x + + + A 2 Aa(T) A 2 P(T) A 2 K(T) T a b l e T stable A 2 A1(T) A 2 K(T) Skolem

A 2 A,(T) A 2 Aa(T) A 2 K(T) T stable A 2 Al(T) A 2 K(T) Skolem

funations

funations

A 2 IT1 funations

funations x

A 2 P(T) A 2 P(T) A 2 K(T) T stable x X ? + A 2 A,(T) A 2 A,(T) A 2 K(T) T stable x X ? + + + T stable T stable + X t X

( 3 ) Let ~ ( 5 " ) < A , F i satisjies Ax(X.l) and (X.2). Then A is Fi-saturated ifs A is the universe of a A-compact model.

Proof. Easy.

We can sum up (most of) our results (see Table 1) which includes also results on F s which will be defined later. In each place we write a plus if the axiom is true for that F and a cross if there is a counter-example. In the remaining cases we give a sufficient condition for the axiom to hold or just put a question mark. On Fi see VII, 4.4, Exercise VII, 4.2. See also 4.4-0.

The exercises, usually, show that appropriate lemmas cannot be improved, and are ordered approximately in the same way. Whenever h is singular h = zfeX A, where A,, K < A, K = cf A.

EXERCILSIF 2.1: Show that in Lemma 2.1, for unstable T, not necessarily (3) e (4). [Hint: Let M be a model of Find (see 11, 4.8(2)). R ( z , y, z) = P(z) A P(y) A -,P(z) A [zEz = zEg], N = (IHl, P, R),

PRIMlO MODELS [OH. Iv , 8 2 170

T = Th(N) (so N < Q) , and choose a, c such that N k P(c) A ,P(a). Then B = 0, A = {a}, ;E = (c) form a counterexample.]

Deduce in (Ieq a counterexample to Ax(IV), Fi,.

EXERCISE 2.2: (1) Show that for every A for some T , p(FA) = p(F1) =

(2) Show that if FA satisfies “every FA-type over A has a FA-iSOht0d

(3) Show ,u(F:) = A , ,u(F:) 2 A and for A > KO, p(F,”) = A. (4) Show that for some stable T, p(F&) = 8,. (5) For every K , show that there is a F,-saturated, not 8,-compact

8 0 .

complete extension over A” then p(Fi) -< A.

model.

Remark. See Lemma 2.2(4). [Hint: (1) Take Tin* (see 11, 4.8(2)) choose A E I“( , ( A n P(q1 = ( A - P(Q)I = A,andlet T = Th(C,. . .,a,. . .)aeA.

(2) (3) By 4.10 and VIII, 4.8 for each regular p < A, T has a p- saturated model which is not A + -saturated.

(4) Let T = Th(’2,. . . , E n , . . .) when qEnv i f f q n = v n. (6) like (l).]

EXERCISE 2.3: For h singular, Ax(V.2) may fail for FA even for stable T (by 2.2(8), 2.8(2), A 5 IT1 of course). [Hint: Let Q, Pt (i < K ) be disjoint one-place predicates, Pf a (partial) one-place function, from Q onto P,. E t (a < A,) are independent equivalence relations on Pf (i < K ) .

For every G,EP,(Q), {Pi(%) = cf A &@): i < K } is realized in Q. Let C, E P,(Cr), a E Q(Q), P&) # ct, A = B = 0, C = {cf: i < K}, CZ = u.]

EXERCISE 2.4: Show that for A satisfying cf A 5 I TI, Ax(V.2) fail for Ff, (when of A > IT1 it holds by 2.2(8), 2.8(2).) [Hint: Let I be indis- mrnible, IIl = A, ~ E I , C = I - {a}, and B = A = 0. Remember h ( q ) = A+ + IT1 + by 2.2(3).]

QUESTION 2.5: Show that Ax(V1) may fail for Fi, when A is singulax CfA 5 pi .

EXERCISE 2.6: Show that for A singular Ax(VI1) may fail for FA, (notioe that by 2.2(8), h 5 12’1). [Hint: As in Exercise 2.3 but P,(a) = c,, B = 0, A = {at: a < &, i < K}, afr E Pi(&) and Efr(c,, a;) iff a = 8.1

QUESTION 2.7’: Show that for A singulax Ax(VI1) may fail for F:, with IT1 = K (=of A),

OH. IVY 8 21 EXAXPLES OF F 171

EXERCIBE 2.8: Show that Ax(VI1) may fail for Fa when T is unstable. [Hint: As in Exercise 2.1, and let b E R(Q), Q C - , E ( b , c, a). Let B = 0, A = {b}, c = {a}, B = (c}.]

EXERCILYE 2.9: Show that Ax(1X) may fail for Fi, when cf A 5 I TI, and we can choose I TI = cf A. [Hint: Let L( T) consist of equality only; af (i 5 A) be distinct elements. B, = A, = {a,: j < i} for i < h and P = tP(aA, Ul<A

EXERCIBE 2.10: Show that Ax(VII1) may fail for F!, when cf A I IT I, and we may choose 1'111 = cf A. [Hint: Let P, (i < K ) be infinite pairwise disjoint one-place predicates B, E P,(@), lBfl = A, 6 = K , A, = U,<, B,, a 4 u1 PdQ), P = tP@, UI W I

EXERCILYB 2.11: Show that Ax(X.1) (hence (X.2)) may fail for FZ x = t , 8, a even for stable T, A = ITI. [Hint: (1) z = t : T will have independent one-place predicates P, (i < A). A = 0, Q = (a: = x) contradicfs Ax(X.1).

(2) z = 8: P, (i < A) and Q will be disjoint one-place predicates, IP,(Q)) = 2, P, (i < A) one-place partial function from Q onto Pi, such that for every cf E P,(Q), {P,(x) = c, A &(%): i < A} is realized. Then A = ut<A pi@), cp = &(x) contradicts Ax(X.l) .

(3) x = a: the example of Exercise 2.3, with A instead of K , deleting the Ek's, A = (q: i < A}, Q = Q(x) . ]

EXERCIBE 2.12: Show that Ax(XI.1) (hence Ax(XI.2) may fail for FZ, x = t , 8, a, even for stable T, IT1 = A. [Hint: (1) x = t : Use T from Exercise 2.11(2), B = A = 0, C = Uieh P,(Q), p = tp(a, A) for any a E Q(Q)*

(2) Z = 8: asin 1. (3) x = a: T aa in Exercise 2.11(3), B = A = 0, C = {c,: i < A},

p = tp(a, A) for any a E Q(Q).]

EXERCISE 2.13: Show that Ax(X.l) may fail for Fi (IT1 = H,, OT T stable, but IT1 > A; in fact we can .choose a superstable T). [Hint: (1) Tin*, (2) Let T be TA+ but the counterexample is for Teq (for TA* see Exercise 11, 2.3, for "eq" see 111, Section 6). Clearly ~ ( 2 1 ~ 9 . ) = A + + , ITe'] = A + . Let I, (a < A + ) be the set of one-to-one functions from u into A, Z = U,Z,, and choose a, E Q for S E Z such that a$,at iff

172 PRIME MODELS [OH. Iv , 4 2

8 1 i = t 1 i; then A = {a8/Ei: 8 E I , i < A+} c P, 'p = P = ( z ) form a counterexample,

(3) Let A be regular

T = Th(N), M = (AA x 0 , . . ., P,,, . . ., . . ., Ev, . . . ) , , s~>h ,vs~A

where P, = {(p, n) E 1611: r) Q p} and (p , n)E,(r), m ) if€ for some a p t a = 7 1 a = v 1 a, p(a) = r)(a) # v(a) or p = r). The counterexample is for TW (assuming N < a): A = {(p, n)/Ev: v E AA, ( p , n) E 1N1, v # p}, and 'p = P=WI

EXERCISE 2.14: Show that Ax(X.2) may fail for Fi, T stable, I TI = A. [Hint: Let T = TA, A regular (see Exercise 11, 2.3), choose u,E(L: for 7 E + 1) such that a,,Eiav iff r ) r i = v i. The counterexample is for Teq: A, = 0, A , = (a,,/#,: i < A, r ) E "}, A, = A , u (a,,: r) E A(A + l), (A + 1) E Range(r))} and 'p = PI(z).]

EXERCISE 2.15: Show that Ax(XI.l) (hence (XI.2)) may fail for FL, T stable, A = ITI. [Hint: Let P, (i < A) be pairwise disjoint, infinite one-place predicates c E 1C1 - U, Pi(a), ci E Pi( a), A = 0, C = {q: i < A}. SO tp(c,A) is F&-isolated, it has a unique extension in S(C): tp(c, C), which is not Fi-isolated.]

EXERCISE 2.16: Show that Ax(XI1) may fail for (T unstable). [Hint: Use T, a, b, c from Exercise 2.8: we let C1 = 0, B = C, = {a}, A = {a, c}, P = tp(b, A).]

EXERCISE 2.17: Show that (VI) may fail for Fr. [Hint: Use the theory of the rationals as an ordered set, B = C = 9, A = {l}, 6 = (3), a = (2).]

EXERCISE 2.18: Show that Ax(XI.1) (and XI.2)) may fail for FX. mint: Let T be as in Exercise 2.11, hint 2. We let A = 0, B = 0, C = u Pi(C), choose a EQ(&) and let p = tp(a, A). (We use the notation of Ax(XI.l).)

EXERCISE 2.19: Show that Ax(XI1) may fail for Fhp. [Hint: Use the previous example. C, = 0, A = Cp = {P,(a)}, B = Po@), p = tp(a, B).]

EXERCISE 2.20: Prove 2.12 for F( instead @.

CH. Iv, 8 21 EXAMPLES OW F 173

EXERCISE 2.21: Assume Ax(III.1) and (VII). If Ax(X.1) [or Ax(X.2)] holds, then it holds for v = p(P; a) too.

EXERCISE 2.22: Show that A3(T) = KO iff A3(T) = KO. [Hint: Suppose A3(T) > No, then T is not totally transcendental (by 2.15(4)); let M'be a IT1 +-saturated model of T . Then A = [MI, q = {7v(z; 8): R(cp(P; a), L, 2) < 00) exemplifies A3(T) > KO. Now A3(T) > No implies A3(T) > No by 2.16(2), so as A3(T), A3(T) 2 24, we finish.]

QUESTIONZ.Z3:Showthatbetween~~(T), IT[, X(T)A,(T) (1 = 1,2 ,3) there are no connections except those from 2.15(1), (6), 2.16(1) Exercise 2.22. [Hint: (1) T,: Let E, (i < A,) beindependent equivalence relations, each with two equivalence clwes. Then KJT,) = KO, ITl/ = A,, A,(T,) = A1(T1) = AS(T1) = Aa(T1) = KO. A3(T1) = A3(T1) = A:.

(2) T,: Let AI = (A(a)3, . . . , P,, . . . ) where, for r ) E A(2)>2 P n - - {v E *ca)3: 7 Q v}, and A, = 2<A(a) and T , = Th(M). Then K,(T,) = M,, ITa! = 2<"'),A,(T,) = A'(T2) = A S ( T 2 ) = Aa(T,) = K,,A3(T,) = M,, A3(T2) = A(2)+.

(3) T3: Is the T from Exercise 2.11, hint (2). Clearly K ( T ) = No, IT1 = A, A,(T) = X(T) = A+ for 1 = 1, 2, 3.1

QUELYTION 2.24: What happens if we change the definition of A3(T) to: A3(T) is the first cardinal A such that if r is a set of m-formulas, p G r a n m-type, then there is q E r, 1q1 c A, p u q is consistent but for every tp Ere i the rp u q I-v o r p u q I- -p?

(i) Does the value of A3(T) change? (ii) Do we have nicer or better results with this definition?

QUESTION 2.25: What happens if in the definition of ha( T) (Definition 2.5(1)) we replace q 1 (B U C) I- q by p U q 1 C I- q?

(i) Does the value of ha( T ) change ? (ii) Do we have nicer results with this definition ?

EXERCISE 2.26: Prove that T is transcendental iff A,(T) = KO (see Definition 11, 3.4).

EXERCISE 2.27: Suppose T is stable, M is F&,-saturated,p E Sm( ]MI) . Prove that there is a B E 1611, IBI < K ( T ) such that p does not fork over B and p 1 B is stationary.

174 PRWE MODlPLS [OH. Iv, 8 3

EXERCISE 2.28: Show that if F = {(p, B): p is definable over B} then Ax(XI.l) holds for F if for some M, B c IMI c A .

EXERCISE 2.29: Let F = R, (1) If cf h 2 K(T) , F satisfies Ax(XI.l), then F satisfies Ax(XI.2). (2) If h 2 h3(T) + K(T) and cfh > K(T) or h > A3(T), then F

satisfies Ax(XI.2).

Remark. (1) For (X.2) see 1.1(6).

clear. [Hint: Use 4.3(1).] (2) For x = 8 we can get a result using h3( T), but with ha( T) it is not

EXERCISE 2.30: Suppose p = tp(ii, A) is Fi-isolated. (1) If cf A > X,, and FZ satisfies Ax(XI.l), then p is Fi-isolated. (2) If cf h = X, < h but P ( T ) < A (or Fi satisfies Ax(XI.1) for

arbitrarily large p < A), then p is FA-iSOlated. [Hint: see 3.2.1

EXERCISE 2.31: Show that Ax(VI1) holds for Fi when (C( < cf A.

EXERCISE 2.32: F; may fail to satisfy Ax(V1).

Proof. Let P, (i < A ) , P , Q be one-place predicates, P,(E) pairwise disjoint, P,(6) E P(&), P(%) n Q(6) = 0, P, a function from Q onto P,, and for every c, E P,(G), {Q(z) A P,(z) = c,: i < A} is realized. Let B = C = A = 0, b E Q ( ~ ) , a E P(6) - UfeA P,((E). If we add a,: Po --t Pi one-to-one, onto, Q(x) --f U,P,(z) = F,(x) we get T superstable.

IV.3. General properties of F-primary models

We use the notation of Definition 1.2 and Section 1 in general. Before eaoh theorem the axioms used are listed and also assumptions on h(F).

THEOREM 3.1: (1) [Ax(XI.l)] Over every A , for any p there is an (F, p)-p'mry set (so by 1.2(6), if Ax(X.1) holds it is an (F, p)-primry

(2) [Ax(XI.l)'J If p(F) < m, then over every A there is an F-p*mary

(3) [Ax(XL1)]8uppme IBI ISm(B)I I poratleaet IBI < p 3

d l ) .

set .

I{p: p E 8m(B), p is an F-bolated tgpe}l I; p.

CH. Iv, 8 31 PROPERTIES OF F-PRIMARY MODELS 175

If p is regulccr then over every A , IAI I; p there is an (F, p)-primary set of cardinality p, and so it is F-semi-saturated.

(4) [Ax(III.l)] In (3), imtead of requiring p to be regular we can aaeume T stable and; (i) cfp r: A(F) + K(T) or (ii) c fp 2 h(F), and also p E F(B) implia p does not fork over B.

(6) [Ax(X.l)] Over every A, there is an F-mtructible model.

Proof. (1) We define by induction on i, sequences a,, and sets B,, such that pi = tp@, A,) E F(B,) where A, = A u U {Ti,: j < i}. Suppose we have defined Zf for i < u, and A, is not (F, p)-saturated. Then there is a pair (q, C) EF, C c Dom q E A,, lDomq( < p, such that q is not realized in A,. Among these pairs choose one, (q,,C,) with minimal j ( q ) = min{j: j I u, Dom q E A,}. By Ax(XI.l) we can find a pair (pa , B,) E F , q c pa, pa E Sm(A,), and choose a sequence a, realizing

If for some a A, is (F, p)-saturated then clearly we are through. We shall show this holds for some a < ((IAI + 2)f l+ITI)+. The number of pairs (p,C) EF, Domq E A,, lDom qI j). We can easily prove by induction on j < ((IAI + 2)’+ITI)+ that /3, < (IAI + 2)”+ITI)+. Hence there is p* < ((IAI + 2)”+ITI)+, cfp* = p+, and j < p* - p , < p*. Clearly A,* is (F, p)-saturated, so we finish.

- . _ _ _

Pa-

(2) Immediate, by (1) for p = p(F). (3), (4) We shall define Tif for i < p2 (ordinal exponentiation) such

(a) d = (A, {ai: i < pa}) is an F-con8truction. (b) For i < p write & ( p s i ) as U,<” Cj where 1Cj1 < p and u <

that:

* Cf, c Ci. We then require that for each i each F-isolated type in Sm(C:) (j < p) is realized by some sequence in d ( p m ( i + 1)).

If p is regular clearly &(pa) is (F, p)-saturated aa desired. So we have to prove (4) only. If (p, C) E F , lDom pI < p there am q, p E q E Sm(,c9(pa)) and C‘, such that (q, C‘) E F. If cf p 2 h(F) + K( T), then for some u < p, j < p, C‘ c Cf and q does not fork over Cy. Now we can findforeachp,u X,, {a,: u + w I /3 < p} is an indiscernible set, whose average is q. Hence by 111,3.5 some ZB realizes p . The other case ((ii)) is handled similarly.

(6) Easy, like (l), using the Tarski-Vaught test.

176 PRIME MODELS [OH. Iv, 8 3

THEOREM 3.2 [Ax(II.l), (III.l), (111.2) and (VII); A(F) regular]: (1) If C is F-cowtrudible over A t h C is F-atormic over A.

(2) A is F-atornic over A. (3) If tp(6, A) ia F-iaolated then A u 6 is F-atomic Over A. (4) If A , E A, c A,, A , is F-atorni~ over A,, A, F-atornk Over A,,

(6) If for i < 6 A, ia F-atomic Over A,, i < j < 6 - A, c A,, t h then A , ia F-atomic over A, .

u,<6 A, is F-atomic over A,,.

Remurk. For (5) we need no axiom at all, and for (2) we need Ax(II.1) only.

Proof. (1) Let & be an F-construction of C over A. It suffices to prove by induction on a that &(a) is F-atomic over A = d ( 0 ) . For a = 0 this follows by (2), and for a a limit ordinal, by (6). For a = /3 + 1, d@) is F-atomic over A by the induction hypothesis and &(a)[ = &@) u is F-atomic over &(/3) by the construction, and (3). Hence by part (4) &(a) is F-atomic over A.

(2) Trivial. (3) If E E A u 6, then E = 7in6,, where 6, E 6 and i% E A. Let

tp(6, A) E F(B). So by Axiom (111.2) tp(6, A) E F(B u 8). By Axiom (11.1) tp&, A) E F(a u B), hence by Axiom (VII) (as A u a = A )

(4) Let E E A,, hence for some B G A,, tp(8, A,) E F(B). For every 6 E B tp(6, A,) is F-isolated, hence tp(6, A,) E F(c6) for some 0 6 c A,. By Axiom (111.2) and the regularity of h(F) tp,(B,A,)EF(C) (C = U~EB Cb). By Ax(III.l), (111.2), tp(E, A, u B) E F(C u B) hence by Ax(VII), tp,(B U a, A,) E F(C), in particular tp(E, A,) E F(C), so we finish.

tp,(Bn6,, A) E F ( a U B). SO tp(E, A) = tp(8-61, A) E F(a u B).

(6) Trivial.

THEOREM 3.3 [Axioms (III.l), (VI) and (VIII)]: 8 q p e I <a a well- orokred set, & = ( A , {a,: a E I), {BE: a E I)) an F-construction. If J is another well order, with the same universe, and for a E I, U , ( d ) c { t : J C t < a} then &* = ( A , {a,: s E 4, {B,: a E 4) ia an F-construction.

Proof. It suffices ,to prove by induction on a E J (on the order J) that (pa(&*), BE) E F. Suppose we have proved it for every a’ < a and we shall prove it for a. For t E I, let C, = B, u [&(t) n &*(a)]. We prove by induction on t E I (in fact t E I u {a1 I}) that (p,(&*) 1 C,, B,) E F.

CH. IV, 0 31 PROPERTIES OF F-PRIMARY MODELS 177

Notice that C, is increasing with t , and Cal(r) = d * ( s ) so p s ( d * ) 1 C,,,,, = p S ( d * ) , so this will suffice.

For the &st t in I , and in fact when I C t I 8, A u B, E C, E d ( s ) . As by definition ( p , ( d ) , B,) E F , this holds by Axiom (111.1). If t is limit, it follows by Axiom (VIII). So suppose t is the successor of t(1) I C t > 8. If J C s I t(1) then C, = C,(,), so there is nothing to prove. Otherwise C, = C,(,, U &), and the result is an application of Axiom VI *

So we finish the induction on t , hence on 8, hence finish the theorem.

CONCLUSION 3.4 [Ax(III.l), (VI) and (VIII)]: If d = (A , {a,: i < a}) is an F-construction, B E &(a), I BI 2 AJF), then for s m e well ordering I of a, d* = ( A , {a,: i E I } ) is an F-construction, and B is included in d * ( s I B I ) , siB1 the I Blth element of I .

Proof. Immediate by 3.3.

LEMMA 3.6 [Ax(II.l)]: If d = (A , {a,: i c a}) is an F-construction, p I a then d' = ( d ( p ) , {a,: i c a}) is also an F-construction.

Proof. Immediate.

LEMMA 3.6 [Ax(II.l), (III.l), (111.2), (V.2) and (VII); A(F) regular]: If d = (A , {a,: i < a}) isanF-construction,C G &(a), ICl c A(F)then d' = ( A U C, {a,: i i a}) is also an F-construction.

Proof. By 3.2 and 3.6, for every i c a, and E E &(a) pj = tp(E, d ( i ) ) is F-isolated, hence for some Bj c d ( i ) , I Bjl c A(F) , (pj, BE) E F . Let B* = u {Bj: E E C u a,} so I Bil c X(F) by the regularity of A(F). By Ax(III.2) (p:, B,) E F for every i3 E C u a,. So tp,(C u a,, d ( i ) ) E F(BL), hence by Ax(V.2) tp(a,, d ( i ) u C) E F( B' u C). So we finish.

THEOREM 3.7 [Ax(I), (II.l), (IILl), (111.2), (V.2) and (VII); h(F) regular]: (1) If B is an F-primury set over A then (B, a)oeA is A(F)- homogeneous: i.e., i f a,, b, E B for i c a < A(F) and tp*((a,: i < a), A) =tp*((b,: i < a), A ) then for every a, E B there is b, E B such that tp*((a,: i I a), A) = tp,((b,: i s a), A).

(2) #uppose B, is F-primary over A, (1 = 1, 2), f an elementary nwp- ping, A, c Dom f E B,, A, c Range f E B,, lDom f - All < A(F), !Range f - Aal c A(F). Then for every 7i E B, there is f ' extending f, satisfying all the abovementioned conditions and 7i E Dom f '.

178 PRIBfE MODELS [OH. Ivy 5 3

(3) In (2 ) for every C, C B,, IC,I < A(F) we canJind an elementaryf’; Domf’ 2 A , U C,, Rangef’ 2 A , U C,.

(4) Instead of “F-primary”, we can in (2), (3) assume only “(F, p)- primary” where p 2 lA1l+ + A(F).

( 6 ) In (1) it su.ces to assume B i s (F, p)-saturated, p 2 IAI + + A(F), and every sequence from B realizes over A an F-isolated type (i.e., B is F-atomic over A).

Remark. Concerning Theorem 3.7, note that in (2), (3) it suffices to assume B, is ( F , p)-saturated, (F, p)-atomic over A,, p 2 lAll + + A(F). Also in (3), for any C, E B, (1 = 1 , 2) ICJ + IC,I < A(F) there is an elementary f’, Dom f’ 2 A , U C,, Range f ’ z A , U C,.

Prmf. (1) Follows from (2). (2) By 3.6 B, is F-primary over Dom f, so by 3 . 2 ~ = tp(a, Dom f ) is

F-isolated, so for some C E Domf, ( p , C) E F. Hence by Axiom (I), (f ( p ) , f (C)) E F. As B, is F-saturated some 6 E B, realizes f (p). Extend f to f’ by defining f ‘(a) = 6. Clearly f ‘ satisfies our requirements.

(3) Prove by repeating (2) A(F) times. (4), (6) The same proof, in fact, as of (l), (2).

THEOREM 3.8 [Ax(I), (II.l), (III.l), (111.2), (V.2), (VI), (VII) and (VIII); A(F) regular]: (1) Suppose B, is F-primary over A, ( 1 = 1, 2) and f is an elementary mapping from A, onto A, then we can extend f to an elementary mapping from B, onto B,.

(2) The assertion is true also for B, (F, p)-primary over A,, when IAZl < lBll 5 P-

CONCLUSION 3.9 [The same assumptions as in 3.81: The F-primary set over A is unique up to isomorphism over A (i.e., up to isommphitwm which are the identity on A).

Proof of 3.8. (1) Let d, = (A,,{Gf: i < a,}, {Bf: i < a,}) be an F- construction of B, over A,. We define by induction on p < a* elementary mappings fs where a* = max{a,, a,} such that:

(1) f o = f. (2) For /? < a, fa extends fs, and for limit 6, fd = u B < d fs. (3) Dom fs E B,, Range fs E .Bas (4) Dom fs is closed, i.e., for some V;, DomfB = A U U {at: i E 7;)

and i E V i =c B: E Dom fs.

CH. Ivy $31 PROPERTIES OF F-PRIMARY MODELS 179

(6) Range f8 is closed (the same as in (4) for V;) . (6) E Dom faa+,, E Range faa+, (when is defined). For @ = 0, @ a limit ordinal, there is no problem. So suppose f6 is

df = (Dom f6, {af: i < a,}),

defined. By 3.3, 3.6

d!j = (Range f8, {af: i < a,}) are F-constructions. Now we define by induction on y c h(F) elementary mappings f] such that:

(i) f] is increasing in y , fj = fE, fj = (ii) Domf] c B,, Range fJ c B,, (iii) lDom f] - Dom f 6 1 c h(F). Let Dom fJ = Dom f6 U U {a:: j E VjSy} , Range fj' = Rangets U

u {a;: j E VjjY}. For y = 0, y a limit ordinal-the definition is by (i). By 3.6, B,, B, are F-primary over Dom f], Range f] respectively.

Hence by 3.7(2) we can extend f; tof;+ satisfying (i), (ii), (iii) and such that for y even:

f;,

V L + l = G , Y u u {U,(SB,): j E V $ J " {a> ({a} appears only when @ = 2a + 1, a < a,). For y odd, replace 2 by 1, and {a} appears only when @ = 2a, a < a,.

Clearly f8+, = UyeA fSy (A = h(F)) satisfies our requirements. Now fa. is the required mapping.

(2) The proof is similar.

Remurk. In fact the regularity of h(F) is needed for 3.3, 3.6 but not in this proof.

THEOREM 3.10 [Ax(I)]: (1) If B is F-constructible over A, then B is F-primitive over A.

(2) If B i s Fqrimury over A, then B is F-prime over A.

Proof. (1) Let d = (A, {ai: i < a}) be an F-construction of B over A and let B* be an F-saturated set, A c B*. We should find an elementary mapping f from B into B*, f t A = the identity. Define by induction on /3 I CL elementary mappings f8 from .at@) into B* so that f, extends fE for /3 c y. Let fo be the identity over A, and for limit 8, fd = U f l < a f s * Iff8 is defined, ( ~ E Y B4) E F (PE = tp(aE, Axiom (I), (fj(p8)~ f6(B8)) E F, and DomLfs(pfl)]

then by B*, so f8(236) is

realized by some i;B E B* (as B* is F-saturated). Extend f8 to f8 + , by defining f8 + 1(7iB) = 68. Clearly f a is the required mapping.

(2) Immediate.

180 PRIME MODELS [CH. Iv, 0 3

LEMMA 3.11: (1) If A G B c C, C is F-primitive over A then B is F primitive over A.

(2) If A c B E C, C is F-prime over A, B F-saturated then B is F- prime over A.

(3) The same for (F, p) instead of F.

Proof. (1) Let B' be an F-saturated set, A G B'. Then there is an elementary mapping f from C into B', f 1 A = the identity. Let g = f 1 B, so g is an elementary mapping from B into B', g 1 A = the identity. Hence B is F-primitive over A.

(2) Immediate from (1) and the definition. (3) Proved similarly.

CONCLUSION 3.12: Let p(F) < a0 then: (1) [Ax(I), (X.l) and (XI.l)] Over every A there is an F-prime model. (2) [Ax(II.l), (III.l), (111.2), (VII) and (XI.1); h(F) regular] I n every

F-prime set B ouer A, every sequence realizes over A an F-isolated type. (3) [Ax(I), (ILI), (III.l), (111.21, (V.2), (VII) and (XI.1); A(F) regular]

If B is F-prim over A, C E B, ICI < cf h(F) then B is F-prime over A U C . The Same hoZds.for F-atomic.

Proof. (1) Immediate, by 3.1(2), 3.10 and Exercise 3.1. (2) By 3. I( 2) there is an F-primary set B* over A. So by the definition

of F-primeness there is an elementary mapping f from B into B*. f 1 A = the identity. By 3.2 for every 6 E B* tp(6, A) is F-isolated. Hence for every 6 E B, tp(f (6) , A) is F-isolated. But as f 1 A = the identity, 6, f (6) realize the same type over A, so we finish.

(3) By 3.1(2) over A there is an F-primary set B*; and by the definition of F-primeness there is an elementary mapping f from B into B*, f F A = the identity. By 3.6 B* is F-primary over A u f ( C ) , and by 3.10 B* is F-prime over A u f (C). As A uf(C) c f (B) G B* by 3.11(2) f (B) is F-prime over A u f (C). (By Ax(1) the property of being F- saturated hence F-prime over a set is preserved by elementary mapping.) Hence B is F-prime over A u C.

The proof for F-atomic is similar.

LEMMA 3.13 [Ax(III.I), (VI) and (VIII)]: Suppose A,C, , (i < a) are given sets, and for every 5 EC,, for some BE E A, tp(E, A u Uj<, C j ) E F(Bi). Then for every i < a, 5 E C,, tp(E, A u Uj+ , C,) E F(BE).

Proof. By 1.2(2) (and Ax(III.1) and (VIII)) it suffices to prove that for

CH. Iv , 8 31 PROPERTIES OF F-PRIMARY MODELS 181

i < a, E EC,, a E u,*, C,, tp(E, A LJ 2) E F(@). Let a = Z o n e .

where 2, E C j ( l ) 0 s 1 s k, andj(0) < . . . < j ( n ) < i < j ( n + 1) < - - - < j (k). We prove by induction on 1 I k + 1 that tp(E, A u a,-. - . -a, - EF(&). For 1 s n + 1 this holds by Axiom (111.1). If it is true for 1, 1 s k, it is true for 1 + 1 by Axiom (VI). For 1 = k + 1 we get what we want.

THEOREM 3.16 [Ax(I), (III.l), (111.2), (V.2), (VIII) and (XI.1); h(F) regular] : (1) 8uppose A c [MI, M is F-saturated, and F-atomic over A. Letp, ~ S ( A ) f o r i < a, and A* = A u {c: c E IMI, c realizes a p f for an i < a}. Then hf is F-atomic over A*.

(2) For F = q, we can assume thcct pf's are types almost over A , when T is etable.

Proof. (1) Let 7i E 1M1, by Ax(XI.1) a s p = tp(Z, A ) is F-isolated, p has an extension q E ~ ' " ( A * ) which is also F-isolated. So for some C G A*, (Q,C)EF (so ICl < X(F)) and some 6~ ]MI realizes q. Let C = {c,: i < ICl}. By the hypothesis for every E EC u 6, tp(E, A) is F- isolated, say tp(E, A) E F(B,). By Axiom (111.2) and the regularity of X(F), tp,(C u 6, A ) E F(B), where B = u {Be: E E C u 6}. Hence by Ax(V.2), for every a < ICl,

pa = tp(c,, A u 6 u {cf: i < a}) EF(B u 6 u {cf: i < a}).

Now we define, by induction on i, c; E 1M( such that the mapping fa(a s ' ICI), f,(a) = a for a E A, f,(6) = 7i and f&) = c;, for i < a, is elementary. It is trivial to definef, and fa (6 a limit ordinal). For note that by Ax(I), f,(p,) is F-isolated, (w a < h(F)) hence some ci E 1M1 realizes it. Let C' = {c;: i < ICl}; as tp(c;, A) = tp(c,, A), c, E A* clearly C' c A*, and IC'I < X(F). We shall provep* = tp(B, A*) E F(C'), and thus finish. By Ax(VII1) it suffices to prove tp(& A u C' u E) E F(C') for any E E A*. As before, we can extend ficl to an elementary mappingf', whose domain is c [MI, and whose range is C u Rangef,,,. By the choice of C, and Ax(III.I), tp(6, Domf') E F(C), hence by Ax(1) tp(7i, Rangef') E F(C') so tp(Z, A u C' u E) E F(C').

(2) Work in Ceq. By 2.12(2) Meq is F-saturated and also F-atomic over A , hence it is F-atomic over acl A (see 2.1). Now w.1.o.g. each pt is over acl A ; hence, by 3.14( 1) Meq is F-atomic over (acl A ) U A*, and by 2.1 Meq is F-atomic over A* so M is F-atomic over A*.

LEMMA 3.16 [Ax(III.2) and (V.2); X(F) regular or [Ax(III.2), (V.1) and

182 PRIME MODELS [OH. Iv, 3

(IX)]]: If C is F-atomic over A, and IC - A1 s h(F) [or IC - A1 I cf h(F) when A(F) is Singular], then C is F-constructible over A.

Proof. Let C - A = {c,: i < a = ICl}, and let A, = A u {c,: i < j}. It suffioes to prove tp(c,, A,) is F-isolated. As h(F) is regular [or as IC - A1 5 cf h(F)] and Ax(III.2) holds, tp,({c,: j I i}, A ) E F(B) for some B c A, I BI c h(F). Hence, by Ax(V.2) [or by l.l(4)] tp(c,, A,) E F(B u {c,: j c i}).

THEOREM 3.17: (1) [Ax(XI.2) or Ax(XI.l), a c cf h(F), and Ax(IX)] If A, E A, f o r i c j I a then there are sets B, (i s a) Buch t7t.d A, E B,, i < j =- B, c B,, B, U A, is F-constructible over A, U U,<, B, when i I /3 s a, and B, is (F, p)-saturated. (80 B, u A, is F-constructible over A, when i I /3 I a.)

(2) [Ax(X.2) or Ax(X.l), a < cf h(F), and Ax(IX)] If A, c A, for i c j I a, then there are models M,, A, E !Mil, i < j * [Mil c IM,l, [Mil U A, $8 F-constructible over A, U u,<, lM,l when i I B I a.

Proof. Similar to 3.1 (remember 1.1(7))#

CLAIM 3.18: [Ax(I), (IV), (V.1) and (VII)] If Ax(XI.l) [or (XI.2)] are satisfied for l-types then it i s satisfied.

Proof. Easy.

THEOREM 3.19: (1 ) [Ax(I), (X.l) , (XI.l)] Let M be F-saturated. If M is F-minimal over A (see Definition 4.4), then in M there i s no in$nite indiscernible set over A.

(2) We can replace the three azim by the acreunzlption “over any A there ie an F-p~ime d”.

Proof. Just the proof of one implication in 4.21.

EXERCISE 3.1 [Ax(X.l)]: If A is F-primary over B, then A is an F-primary model over B.

EXERCIBE 3.2: In 3.16(1) let p , be an m,-type over A, and A* = A u u, I,, I , = { E E M: E realizes p,}. Show the conclusion still follows. Similarly 3.16(2).

EXERCISE 3.3: Show that 3.6 holds for any closed C (assuming the appropriate axioms),

OH. I v , $41 PELIlldE MODELS BOB STABLE THEORIES 183

QUESTION 3.4: Is there in any of the lemmas of Section 3 an un- necessary axiom ?

EXERCISE 3.5 [Ax(I)]: Suppose B is F-primitive over A, f an elementary mapping, Dom f = A, and Range f C_ C, C is F-aahrated, then we can extend f to an elementary mapping 9, Dom g = B, Range g E C.

EXERCISE 3.6: Suppose A is singular, and Ax(VI1) holds when ICl < cf A, and Ax(II.l), (111.1) and (111.2). Then:

If ICl < cf A, C is F-atomic over A, tp(ii, A u C) is F-isolated, then C v ii is F-atomic over A.

IV.4. Prime models for atable theories

In this section, we wume T is stable, except in the stafied theorems.

LEMMA 4.1*: If (p(Z; a) i8 almost over B, B E A , then there is a fomnula +(Z; 7;) such that

(i) #(Z; 6 ) i8 over A and dmost over B; (ii) k(p(~; a) --f #(z; 6); (iii) if E E A then k(p(~; a) -+ e(z; a) i# I#(z; 6) -+ e(3; a): (Note that i f ?i E A we can take # = (p.)

Procf. Let a,, . . . , a,, be a maximal set, such that Z0 = a, the (p(Z; a,) (I s n) are not equivalent, and tp(i&, A) = tp@, A) (n < No by- 111, 2.3(1), using the fact that (p(Z; a) is also almost over A). Let f(Z; a’) .= Vlsn (p(Z; at). Clearly by 111, 2.3(2), f(Z; 6‘) = #(Z; 6) for some # and 6 E A. It is easy to check (i), (ii), (iii).

LEMMA 4.2: flwppose cp(Z; 8) does not fork over C , C G A . Then there is a f m u l a #(Z; &) such ticat;

(1) #(Z; 6) ia over A , but do@ not fork over C . (2) {(p(E; a), +Z; 6)) f o r b ower A (hence over C ) . (3) For every ~ E A a d 8, I(p(~;a)-+0(3;a) implies k#@;6)+

(4) If C E B E A, tp(& A) &OM not fork over B, t h we can c h e eg; a).

+(it; 6 ) which ie alnzost over B.

Proof. Let B be as in (4); for (l), (2) and (3), B = A satisfies that tp@, A) does not fork over B. By 111, 1.12, there is an indiscernible set

184 PRIME MODELS [CH. Iv, 5 4

I = {a,,: n < w } over A, based on B, 3 = a,,. Hence by III,2.5, for some n, p,,(Z; a*) = Vros2,,-l,lrl=,, Atpro p(E; a,) is almost over B. Cleady by 111, 1.4, there is q E Sm(U 1 U A ) , p(3; a,) E q, q does not fork over C; for every I , q(Z; a,) E q (as q does not split strongly over C by 111, 1.6(1)) hence. p,, E q so pa@; a*) does not fork over C. Clearly for every 1 > 0, {p@; a) , +E, a,)} splits strongly over A, hence forks over A. Hence for every w E 2n - 1 (~(3; a), Vlsw - q ( E ; a,)} forks over A (by 111, 1.1(9)), hence {q@; a), --,p,,(Z; a*)} forks over A. Also, if E E A, t - q ( ~ ; a) 3 d(Z; 17) then t-p(Z; a,) 3 e ( E , E) hence t-p,,(~; a*) --f d(Z; a).

Now apply 4.1 to p,,(Z; a*), B and A. We get #(Z; 6) satisfying (i), (ii) and (iii) from 4.1. It is easy to check that it satisfies the conditions (1)-(4) from 4.2.

THEOREM 4.3: Swppocre p cBrn(A), C E B c A, p doee not fork over C . (1) If p i8 Ey,-b~Wed, t h p 1 B b FA-boWed. (2) If p i8 q-i80kZ&?d, A 2 K=( T) , t h p r B b ~ - h o k d . (3) If p 1 C i8 8tdbWY9 ICl < A, p b F”~-bOlate&, then p r B b

PA-i801ated, provided that A 2 K,(T). (4) If p is Fi-isolated A 2 K,(T), then p rB is Fi-isolated. ( 5 ) If p is F!,-isolated, then so is p r B.

Proof. (1) As p is ~A-isolated, there is q c p , q I-p, Iql < A, and as in 2.2(2) aasume q is closed under conjunotions, and let q = {p,(Z; a,): i < a < A}. As p does not fork over C and pi@; 4) ~ p , by Lemma 4.2 there is #@; 6,), 5, E B satiaf- (1)-(3) from 4.2 (B here corresponds to A there). As p,(Z; a,) and (by (2) from 4.2) {p,(Z; a,), -,#,(Z; 6,)) forks over 0, but p does not fork over C, we have that #,(Z; 6,) ~ p . SO r = {#,(Z; 6,): i < a} E p , SO r E p B; and clearly 1.1 < A. If QZ; 5) By then for some i < a, t-p,(f; a,) + B(3; a), hence by (3) from 4.2, t-#,(Z; 6,) + O(Z; a). Hence r t- p r B.

(2) As p is FR-isolated, for some C1 s A, IC,l < A, there is a type q E p almost over C,, q t- p . For every E E C, there is BE c By I BEI < K(T) such that tp(E, B) does not fork over BE (by 111, 3.2). Let B* = u,,l B, U C; then as A 2 K,(T), p*l < A . Now for every q = q(3; a E q there is +JZ; 4) satisfying 4.2(4) (A, B there correspond to B, B* here). Let T = {+p:rp~q}. Clearly T is almost over B*, and as in part (l), T E p , T I-p rB. As p*l < A we finish the proof.

(3) By (2), for some B* c B, p BEF;(B*). As ICl < A we can assume C G B*. By 2.2(13) p 1 B E Fi(B*).

m. IV, f 41 PRIME MODEM FOR STABLE THEORXES 185

Proof. (4) So suppose q = {tpr(E; st): i < u} E p , u < A, U, < a si, E C* E A , IC*l < A, and q has no extension in Sm(A) forking over C*. W.1.o.g. q is closed under finite conjunctions. We produce #,(3; 6,) (6, E B) as in the proof of 4.3( l), r = {#,(Z; 6,): i < a} c p, and w.1.o.g. p , tp*(C*, B) does not fork over C* n B. Now suppose for some 7i E B and 8, r u { 8 ( E , 5)) is consistent. For each i, not #,(if, Ei) I- a) hence tpr(E; b$ I- -d(?Z; a) fail, so as q was closed under conjunction, q1 = q u { 8 ( E ; si)} is Consistent. By the choice of q, q1 does not fork over C*, so let Erealize ql, tpiE, A) does not fork over C*. As {rp,(iE; $), -,I,$,(?& 6,)} fork over C*, $,@, b,) E tp(E, A), hence r u {8@; a} c tp(8, A).

As p’ = tp@, A) does not fork over C*, also tp(E, B u C*) does not fork over C*, and as in addition tp*(C*, B) does not fork over B n C*; tp,(C* u E, B) does not fork over B n C*, hence tp(Z, B) does not fork over B n C*. So also r u {@, a)} E tp(Z, B) does not fork over B n C*.

S o r ~p r B , C * n B c B,Ir l , IC*nBI < AandforeveryO,aiB, r U { O ( Z ; h ) } consistent implies r U { O ( Z ; ~ ) } does not fork over A ; clearly p I B is Fg-isolated.

( 5 ) Just like 4.3(2).

EXERCISE 4.1: Does a similar assertion hold for Fi (i.e., an assertion like 4.3( 1) or at least 4.3(2)) 1 Prove that the answer is no.

Remark. The question is interesting mainly for h = H,, since if h > KO then Fk-saturation is equivalent to Fi-saturation (by 2.2(6), (7)). Similarly for other concepta (primeness, etc.).

EXERCISE 4.2: Show that in 4.3(2) we cannot omit the assumption that h 2 K ~ ( T ) .

DEFINITION 4.1: For x = t , 8, a, h 2 K , let

F;,= = {(p, B): for some A, m; p ES”(A), B c A , IBI < K , P does not fork over B and p is an Fg-isolated type}.

Note that (p, B) E Ff,= does not imply (p, B) E Ff.

LEMMA 4.4: (1) For x = t , 8 , a, K 2 K(IT), p E @(A), p is e-i8olaterl i#

(2) I n tlce following concepts for x = t , 8 , a; it does not matter w&kr F = V, or F = Ff,%: F-construction <A, (ar: i < a)), F-saturated, F- primary, F-primitive, F-prime and F-atomic provided that K 2 K( T) .

p i8 F,,-GO&ed.

186 PRIME MODELS [m. IVY 5 4

Proof. Immediate.

Remark. ButforF-construction(A,(a,:i < a ) , (Bt:i < a))itmatters.

THEOREM 4.6: If x = t ,a , K = K,(T) < d~ then E,ls aatiaJie~ the foZlozoing aziomCr; (I), (11.1) [Z = 1, 2, 3, 41, (III.l), (111.2), (IV), (V.1)

for 6, cf 6 < cf h and (IX) [when cf h 2 K] and (XII). Also h(Ff,,) = K

a d (v.2) [When Cf h 2 K(T)], (W), (VII) [When Cf h 2 K(T)], h(VII1)

and P(XrC) = P(F3.

Proof. Most of them follow by combining the results on Fg, and FL. Ax(III.1) follows from 4.3. For Ax(V.2) notice Lemma 2.8(2), &B

ICl < K 5 cfh. For Ax(VI1) note that ICl < K 5 cf h so there is a B' c A such that tp*(C, A) E E(E) and the rest follows by Ax(III.2). For Ax(VI), use Ax(IV) and (VII) to show tp(sin6, A) is Fz-isolated, and (V.1) to show tp(si, A u 6) is E-isolated.

THEOREM 4.6: If x = t ,a, A 2 K = K,(T), Fg 8ati8jEeS Ax(X.l) [Ax(XI.l)] then also Fgels 8atiafiecl Ax(X.l) [Ax(XI.l)]. The same holds for Ax(X.2) and (XI.2).

Proof. Immediate.

CONCLUSION 4.7: If K(T) 5 cf A, A is F$-conductible over By x = t , a, then A is Ff-atomic over B.

Proof. Immediate, by 4.4 and 4.6, applying 3.2(1) to Fg,E ( K = K r ( T ) ) .

CLAIM 4.8: Let F% 8atisfy Ax(X.l), (XI.1); x = t , a, cf h 2 K(T) .

over A u C.

ICl < cf A, then .d = ( A u C, {a,: i < a}) is an ~-mtrruct ion.

(1) If M is Fg-prime over A , C E IMI, ICl < cf A then M is Fg-pime

(2) If .d = (A, {al: i < a}) i s an Ff-MWition, C E &(a),

Proof. (1) Like 3.12(3), using (2) instead of 3.6; (2) like the proof of 3.6, using 4.7 instead of 3.2, and 2.8(2) instead of Ax(V.2).

THEOREM 4.9: Suppose z = t , a, F = Fg satie$ea Ax(X.1) and (XI.1). M is an F-pime model over A, and I c IMI b an inJinite indiscernible set over A. (So w.1.o.g. I is muxid in M.)

OH. Ivy f 41 PRIME MODELS FOR STABLE THEORIES 187

(1) If cf h 2 K ( T ) , then dim(I, A , M ) 2 h ezcept, poseibly, when: (i) cf h = No < A, and

(ii) dim(I, A, M) = No, and (iii) for no B c A u I and J E I, IJI = No, is Av(J, A u U J) E

(iv) if x = a, for no B -c A, tp*(U I, A) E F(B). (2) dim(I, A, M ) 5 h when cf h 2 K,(T). (3) If cf h 2 K(T) + N, OT h = KO = K(T), then Av(I, 1M1) is

(4) If cf h 2 K(T) + N,, then M is h-hmmgeneoua over A (as in

(5) dim (I,A,M) < A when h is regular, A+ 2 K(T).

F(B), a d

Ff+ -iadated.

3.7(1)).

Proof. (1) Assume dim ( I , A , M ) < A and we shall get a contradiction. Clearly, h > KO. We can assume 111 = dim(I,A,M). Choose distinct ~ , E I (n < w ) . By 3.12(2) (using 4.7 instead of 3.2) and Section 2, there are B, c A such that tp(gon.. . ^ ~ , , , A ) E F(B,). So when cfh >KO, B = U,,,B, has cardinality < h hence, tp,(UI, A ) E F(B) (by the indiscernibility), so when x = a, we get a contradiction to (iv). Similarly, when x = t let J = {G,,; 0 < n < ~},~obyAx(V.1)tp(G~,A~Uln_~Z~)~F(BuU~~~li~), hence tp(lio, A u U J) = Av(J, A u U J) E F(B), contradiction to (iii). So (iii) [(iv)] implies cf h = No (i.e., (i)); but they follow from 4.10(1) below. So we can assume cf h = No hence K(T) = No, and have to prove (ii); so suppose 111 > KO. Choose disjointJoy J , s I , lJol < X,, I Jll = KO, such that Av(I, U I) does not fork over U Joy Av(I, U J,) is stationary. By 4.8 M is F-prime over A u U Joy so w.1.o.g. J, = 0. Let N be F-prime over A u u J, , ao there is an elementary embedding f of M into N, f 1 A = the identity. As 111 > No for some li E I f (a) realizes Av(J, A u U J , ) , so by 4.7 Av(J, A u U J , ) is F-isolated, hence by 4.10(1) below dim(I, A, bl) 2 A.

CLAIM 4.10: (1) Under the assumption of 4.9(1) if 2 = a, tp*(U I, A) E F(B) or for s m injnite J s I , Av(J, A u U J) is F-isolated, then Av(I, A u (J I) is Fz-isolated where p = h + 111 +; so 111 2 A.

Proof. For simplicity we concentrate on 2 = a, tp*(U I , A) E F(B). By the indiscernibility of I , Q = Av(I, B u U I) k Av(I, A u U I).

So it suffices to prove that Av(l, B u U I) is FE-isolated. By the indiscernibility of I and Ax(V.l), for every finite J E I there is an

188 PRI116E MODELS [OH. Ivy 8 4

F-type qJ c q almost over B u U J, q, t- Av(I, B u U J). Clearly q = U {qJ: J c I, J finite} is an FZ-type almost over B u U I, q t- Av(I, B U U I), so we finish.

For 4.9(2) we prove:

CLAIM 4.10: (2) If U J c B E C, J an injinite i?zdiecernibie set, IJI 2 A 2 K,(T), c i8 F-constructible ower B, F = F;, x = t , 8, a then Av( J, B) t- Av( J, C).

(3) The claim holds for FX,h >, K,(T), too.

Proof. Let S = ( B y {6;: i < u}, {B;: i < a}) be an F-construction of C over B. We prove by induction on i that Av( J , B) k Av( J , B(i)). For i = 0 it is trivial, for i = 6 a limit ordinal it is immediate. For i = j + 1 , B, c S(j), IBjI < A, so by 111, 3.6, for every a E S(j) there is Jd E J. I J,J < lB,l+ + KJT) I A s I JI, such that J - J,is indiscernible over u B, u U Jd. Let B realize Av( J, B(i)). In all cases there is an F-type

p c p , = tp(6,, S(j)) [almost] over B,,p t-p,. Nowp t- tp(6,, S(j) u a) (for otherwise for some E S(j) C q[6,; By Z] but p u {-,p(E; E, a)} is consistent. So for any 5’ E J - Jay p u { - , q ( E ; E’, a)} is consistent, and, of course Cq[6,; E’, a]. Contradiction to p t-p,). By Ax(V1) for FL

Av(J, a(j)) = tp(8, a(j)) I- tp(E, S(j) U 6,) = Av(J, B(i)).

By the induction hypothesis Av( J , B) t. Av( J , a(j)).

i = u we get our result. Hence Av(J, B) t- Av(J, B(i)). So we finish the induction, hence for

(3) Because an Fi-construction is an F;-construction.

EXERCISE 4.3: In 4.10(2) for x = t show that A 2 K ( T ) is sufficient.

Continuation of 4.9. (2) Suppose not, so we can assume IIl = A+. By 111, 4.17(4) for some J E I, I J1 < min{K(T), X,}, p = Av(1, 1611) does not fork over U J andp (U J) is stationary. By 4.8 M is F-prime over A u U J. So we can replace A by A u U J , I by I - J , retaining all our previous hypotheses and in addition p A is stationary, p does not fork over A. Let I = {gf: i < A+}, M* be an F-primary model over A u U I, where I, = {al: i < A}, H* exists by 3.1; and by 4.10, Av(1, A u U I , ) t. Av(I, 1M*I). As A c 1M*1, H* is F-saturated, there is an elementary mapping f: M --+ M*, f t A = the identity. So {f(af) : i < A + } is an indiscernible set based on p r A, hence equivalent to I. As K,(T) 5 A, for some if(af) realizes Av(1, A u u I,) contradiction to Av(I, A u U I,) t- Av(I, lM*I).

CIE. Iv, 5 41 PRIME MODELS FOR STABLE THEORIES 189

(3) Here the hypotheses of (1) and of (2) hold. We can assume Av(I, IMI) does not fork over A, Av(1, A) is stationary, as in (2) [by (1) we can assume A I 111, hence I JI < 8, I cf A s A when cf A 2 K( 2') + K,, and IJI < KO = 111 when A = No = K(T)]. We can have I = {af: i < A} (by (1) and (2)), and define M*, f as in (2). Let the image of M by f be N. So it suffices to prove Av(I, INI) is Fg+-isolated. By Claim 4.10 Av(I, A u U I) is Fg+-isolated, but Av(I, A u (J I) 1 Av(I, 1M*1) hence Av(I, IN*!) is Fg+-isolated. Now as Av(I, IM*I) does not fork over A c IN!, by 4.3, Av(I, INI) is Fg+-isolated.

(4) Let f be an elementary mapping, A c Dom f c I MI, Range f E [MI, f 1 A is the identity and JDomf - A1 < A; and a EM. We have to find an elementary mapping fl extending f, Dom fl = Dom f u {a}, fl(a) E IN). Let M l be FZ-primary over Domf, so we can assume IM'I E 1M1.. As M is Fg-saturated, by 3.10(2), Exercise 3.6 we can extendftoanelementarymappingfl, Domfl = lM1l, Rangefl c !MI. Choose C c lMll, C < ~(2') such that tp(a, lM1l) does not fork over C. We now define by induction on u < w + w , a, E IMII realizing tp(a, A u C u {af: i < u)). (Remember M is Fz-atomio over A by 4.7, so the above mentioned type is Fg-isolated by 2.8(2), as ICl + KO < K(T) + Kl I cf A, hence it is realized in M1.) So let I = (a,; w 5 a < w + w}; clearly I is indiscernible over A u C, Av(I, lM1l) = tp(a, IM'I). By 4.9(1) A I dim(I, A, M), so let J 2 I be indiscernible, I JI = A. Now some b E J should realize tp(a, Domf); so we can define fl by: fl(a) = fl(b), fl 2 f.

(5) Since F satisfies Ax(X.l), (XI.i), there is over A an F-primary model N . Since M is F-prime over A , w.1.o.g. M < N . Suppose I C M is indiscernible over A , 111 = A+. As in the proof of 4.9(2), w.1.o.g. Av(I,A U I ) does not fork over A,Av(I,A) is stationary. As A+ 2 K(T) clearly

(*) For every El all but d A of the members of I realize Av(I,A U C).

By 3.4, since I C N, there is an F-construction over A , &' = (Al(ai:i < A + ) ) , such t h a t 1 S &'(A+). Choose a < A+, c f a = A, such that )I n &'(a)l = A and for every /3 < a some CEI n &'(a) realizes Av(I, d(/3)) (this is possible by (*)). Again by (*) some J E I realizes Av(I, &'(a)) butby3.2,&'(A+)isF-atomicover&'(a) (and2EI c &(A+) ) so tp(d, &'(a)) is F-isolated. So for some D c &'(a), tp(d, &'(a)) E F(D), so ID1 < A. Since cfa = A for some /3 < a,D G &'(/3). But by the choice of a some J 'EI belong to &'(a) and realizes Av(I, &'(/3)), i.e. it

190 PRRtfB MODELS [a. w, 8 4

realizes tp(d, d(/3)). This contradicts tp(d, &(a)) EF(D),D G &(p) (for x = a remember Av(I,A) is stationary, hence tp(d, d ( p ) ) is stationary).

We now prove some preliminaxies to the characterization theorems.

LEMMA 4.11: h4 x = t , a . Assume 8&i8@3 Ax(X.1), (xI . l ) , h 2 K ( T ) and cf h > No.

If M is e-cumpact and Fz-atomic over A , I E ]MI an infinite indiscernible set over A and p = Av(I, IMI) is c+-ieolated then there is an indiscernible set J ouer A , equivalent to I , J c 12111, IJI = h and Av(J, A u U J) k Av(J, 1611).

Proof. By 111, 4.17(4) there is a C E IM], such that p = Av(I, 12111) does not fork over C and p tC is stationary, and ICl < 8,. By a hypothesis there is B E IN], IBI = A, ( p , B) E Ff+. We can aasume C E B [by Ax(III.2)] and let E realize p , So tp(E, B) k stp(E, B) (by 111, 2.14) hence tp(E, B) k tp(E, lM1) = p (see 2.1 for x = a).

Let {6,: i < A} be a list of all finite sequences from B, each sequence appearing A times. Now define by induction on i < A, a, E 12111 such that a, realizes Av(I, A,) where A, = C u A u {a,: j < i}.

If at is defined for i < a then as in the proof of 4.9 above, there is an F-type q a c p overC u A u U {a,: j < a}, q, k p t A,. If q, k p 6=, 8, will be any element of which realizes q, (there is one, aa M is F-compact, q, an F-type). Otherwise there is a formula ~ ~ ( 3 ; 6,) such that qk = q, u {tp,(Z; 6,)3 is consistent for t E (0, 1). As p 1 C E q, p t C is stationary, for one such t, q: forks over C. W.1.o.g. it occurs for t = 0, and let G, E [MI realize q:. As q, k p 1 A, and p 1 C is stationary, J = {a,: i < A} is an indiscernible set over A equivalent to I (as both are based on p C) by 111, 1.10.

As h 2 K(T), by 111, 2.4 for every 6, there is J’ c J, I J’I < K(T) 5 I JI such that J - J’ is indiscernible over 6, u C u A u U J’. Hence for every 6 E B the set {i < A: 6 = 6,, there is tp@; 6,) aa mentioned above} haa cardinality < A. Hence for some a, 6, = 6 and q, k p t 6,, hence p (A u U J) k p 6, (q, c p (A u U J) of course). Aa this holds for any a, p t (A u U J) k p B, but p 1 B k p by the choice of B. Hence p (A u U J) k p so we finish.

THEOREM 4.12: (1)* Swppose tp@,, A ) tp(a,, A,) where for 8 E p, A , = A U U{it,: j~S} (notice i = {j:j < i}). (The iti’$ m y be inJinite, in fact).

OH. I v , 8 41 PIlfME MODELS FOB STABLE THEORIES 191

Assurne case (i): for no i < p does tp(6, A) k tp(6, A u a:), or case (ii): No < pandfor nos E p, IS1 < pdoes tp(6, A,) t- tp(6, A#).

(2)* Suppose p 2 h and that in the hypothesis of (1) we udd to: case (i): tp(6, A ) is Fg-isolated (z = t , s), case'(ii): for every finite S G p, tp(6, A,) is F$-isoZuted; and (A)

Then p < ha(T) (see Definition 2.6).

p > A, or (B) p = h is regular ( > No). Then Ff (z = t , s) does not satisfy Ax(XI.1).

(3) The same a8 (1) replacing tp by stp and the conclzcsion is p < K(T). (4) If in (3) p 2 h and in the hypothesis we add to: case (i): tp(6, A) is Ff-isolated, case (ii): for every finite S E p, tp(6, A,) is B'g-isolated; and (A)

p > h or (B) p = h is regular > KO. Then Ff does not satisfy Ax(XI.1).

Proof. (1) Suppose case (i) holds but p L ha(T). Then there is q E &'"'(A,), p = tp(6, A ) E q and a < p such that q 1 A, k q (this can be done by the definition of P ( T ) and by renaming the ail's which is possible by 3.13 applied to F&). By renaming, we can assume 6 realizes q, without changing the hypothesis.

By Ax(V1) for Fm, w tp(6,A) Vtp(6, A u a , ) a h tp(iZ,, A) Y tp(iZ,, A u 6). So there are E E A and 'p such that b[6; a,, i5J but tp(iZ,, A) u {-,'p(6, f, E)} is coneistent, and let iZi realize it. By assumption, there is an elementary mapping f,f 1 A, = the identity, f (a;) = a,. We can extend it to an elementary mapping g whose domain includes 6. Then g(6) realizes q A, but not q 1 A,+1, a contradiction.

So we are left with case (ii), and we reduce it to case (i) as follows: Let So = {a < p: tp(6, A,) If tp(6, A,+l)}. Now w.1.o.g. So = p. [we define by induction on a 5 p sets &a) E p, S(a) increasing and continuous, lS(a + 1) - S(a)l < KO. If S(a) is defined tp(6, As(,)) Y tp(6, A,) 80 for someS(a + I), #(a) E S(a + I), IS(a + 1) - S(a)l < No, a €#(a + I), tp(6, A,(,)) br tp(6, As(,+1,), and S(a + 1) is minimal under those conditions. Let S(a + 1) - &a) = {~,(,.o), - . . , %&k(a))}, a,* = %a,o)na,(=.l)

now the az's satisfies all the hypotheses on the iZ,'s, and for them So = p.] So for every a there a m n(a) < o, i(Z, a) < a (for I < n(a)), formulas 'pa, and 5, E A such that C'p,[6, a,; 7il(o,,), . . . ; E,] but tp(6, A,) u {-,'pa(%; a,, . . . , E J } is consistent. If KO < x 5 p, x regular, by 1.3(1) of the Appendix there is a stationary Sx E x , such that a € S X * n(a) = n, i(l, a) = i(Z) (for I < n). Then let A' =

-. . .n-

192 PRIME MODELS [OH. Iv, 0 4

A u U1<,, a,iI, and hypothesis holds for a:, A', 6 (a < x ) hence by (i) x < ha(T).

is a#, where fl is the ath element in dx. Clearly our

So we are left with the c&88 p = ha(T) > No is singular. Let p =

For every regular x < p, find Is,, n = n,, i(Z) = i,(Z) aa above. We can aasume that for 5 # < cf p, ix(& 4 dHc). We let A' = A U

(a,: for some 5 < cf p. Z < nx(o, i = iXto(Z)}, and {a:: a < p) be

2idId X W , cfp < x ( 4 < t c 9 x ( i ) Wular.

{a,: i Ed,(o, 5 < cf p}. (2) The same proof. (3) The same proof as (1). (4) The same proof as (2).

THEOREM 4.13: Suppose Ff = F, z = t , s , a , F satisJies Ax(XI.l), and

(A) h > 8, is regular, or (B) z = t , 8; cf h 2 ha(T) + HI, OT

(C) z = a, cf h z K(T) + N,. If for every i < 6, E E C,, for .9ome @ E A tp(E, A,) E FS(&), where

c,, and for i < 8, tp(6, A,) i8 F-i80latfd, then tp(6, A,) i8 A, = A U

F-?kO&e&.

Prmf. W e prove it by induction on cf 8. If cf 8 < cf A, thia is immediate by Axiom (IX). Suppose the conclusion fails. Then we can eaaily define by induction on i < cf 8 , a(i) < 6, and a, E C,,, such that

(1) j < i 3 a ( j ) < a(i), (2) tP(6, A,{)) LJ tP(6,AM:) u 4) for 5 = t, 8,

(3) stp(6, A,,)) Y stp(6, A,,, u a,) for x = a. W.1.o.g. 6 = cf6 (by renaming), so we already know 6 2 cfh. If A

is regular, z = t , s we get by 4.12(2) case (ii) (for p s f 6) that Ax(XI.l) fails for Ff contradicting an assumption. If h is regular z = a, we get a similar contradiction by 4.12(2) case (ii). So we have proved case (A) of 4.13. In case (B) of 4.13 apply 4.12(1) case (ii) and in case (C) of 4.13 apply 4.12(3) case (ii).

Remark. In 4.13 for x = t , s we need not assume T is stable.

DE~MTION 4.2: Let cfh 2 K(T) + N,, z = t , 8 , ( t . Then M is d e d Ff-admissible over A if:

(1) bd is q-saturated. (2) A c IM(, and M is F$-atodc over A.

OH. Iv, 8 41 PRJME MODEL9 FOR STABLE THEORIES 193

(3) For any infinite indiscernible set I (of elements) over A , I G ]MI, Av(I, 1iK1) is F,"+-isolated.

Note. If there is no I aa in (3) then of come (3) holds vacuously.

THE FIRST CHARACTERIZATION THEOREM 4.14: Suppose cf h 2 K ( T ) + K,, x = t,8,a, F = FZ, F 8Udi8fies A X ~ M (x.1) and (xI.1). If h i8 8i747UhrY 2 = t, U%8U??M 8Udi8$&9 Ax(XI.1) fc" arbitrarily large regular p < h. For x = t, h 2 h3(t) or at leaet Pi-saturated implie8 h - v t . Then:

(1) i8 Fprime over A M i8 F-admissible over A . (2) The Fprime model over A is unique up to i s m p h i m over A . ( 3 ) If M I is F-adrnissible over A,, 1 = 1,2; f an elementary mapping

fr.m A , onto A, then we extend f to an i 8 0 t ~ ~ p h ~ s n ~ from M1 onto Ma.

Remark. In the proof, if x = a, you should replace tp by stp in many cmes.

Proqf. (1) If M is F-prime over A , then M is F-saturated, by definition, iK is F-atomic over A by 3.12(2) (using 4.7 instead of 3.2), and if I c ldll is an indiscernible set of elements over A then Av(I, !MI) is Ff+- isolated by 4.9(3). Hence M is F-admissible over A. As there is an F-prime model over A by 3.12(1), the other direction follows by (3).

(2) Follows by ( l ) , (3) and the existence of an F-prime model over A by 3.12(1).

( 3 ) Assume x = t or x = a (the result for x = 8 follows by x = a as h > KO (see Exercise 2.30, 2.2(6), (7))). Let h = & A, be such that if A is regular, x = K,(T) + K,, A, = A ; and if A ,is singular then A, is regular, x = cf A ; and if x = t , A singular then Fi(,) satisfies Ax(XI.l) ; and h(i) = A, > lil, A, 2 z,<, A, + K ( T ) + K , . For every C E lMIl 1 = 1, 2, there is A, = h(E) such that tp(E, A,) is F&-isolated (we can assume 1M,1 n IiKaI = 0) . We shall define elementary mappings fa (a < x ) such that

(i) ,!? < EL implies fa extends fs, (ii) A, c Domf, = A: c lM1l, A , E Rangef, = A: c IMaI,

(iii) M,, M , are F-admissible over Dom fa, Range fa respectively, (iv) if 2 = 1, 2 E E lMI1, then tp(E, Af) is FE-isolated, where p =

For the checking of F-admissibility in (iii) notice that by (3) conditions (1) and (3) &om Definition 4.2, are immediate; hence only (2) (on being F-atomic) needs proof, and it follows from (iv).

A, + h@).

194 PRIME MODELS [CH. IV, Q 4

Case 1. a = 0.

Let fo = f; trivially all conditions hold.

Case 2. a = 8, a limit ordinal.

Let fd = U,<df,. Then (i) and (ii) are immediate. For (iv), as 8 < cf Ad this follows by Ax(IX).

Case 3. a = 8 + 2n, 8 is 0 or limit, fa is defined; we shall define

Choose by induction on i elements a: = a:,, E lM,l - At, such that for some B:,, E A:, tp(a:,,, A: u {&:j < i}) E F;(B:,,) where p = A,. The first i for which we cannot find such a:,, is i(a). Now we define a:,, E 1bl,1 which will realizef,[tp(a:,,, A: u j < i } ) ] by induction on i < i(a) (this is possible as bl, is F-saturated). We extendf, to by letting is an elementary mapping and (i) and (ii) am clear. As for (iv), if E E 1bl,1, by the induction hypothesis tp(Z, A;) is FE-isolated, p = A, + h(E), and the h ( j ) are regular > KO. So (iv) follows by 4.13(A), by induction on i(a).

(a:,,) = a:,,. Clearly

Case 4. a = 8 + 2n + 1, ( 8 = 0 or 8 limit) fa is defined and we shall define f a + 1'

We define by induction on i infinite non-trivial indiscernible sets (of

(A) For some infinite I:,, E A:, I:,, u J:,, is indiscernible. (B) For any n, a,, . . . , a,, E J;,,, for some Q,, c A:,

elements) over A:, Ji , , such that:

(C) Av(J:*,Y A: u u JL) AV(Ji,,Y 1~11).

Let i(a) be the first i for which there is no such J:,,. By 3.13 clearly J:,, is indiscernible over U U,+, Jk,, v A.

Now we prove by induction on i that when p = A, + h(E), then tp@, A u U U,<, J:,,) is Fg-isolated. For i = 0 this is by the induction hypothesis on a, for i limit, by 4.13, and i successor by 4.16 (see below) (as A, 2 K ( T ) + K,, Aj is regular). So first we prove

CLAIM 4.16: Swppme cf A 2 K(T) , A u U J v 5 i s F:-atomic Over A

OH. Iv, 8 41 PRIME MODELS FOR STABLE THEORIES 195

(z = t , 8 , a) and Av(J, A u U J) t- Av(J, A u U J u E), and J ie indiscernible over A. (1) tp(8, A U u J) i8 Fg-iSOlated, Fd&d t h d cf r\ 2 K(T). (2) If J is bmed on A, Av(J, A) i s dationary, tp(E, A u U J) doee not

fork over A , then tp(E, A) k tp@, A u U J ) ; stp(E, A) k stp(E, A u U J).

CLAIM 4.16: SuryHEse B s A , tp(?i,, A) dim nd fork over B, and tp(sil, B) i8 StatiOnaY, 1 = 1, 2. (1) I f tp(7i1, A ) k tp(81, A u a,) t h tp(a1, B) k tp(ai,, B u a,). (2) If stp(a,, A ) k stp(lZl, A u a,) then stp(a,, B) k stp(a,, B u Z,).

Remark. As Ax(V1) is satisfied by FO,, the assumption and conclusion of (1) and (2) are symmetric for al, a2. See also Ex. 4.10.

Proof of 4.16. (1) Suppose that the conclusion fails, so there are 8 E B and a formula v, such that tp(Si,, B) u (cp(Z; a,, Z)t} is consistent for t E (0, 1). So there are $, t E (0, l}, such that tp($, B) = tp(a,, B), hp[ii;, Zi,, 8It. By 111, 2.6(3) there me sequences 6p, 6!, 6? such that tp(6pn6:n6,, A) does not fork over B, and stp(6pnb:^6,, B) = stp(Bpnti:nB,, B). Hence stp(6,, B) = stp(a,, B). But the last type is stationary over B, by 111, 2.9, hence tp(6,, A) = tp(a,, A). Also tp(6:, B) = t p (4 , B) = tp(a,, B) (i = 0, 1) and the last type is stationary, by hypothesis. So tp(6i9 A) = tp(7il, A). But clearly b[@, 62, Elt (t E (0, 1)). SO tp(7il, A) = tp(@:, A) It tp(6!, A u 6,). AS tp(B,, A) = tp(6,, A), tp(8,, A) If tp(a,, A u a,), so the assumption fds.

(2) If stp(Z,, A) k stp(a,, A V a,) then stp(a,, A) k tp(a,, A u a,), Hence tp(8,, A U a,) does not fork over A, and even over B. As tp(al, B) is stationary, by 111, 2.14, tp(8,, A) k stp(a,, A u a,) hence tp(G,, A) k tp(a1, A u a,). So by (l), tp(al, B) t- tp(a,, B u a,). Again by 111, 2.14, tp(a,, B) k tp(Zi,, B U a,) t- stp(8,, B u a,), so we finish.

Proof of Claim 4.15. (1) Choose I G J, I I1 < K(T) such that tp(8, A u U J) does not fork over A u U I , Av(J, A u U J) does not fork over A U UI, and Av(J,A U UI) is stationary (by Theorem 111, 3.2 and 111, 4.17(4)). So 11 U 4 < K,(T) hence by Ax(V.2) for F:,x(T,), i t suffices to prove (2).

(2) For simplioity, let z = 8, t . Let J = {ai: i < p}, a,, realizes Av(J, A u U J u a), and A, = A u U {a1: i < a}. By assumption tp(iZ,, A,) k tp(a,,, A, u a) hence by Ax(VI) (for P",) tp(Z, A,) k

196 P&IME MODELS [OH. m, 5 4

tp(Z,A uSi,).ByClaim4.16(l)foreverya < p,tp(Z,A,) ktp(Z,A,,ua,,). As {B1: i s p} is indiscernible over A u B (by 111,2.10) tp(iZ,, A, u a) = tp(Z,,, A, U a) hence tp(Z, A,) k tp(Z, A, u a,). So by induction tp(Z, A ) k tp@, A#). (For x = a, use 4.16(2), and 111, 2.14.)

Continuadim of the pmf of 4.14. AS in Cam 3, we can define J:,{ c lMal [i < i(a)] such that Jiel is indiscernible over A:, and for every distinct a:, ..., U",J:,~ a; ...â% realizes f,[tp(a: ^...â:, B:,,)] (if 2 = a write stp instead of tp) and Av(Ji,,,A: u UJi , { ) kAv(Ji,,, lM1l). Such J;,{ exist by 4.11. So IJ:,{l = A = IJ:,,], so we can extend fa to fa+,, with domain A: U U U1 J:,{, which maps J:,{ onto J:,{. By 3.13 fa+, is elementary. We have proved that for B E lMll, p = A, + h(Z), tp(Z, A u U u1 J:,J is FE-isolated. A similar claim can be proved on any Z E 1M,1. So (iv) holds.

So we fmish the definition. Notice that if we interchange bl, with Ma, A, with 8 2 , a:,{ with at,{, J:,{ with J:,{ fa with f l l all conditions still hold. Also f, = U,<, fa is an elementary mapping, Dom f, c IM,l, Range f, c IMSl, f, extends f. So by symmetry it suffices to prove lM,l = Dom f,. Let c E IMII - Dom f,, and we shall get a contradiction. Now we prove

(*) If a < x , p E #(A:) is realized in MI, then it is realized in A;.

As M, is F-atomic over A:, p is F-isolated, so by Ax(XI.1) there are q E #(A:), p E q such that q is F-isolated. As of x 2 K( T) q does not fork over A: for some f l < x . Clearly for some j, q is e(,-isolated.

If A is singular choose y = h(i) < A, such that a,b < i, A ( j ) < y . So by 4.3 q' = q 1 A: is q(,,-isolated, so for some B G A:, (q', B) E PgCn, and aa A ( i ) is regular > A(j), B E A: for some f < y. We can aasume is even so we could have chosen as an element of M, mlizing q', except when q' is realized in A:, so we get the conclusion of (*). If A is regular the proof is similar but easier. As c E lMll - Dom f,, for some a(0) < x, tp(c, A:) d m not fork over

A:(o). By (*) we can define by induction on i < w + w + w, a, E A:,,,, a(i) < x incmaaing, a(i + 1) 2 a(i) + w (remember cf x > Ho), so that a, realizes tp(c, A:(#,), a, E A,(, +,). By 111, 2.12 and 111, 1.10, I = {a1: w s i < w + w} is an indiscernible set, tp(c, A&@)) is stationary, andAv(1, A:) = tp(c, A:). Letfl = U1<oD+oD a(i) (eoflisalimit ordinal). By 4.11 there is an indiscernible set J E 12111 over b a d on

OH. IV, 8 41 PRIME MODELS FOR STABLE THEORIES 197

tp(c, A:(,)), Av(J, A; +, u U J) k Av(J, 1 MII). (J is non-trivial as c $ A:.) As J is not J j +,,,(,,,), for some distinct b,, . . . , b, E J tp(6, A; + 1) Y tp(6, A; +,) where b = {b,, . . . , b,), w.1.o.g. n is minimal. h A j + , = A;+, u U Uj<r(,+l) Jj+l,j, thereisaminimalt a i@ + 1) such that tp(6, B) br tp(Zi, B u Jj+l,c) B = A;,, u U U,,, J;+ ,,,.

But (aa+a+l,. . - 9 aa+D+n-l, C) mliZe8 tp(6, B),

kt~((aa+a+~,.--,aa+a+n-l),Bu J~+I,,)

by the minimality of n, and

by Claim 4.15(2), a contradiction by Ax(VI1) (for A singular reduce it to some regular A' < A).

DEFINITION 4.3: M is F-*admissible over A (x = t, 8, a, F = Fg) if: (1) M is F-saturated. (2) A s 1 MI and M is F-atomic over A. (3) If I c ]MI is an infinite indiscernible set (of elements) over A,

then dim(1, A, M) 5 A. (4) If I = {a,: n a w) c lMl is indiscernible' over A, but

tp(ao, A u {a,: 0 < ?a < w)) is not F-isolated then dim(1, A, M) 5 No (this can happen only when cf A = No, and is not needed for A = No). We demand (4) only when KO = c f A < A.

CLAIM 4.17 : Let F = F,Z, x = t , s, a and M be F-*admissible [F-atomic, F-saturated] over A, and pi (i < a) types over A (if x = a almost over A), and A* = A U {G: ti€ JMJ, @ realizes eome pi, i < a) U B, where B E W I , IBI < c f h and for some regular p < A, tp,(B, A) E F,"(C), (CI < p. Then M is F-*admissible [F-atomic, F-saturated] over A*, provided at least one of the following conditions holds :

(1) A ie regular and F eatis* Ax(X.1) and (XI.1). (2) A is sing&?', for arbitrarily hrge regular p s A, % satie$a

Ax(X.l), (XI.1) and x = a a cf A r K(T), x = B =- cf A 1 Aa(T) and x = t a cf A r Aa(T).

(3) A is eingular, for arbitrarily large regular p 5 A, satiefi Ax(X;l), (XI.l), and x = t, and for some p a A each p, is a FL-type (or an FL-isolated type).

198 PIlIME MODELS [OH. Iv, 5 4

Proof. Clearly it suffices to prove the version with “F-atomic”, and WSuming B = 0.

(1) Immediate, by 3.15. (2) Similar proof, e.g., if z = a, and 5 E 1611, then for some B, c A,

lBll < A, stp(Z, B,) C stp(E, A). By 2.16(2), (iii), we can find B, c A*, lBal < ~(2’) and E’ such that E’ realizes stp(Z, A), and stp(E’, B, u B,) I- stp(E’, A*). As lBal < cf h we can continue = in the proof of 3.16. (3) Let E E 1611, and we must prove that tp(E, A*) is F-isolated. By

the hypothesis, there is a regular p < h such that FL satisfies Ax(X.1) and (XI.l), tp(E, A), and p , (i < a) are FE-isolated. So tp(E, A) has a FL-isolahd extension q E #“‘(A*) realized by 5’ E !MI. So for some r c q, Irl < .p, r C q, and let r = {q,(Z, a,): i < /3 < p}. For each i, there are sit, a?, p6, . . . , phcr, such that a, = at%?, at E A , si; E A, = U {a E M: a realized pi, 1 I n(i)} and p{ E {p,: j c a}. By 11, 2.2(8) for every i there is 8 , ~ A, and 6, such that for every A,, C q , [ E ’ , i Z t , a] E O,[si, at, 6,]. As M is Fi-saturated, pi is FL-type, there are I,!If(#, 2:) ~ p { , such that the above mentioned fact holds for every 7i E A; = U {a E M: 7i realizes I,!Ii for some 1 I n(i)}. So we can define by induction on i < /3, 6; E A, such that tp(6, A u E’ u {6,:j < i}) is q-isolated, and for every si E A;, Cq, [E’ ; sit, a] EE O,[si; a:, 6iJ. So clearly we can assume r is over A u (6:: i < /I} and the rest is as in 3.16.

THE SECOND CHARACTERIZATION THEOREM 4.18 : Suppose K ( T ) = KO, F = F;, z = t, 8, a and F eatisfies Aziome (X.1) and (XI.l) and when A is singular, for arbitrarily large regular p < A, F; satiefiee h(x.1) and (xI.1). Aeeume

(i) z = a; or (ii) x = s , t and h > No; or (3) z = 8, t, T totaliy tramcendentul.

(1) M is F-prime over A iff M $8 F-~admis~ble over A. (2) The F w m e &l Over A is unique up to ~eornorlph~sm, Over A. (3) If M , ie F-*admissible over A,, 1 = 1, 2, and f is an elementary

mapping from A, onto A, then we can extend f to an i8omOr.ph~87?~ from

Then

Mi onto Ma.

Proof. As in 4.14, it suffices to prove (3). For simplicity restrict ourselves to thecase 5 = a. Clearly it suffices to prove Claim 4.19(2) below, by applying it to qo = {z = z}, y = 1.

CH. m, 6 41 PRJME MODELS FOR STABLE THEORIES 199

CLAIM 4.19: 8uppoee M , , A, (1 = 1, 2) and f are as in 4.18(3). (1) If a: E lMll, tp(a:, A,) = f[tp(ar, A,)] then we can extend f to an

elementary myping from A , u {a E lMll: a realize0 stp(a:, A,)} onto A , u {a E 1M,1:

(2) If qf (i < y ) are l-typea Over A,, then we can extend f to an elenten- t a y mapping from A , u {a E lMll: a realize0 8ome qi} onto A , u {a E 1M,1: a real iz~ f(qf)}.

realize0 stp(af, A,)}.

Proof of 4.19. We prove by induction on a that: 4.19(2) holds, when D(q,, L, 00) < a and 4.19(1) holds, when D[stp(a:, A,), L, 003 s a.

Caae I. a = 0.

Then (2) is empty, and in case (1) by definition of degree stp(ar, A,) is algebraic, hence is realized by a* only, so extend f by lettingf(a:) = a:.

Case I I . a > 0, we have proved (1) and (2) for all fi < a, and we now prove (2) for a.

Choose UZEIM,I (i <c) such that for i < j < c, stp (a: ,Al ) f stp(a:,A,), a: realizes some q, and [ is maximal. Let A; = A, U (U:UEIM,I , a realizes some stp(a:,A,), i <j} (so for 1 = 2 it is defined only when (a::i <j) is defined). We define by induction o n j elements a: E 1M,1 (i < j) and an elementary mapping f, from A{ onto Ah, such that fo = f, i < j implies f, extendsf, and f,+,(a,’) = a;. So clearly fc is the mapping we need. Notice that by Claim 4.1.7, M , is F-*admissible over A{.

For j = 0, j limit ordinal, there is no problem, so let j = i + 1. Choose a: E IMsl, which realizes ff(tp(at, At) ) (possible aa M,, Ma me F-*admissible over Dom fi, Range ff, respectively). By the induction hypothesis we can extend fi to an elementary mapping f,+, from A\+, onto A:+’ (first, since M,,M2 is F-admissible over Dom f,, Range fi, respectively, we can extend fi to an elementary mapping f: with domain A: U a:, qgff:(a:)~M2; second, by 4.7 and (2) for D(stp(@,Ai),L, 00) we can find!,,,).

CaseI1I.a > O.Wehaveproved(1)foreveryp < a,and(2)forfi s a, and we shall prove (1) for a.

200 PRIME MODEL8 [CH. m, 8 4

Let I, E lM,l be a maximal independent set over A, whose elements realize stp(a:, lM1l). Let I, = {a;: j < a,} and a; = a: so if a, 2 w, I, is indiscernible over A, (by 111, 1.10 and 111, 2.9(1)). By Definition 4.3, 4.9 and 4.10, we can Mume that

(A) w1 I a, so, for some regular p s A, for every i < A tp(ai, A, u {a;: j < i}) is FUO+lil +-isolated hence w.1.o.g. a, = A, or

(B) a, I w. By F-homogeneity of (Y,, a)croAl it is easy to see that the same case

If (B) holds, define by induction elementary mappings f,,, n < w,

(i) fo = f, f,,+l extends f,,, M,[M,] is F-admissible over Domf,,

(ii) ut, i < n belong to the domain 0ff3,,+,. (iii) at, i < n, belong to the range 0ff3,,+~. (iv) Domf,,,+, = Domf3,,+, u {a E lYll: a realizes stp(af, A,), and

tp(a, Dom fsn + forks over A l}. This is easily done. Iff3,, is defined, tp(a:, Domf,,,) is F-isolated (by

Definition 4.3, part (2)), hence f3,,[tp(4, Domf,,)] is F-isolated, so by Definition 4.3, part (2), it is realized in Y,, by some bi , and letf3,,+,(u;) = bi , D0mf3,+1 = Domf,,, u {a:}. If n = 0 we can let bi = a% = a:. The definition of f3,, + 2 is similar by Definition 4.3( 2) and that of f3,, + is by the induction hypothesis on 3n + 2, by (1) [notice that if a realizes stp(a:, A,) then tp(a, Domfl) k stp(a:, Al); and by 111, 1.2(4), forking implies decreasing in degree]. Now U,,<@ f,, is the desired mapping (by the maximality of I,).

So wume (A) holds, i.e., a, = A. So I, is indiscernible over A,, and, using 4.17, for every i < cf A, M , is F-*admissible over Bf = A, u {a;: j < i}. Let Ci = A, u {a E Y,: a realizes stp(af', A,), tp(a, Bf) forks over A,}. Clearly for every i I A, we can extend f toft by letting &(a;) = a;(j < i), and if i < of h then extend it by 4.18(1) to an elementary mapping fr from 0: onto q.

If tp(ai, 0;) forks over A,, then by F-homogeneity it is eaey to prove that if a mdizea stp(a:, A,), a E M, then stp(a, CL) forks over A,. Hence we can prove (1) as in (B), (with C5, u {a E liK,l: tp(a, C5,) forks over A,} included in Domf3,,, &ngef3,, for I = 1,2 reap.

So assume tp(ai,Ck) does not fork over A. Then, again, tp(at,Cf) does not fork over A, for i < A. Also if F E Cf (i < A) then stp(c,B;) I- stp(c, B;+J (by the non-forking assumption, i.e. otherwise, sinceM, is F-saturated, we can find Z E M , realizing stp(F,B;), but not stp(c,

occurs for I = 1, 2.

such that:

[Range f n l .

OH. Iv, 8 41 PRIME YODELS FOR STABLE THEORIES 20 1

B:+J but necessarily C’E C: and tp(af, C:) fork over A;, a contradiction) hence stp(C,B:) I- stp(C,Bi). In fact for every CE W J , for some finite J E I,, stp(C, A, U J) t- stp(C,A, U I). So Ml is F-atomic over Bi, and we can easily show that M , is F-*admissible over Bi (notice we can reorder the ai’s). So by 4.18(2) (for a) we can extendf, to the desired mapping.

8 k h of the proof for other ma. Let x = 8; by 2.2 FA-primenesa (seturation) is equivalent to R-primeness (saturation). By Exercise 4.8 FA-*admiasibility implies F+dmLsibility. So as we have proved 4.18 for x = a; it suEces to prove that over any A there is en FA-*admisaible model, but this follows by Ax(XI.l) and Exercise 4.14 and 4.16. If x = 8, t T totally transcendental, use R ( p , L, 2) instead of D(p, L, m), and the proof is similar.

If x = t, h > KO, in proving 4.19, case 11, we first, as in the proof of 4.14, extend f to f‘, such that (bl,, Dom f‘), (Ma, Range f’) are F-*admissible, and Dom f’ is the universe of e model; so each p, is stationary.

If x = 8 , t h > of h = KO, in proving 4.19, case III(B) let h =

z n < a p n , pn < pn+1, F,. satisfies Ax(X.1), (XI.l), pn regular, and

(iv)’ Dom f3n+3 = Domf,,+a U {a EM,: a realizes stp(a:, A,) and

In cases 11, III(A), the change is similar, and when cf h > KO-is

tp((a& . . . , a;>, A) is F,I-isolated, and e.g.,

tp(a, Dorn fsn+a) forks over A, and is F,,n+,-isolated}.

easier.

DEFINITION 4.4: bl is F-minimal over A, if M is F-saturated, but there is no F-saturated model N , A G IN1 E IMI, IN1 # ]MI .

LEMMA 4.20: If Over A there are an F-prime model and an F-minimal model, tlren, up to ietnnorphim Over A , there i8 a unique F-prim model Over A, and a unique F-ncinimal model over A and they are ietnnorphic Over A .

Proqf. Let blo be F-prime over A, No F-minimal over A. By the d e h i - tion of F-prime there is an elementmy mapping f: M, --t N o f 1 A = the identity. By the F-minimality of No, f ( M o ) = No, so they are isomorphic over A.

Hence if MI, bl, am F-prime over A, they me isomorphic to N o over A, hence isomorphic over A. Similarly any two F-minimal models over A am isomorphic over A.

202 PRIME MODELS [CE. Iv , 8 4

THEOREM 4.21: Let F = Ff, x = t, a, a, of h 2 K(T) , F satisfies Axioms (X.1) and (XI.l), and: T is totally transcendental or x = a =- h > KO, x = t =- of h > No. Suppoae M is F-aaturated and F-atomic over A. Then il4 i a F-minimal over A if in M there &3 no in$nite non-trid indkcemible aet over A.

Proof. suppose I C 1M1 is an infinite non-trivial indiscernible set over A. We can assume I is a maximal one, and let I = {gi: i < a} (a 2 w) . Let N be any F-prime model over A u U I (exists by 3.1). We can assume N < il4, hence N omits Av(I, A u U I), by the maximality of I . Hence also any F-prime model over B = A u {a,: 0 < i < a} omits Av(I, I?); and let N , E il4 be one such model. So N, is F-saturated, A c INl/ E 1iK1, but ao$ IN1]. We conclude

hf is not F-minimal. Now assume M is not F-minimal, so there is an F-saturated N,

A E IN1 E ldll IN1 # 1il41. Choose C E 1M1 - INI, and C c IN!, ICl < K(T) such that tp(c, INI) does not fork over C. For simplicity let x = a, choose c, E IN1 which realizes stp(c, A u C u {c,: i < n}) (exists, &B bl ia F-atomic of A, and by Ax(V.2) (or 2.8), tp(c, A u C u {ci: i < n}) is F-isolated). So clearly {c,,: n < o} c is an infinite non-trivial indiscernible set over A (see 111, 1.10 and 111, 2.9).

EXERCISE 4.4: Prove 4.18, when x = 8, h = No, and T is transcendental (see Definition 11, 3.4).

QUESTION 4.5: In the characterization theorems, can we weaken the assumption “F satisfies Ax(X.1) and (XI.1)” by “over A (A,) there is an F-primary model”? (see 6.6).

EXERCISE 4.6: Let M be F-**admissible over A, F = Fie, x = a, t, a if iK is F-saturated, F-atomic over A, and for every finite B c 1M1, and p ES(A u B), dim(p, A u By M) is s No. Prove a characterization theorem parallel to 4.14, 4.18, for countable superstable T . (Instead of countable, you can assume h*(p) s No for every Fio-isolated complete typep; h*(p) is definedin Definition 111,6.1). Generalize similarly 4.21.

QUESTION 4.7: In 4.12, try to replace K(T) by K,,,~(T), at least for stable T .

EXERCISE 4.8: Prove that if 211 is Fi-admissible over A, h > X, then

OH. Iv, 8 41 PRWE MODELS BOB STABLE THEORIES 203

M is F+dmissible over A ; and similarly for -*admissible and -**ahis- sible. [Hint: Use 2.2; note that Fi-isolation (atomicity) implies R- isolation aa for Definition 4.3(4) note that tp(ao, A u {a,,: 0 < n < w}) I- stp(ao, A u {a,,: 0 < n < a}), so if tp(ao, A u {a,,: 0 < n < w}) is not Pi-isolated, i t is not Fi-isolated.]

QUESTION 4.9: In the proof of 4.14, A regular, can we choose x = K,(T) for A singular, can we choose x = cf A ?

EXERCISE 4.10: Show 4.16(2) holds also when tp(a,, B ) is not stationary.

PROBLEM 4.11: In the characterization theorem 4.14, can we weaken Definition 4.2(3) to “dim(I, A, M) 5 A ” ?

EXERCISE 4.12: Show that 4.18 may fail for x = 8, t , A = No. [Hint: T = Th(M), M = (m2 x w , . . . , E n , . . .), where <?I, k ) E,,(v, I ) iff 7 1 n = Y 1 n. This model contradicts 4.18(1).]

EXERCISE 4.13: Show that 4.14 may fail if A < K(T) . [Hint: Use T’ (see Exercise 11, 2.3).]

EXERCISE 4.14: Suppose K ( T ) I cf A, A is Fi-constructible over B then A is Fi-atomic over B, provided that

(A) Ax(XI.l) holds, but not cf A = No < A or (B) A is a limit cardinal, and Ax(XI.l) holds for Fi for arbitrarily

large p < A. [Hint: See 4.7; for (A) use Exercise 2.30. For (B) let A = 2,ccfA A,, 2,<, A, < A, < A, Fil satisfies Ax(XI.1). Now we can define inductively pAl-saturated models M,, IM, I Fi,-constructible over A u U,<, M,, and let M = Uf<cfA M,:By 111, 3.11 A2 is Fi-saturated, by 1.3( 2) M , is F$-constructible over A, hence F:A,+,-COnSfI’UCtible over A, hence by 3.2(1) lM,l is FfA:,-atomic over A, hence M is Fi-atomic over A. Now as in 3.12(2) we can prove A is Fi-atomic over B.]

EXERCISE 4.15: Suppose (a) A is regular and PA satisfies Ax(XI.1) or (p) A is singular, and for arbitrarily large p < A FA satisfies Ax(XI.l). Prove that 4.9 holds for x = 8.

EXERCISE 4.16: Use for 4.14, 4.18, eto. (i) or (ii) from Exercise 4.16, instead “for arbitrarily large regular p 5 A satisfies Ax(XI.l)”.

204 PRIME MODELS [m. IVY 6 6

IV.5. VariouS results

THEOREM 6.1: Swppwe IT1 = A, p s A, p replur. T h (1) == (2) (3) + (4) where:

(1) ID(T>l < 2'. (2) There arenozand formulascp,(Z) (7 E "'2) (ud?mutpranz&rs)mh

that for every 7 E '2, {qnlt(Z)qrt1: i < p} is &t& with r . (3) For every m-type p over 0, IpI < A, there is an m-type q over 0

m h t W Iq1 < 1 1 , a n d p u q ~ r f o r s ~ r ~ D , ( T ) . (4) (A) T hm a model A2 of cardinality A, which is FA-conetructible (over

0 ) hence FA-pirnitive and FA-at0mic (over 0). In fact there is such a model over any A, IAI < A.

(C) I f A = Ash, (B) I f A<" = A, we can have M ~-COm2)aCt (Irence Fx-eaturated).

T hm a A- model of cardinality A, t h M ie F A @ ~ y and therefore FA-p+-i~ and A - p i m e .

Remark. By 111, 6.14 we can assume IT1 s ID(T)I.

Proof. We leave it to the reader, as it is similar to previous proofs, except (4) (C), where for h regular it is easy; for h singular, see Ex. 5.17, 5.18.

Remark. Notice that every Fi-saturated model has cardinality 2 A. A converse to 5.1 is :

THEOREM 6.2: Suppose IT1 = A, and T hae a A-compact m d e l of cardidity A (e.g., when A = A<h), and A is regular.

(1) Among the h-compact models, there is a prime one iff condition 5.1(3) holds for p = A (see Exercise 5.16).

( 2 ) I f there is a k p i m e model then it i8 uniprce (up to isomorphisnz) and A-homogeneow. Bee DejEnition 1.6.

(3) I f there ie a Awme model M , it is characterizerl by (A) M i8 P",-saturated, (3) b l i 8 F A ~ k , (C) 11Ml1 = A.

Proof. ( 1) By 6.1 , if 6.1 (3) holds, T has a A-prime model of cardinality A. Now suppoae M is a. A-prime model, but the m-type p is a. counterexample to 6.1(3). As 1p1 < A, M A-compact, some Z E 1x1 realizes it, thus po = tp@, 0) is not FA-isolafed. The following theorem, 6.3, shows

OH. IV, 5 63 VlCIlIOUS RESULTS 205

that there is a A-compact model which omits po, thus M cannot be elementarily embedded into it, contradiction. The rest of the proof will follow Theorem 6.3.

THEOREM 5.3: 8uppose pi ie an m,-type mer A for i < a s A and for no mi-type q Over A , q kpi, 1q1 < A. Assume IAI + IT1 = A.

(1) T hua a model M, ( A s 1M1) of cardidity A omitting each pi (i.e.,

(2) If A = A < A , or T has a A-compact model of power A , A = 8, then in (1) we can choose the model to be Fi-saturated, and even A-compact .

7u) 8epWW tb model ~ f X d ~ Z e 0 it).

Proof of 5.3. (1) For simplicity let A = 0. Let {&: i < A} be a list of all finite sequences from {z,: i < A}, each sequence appearing A times and let {tpi(z, fjMi)): i < A} be a list of the formulas of the form ~(z, g,) (9 E L , j < A) such that &, gd0 E {z,: j < i}. We now define by induction on i < A a consistent set I', of formulas (with no parameters) such that in I', appear only the variables {z,: j < 2i}, and lI',l < lil+ + No. Let I', = 0, I', = Ufcd I',, clearly the induction assumptions hold. So suppose I', is defined, and let I': = I'f u "z)Pi(z, 37(1(f,)l + P?f(%:arY & f d u {%i+l z x,: 5 2;).

Trivially I': is consistent, and the variables that appear in it are only from {zi: i < 2(i + 1)) and its cardinality is < l i l+ + No. Let I'F be the closure of I': under conjunctions, and let

r, = {(3)v(&, Z): 2 disjoint to &, but ~{z,: j < 2(i + l)},

and v(&, Z) E ria}. So lril < A, r, an Z(&)-type. Hence when Z(BJ = m,, j < a, r, Vp, (changing the names of variables accordingly). Suppose & is the j th appearance of j i , in {gB: fl < A}: i f j 2 a or Z(&) + m,, let Ti+, = ria; otherwise for some $&) ~ p , , ri u {--&(&)} is consistent, so let I',,, = I': u {+,(&)}. Clearly the induction assumptions are satisfied, and I', = U i < A pi. As FA is consistent there is an aasignment xi H a, realizing it. By the Tarski-Vaught test {ai: i < A} is a universe of a model M , which by the construction has cardinality A and omits the Pi's.

(2) Similar proof.

Remark. For oountable T the condition on the p,'s is also neceseary for (1). Similarly if A = A < A it is ne(feBB&Fy for (2).

206 PRUKE MODELS [a. IV, 8 6

Continuatiotc of the pmf of 5.2. (2) If there is an F,-prime model M1, by 6.2(1) condition 6.1(3) holds, hence by 6.1 there is an FA-primary model AM,. As Ml can be elernenfitwily embedded into M,, and by 3.2( 1) M , is E*,-atomic, dso Ml is F',-atomic. So by 3.16 Ml is Fi-construotible (over 0), hence Ml is F , - p h q too. So by 3.9 MI, M, are isomorphic. As we could have replaced Ml by any F,-prime model, the F,-prime model is unique. Ite homogeneity follows by 3.7.

(3) Follows from the proof of (2). (Note the remark before 6.2.)

CONCLUSION 6.4: Let T bs countabZe. T ?ms an F&Wm 7nodeZ ifl for every formulcc cp(Z), (3Z)cp(Z) E T there is a form& #(Z), (3Z)[cp(Z) A #@)I E T , and cp(Z) A #(Z) k r E D,(T) (Z(Z) 3: m) for 8ome r.

THEOREM 6.6: 8wppose A+ = 12'1 s ID(T)I, A regular. Then (1) * (2) * (3) where:

(1) lD(nl < (2) For every m < O, and m-type p , 1p1 < A, there i s an m-type q,

1q1 < Asp u q i e &tent, andp u q k r E D,(T) for some r . (3) T hae an F$,-atonCic h l .

Remark. In (3) we cannot demand that the model has cardinality A+ even if A < (D(T)I, see Exercise 5.13. If h = A<A we can demand in (3) that the model is h-compact.

Proof. (1) =r (2) similar to 2.16.

(*I (2) (3) It is easy to see that it sufiices to prove:

If A is Pr,-atomic, IAI < A + , a0 E A, C(3z)p(z, a0), (where 'p E L( T)), then there is b, such that A u {b} is E*,-atomic

For E A , let q4 be an @)-type over 0, q,, k tp(7i, 0), lq61 < A, q,, closed under conjunctions. Let us prove (*). Let {ai: i < a s A} be the set of all finite sequences from A. We now define by induction on i, a l-type pi over A, lptl < A. Let po = {cp(z; a)}, = Ui<dp,. If pi is defined let pi be the olosure of p , under conjunctions and

and bP[b,aol.

Qt = {(3@[&, %s z) A # ( B s z)]: ~p(z; 8,s E ) E pis C disjoint to ?it, #(#, E ) E q,,,-e,

C E Dom pi} .

Clearly lqil s Ip;I + z{lqh-el: C C_ Dompi} < A and qi is consistent.

UH. IV, 8 61 VARIOUS RESULTS 207

So by (2) there is r, = r,(x, g), q, c r l , lrll < A, r , consistent, and r, I- r' E D,(T) for some r:, m = l(zi,) + 1. Let

Pf + 1 = P: u {1G(x, a): 1G@, @ E 4 It is easy to check hence the theorem.

is consistent, so if b realizes pa, we finish (*)

THEOREM 6.6 (The Uniqueness Theorem): (1) Let T be stable and h regular, A+ 2 K(T) . If T hua an PAprimary model over A then all the EY,...prinze models over A are &~nmphk Over A .

( 2 ) Let T be stable, p + 2 K(T) . Let M be an Fg-pimary model Over A (x = t , a) for which there are p r e l u t i m R, on m h that C E B, IBI s p , (B, R,),.,, < (JMl, R,),<,, =r C u A is Ft,-constructible over A . Then all the FA-pi~ models over A are isomorphic Over A .

Remark. Notice we do not demand that our F aatisfiesAx(XI.1) or (X.1). In [Sh 79a] a simpler alternative proof appears. Theorem 5.6 is a generalization of:

THEOREM 5.6': If IT ie stuble and m d u b l e and T hua a primary d l over A , then dl the prime models over A are i s m p h i c over A .

QUE8TION 5.1: What about x = 82

Proof. In (2) we can wume p 5 A. Let p be h and x be t in (l), p be p and z be t or a in (2). In (1) let M be an Fi-primary model over A.

(1) Let N be q-prime over A; we shall prove it is isomorphio (over A) to dl. We can assume, by the definition of F-primeness, that A E IN1 c IMI.AaMisFg-prirnary,letd = (A,{a,:i < cc},{Bi:i < a}) be a Fg-construction of [MI. We can assume a, # A u {a,: j < i}. Clearly tp(a,, d(i)) does not fork over B: (by 111, 4.4). So IB:I 5 p. For B, C s IMI, call the pair (B, C) good if

(i) a, E B [a, E C] implies B: G C[B E B:]. (ii) for every 6 E B, tp( b, I NI) does not fork over B n I NI , (iii) for every 6 E B n C, tp( 6, IN I ) does not fork over B n C n IN I. If a set D E dl satisfies (i) we call it closed (as we have a fixed

construction at).

Now we prove

CLAIM 6.7: Suppose (B, C ) is a good pair, I BI > p. Then we can $nd sets B,, i < IBI m h that B = Uf B,, lB,l s p and former y i -= IBI,

208 PRIME MODEL8 [m. Ivy f 6

Proof of 5.7. We prove it by induction on IBI > p. Suppose we have proved the assertion for every cardinality < IBI (and >p). Let B = {b,: i < IBI}, and define by induction on i, sets B, s By such that

(1) j < i =s. b, E B,, and Bj E B,; Bd = uj<d B, for limit 8. (2) p 5 lBfl 5 lil + p, B, is closed. (3) If 6 E B,, then tp(6, B n INI) does not fork over B, n INI. (4) If ~ E B , n C then tp(6, B n C n INI) does not fork over

B, n C n INI. (6) If 6~ B, then tp(6, B n C) does not fork over B, n C;

tp(6, (B n INl) u (B n C)) does not fork over (B, n IN/) u (B, n C). As p+ 2 K(T) , lBfl I p this can be done.

Let C, = C u B,, and we shall prove (B,, , , C,) is a good pair. As for (i), B,,, is closed by (2), and C, as a union of closed sets. For (ii), let 6 E B,,,, then tp(6, INI) does not fork over B n INI, as (By C) is good; tp(6, B n IN] ) does not fork over B,+, n N by (3), hence by 111, 4.4 tp(6, INI) does not fork over B,,, n IN[ , so (ii) holds. So we are left with (iii). So let ~ E A * where A* = B,,, nC, = B,,, n (Cu B,) = B, u (B,, , n C), and we should prove that tp(6, I N ] ) does not fork over A* n 1371. Note that:

(a) For every Z E B,, tp(Z, B n I N [ ) does not fork over B, n IN1

(b) For every Z E B,, tp(Z, B n INI) does not fork over A* n 1x1

(c) Forevery7iEBi+, nC,tp(a, INJ)doesnotforkoverB n C n IN1

(d) For every ?i E B,,, A C, tp(?i, B n C n INI) does not fork over

(e) For every 7i E B,,, n C, tp(ti, B n INI) does not fork over A* n IN1 [by (c) and (d), using the monotonicity and transitivity (111, 4.4) of forking, remembering B,+, n C n IN1 c_ A* n INI; B n C n

NOW as 6 EA* = B, u (B,, , n C) let 6 = Zna, ZE B,, ii E B,+, n C. ( f ) tpJZ u (B, n C), B n IN11 does not fork over A* n IN1 [by (b)]. (g) tp[Z, (B, n IN] ) u (B, n C] does not fork over (A* n INI) u

(h) t p + p u (B, n C), B n IN11 does not fork over A* n IN1 [by

[by ( 3 ~ .

by (a) and monotonicity because B, c A*].

my the goodness of (By C) as B, + , c B].

B , + ~ n c n INI PY (411.

IN1 E B n IN1 c pi].

(B, n C) [by (f) using Ax(V.2) for FL].

(@I.

CH. Iv , 8 61 VARIOUS RESULTS 209

(i) tp[& (B n INI) u (B n C)] does not fork over (B, n INI) u

(j) tp[z, (B n INI) u (B, n C) u a] does not fork over (B n IN]) u (B, n C) p y (i) and the monotonicity of forking].

(k) tp[E, (B n INI) u (B, n C) u a] does not fork over (A* n INI) u (B, n C), hence over (A* n INI) u (B, n C) u a [by (g) and (j), using the transitivity of forking 111, 4.41.

(1) tp& u a u (B, n C), B n IN!] does not fork over A* n IN1 [by (h) and (k), using Axiom VII for F',].

(m)tp(6, B n INI) does not fork over A* n IN1 [by (l), aa 6 = zniz E a u a u (Bf n C)].

(n) tp(6, IN]) does not fork over A* n IN1 [by (m) and (ii), using III, 4.41.

Thus (B,+,, C,) is good. If IBI = p+, then lB,l c- p and we finish. Otherwise we can a,ssume IB,I > y, and let a(0) = 0, a(< + 1) =

a(i) + lB,l (ordinal addition), a(6) = Ulc6a(i), (so a(i) < IBI for i < 1231) and use the induction hypothesis on (B,+,, C,) to define B;, a(i) I; j < a(i + l), and the Bj show the claim holds.

(Bi n C) [by (611-

Continuation of the proof of 5.6. Now clearly (1M1,0) is a good pair. Assume 11M11 > p, as llMll s p is a trivial case; so by 6.7 there are B,i < 11M11,such that lB,l s p,andlettingC, = U,<, Bj,C,,M,, = 1iK1, B d c c d (for limit 6) and (B,+ ,, C,) is a good pair. By 1.3(2) it suffices to show that IN1 n [A u C,,,] = IN1 n [A u C, u B,,,] is Fg-constructible over IN1 n [A u C,]. Now

(p) tp*(B,+, n C,, "1) does not fork over B,+, n C, n IN1 [by the

(9) tp,(B,+, n C,, IN1 n [A n &+,I) does not fork over IN1 n [A u (B,,, n C,)] (by (p) and monotonicity of forking].

(r) tp,((NI n B,+,, [A u (B, , , n C,)]) does not fork over IN1 n [ A u (I?,+, n C,)] [by (9) and Ax(V1) for F{ with A* = B* = C* =

IN1 n [ A u V4+, n C,)l, and any a E IN1 n B,+,, 6 E (B,+, n Cdl. (a) d is also an F',-construction and B,+,, C, are closed for this

construction my Bt's definition and (i)]. (t) tp*(B,+,, A u C,) does not fork over A u (B,, , n C,) [by (a) 3.3

and Exercise 3.61. (u) tp,(lNI n Bi+l, A u C,) does not fork over IN1 n

[A u (I?,+, n C,)] [by (r), (t) and transitivity of non-forking 111, 4.41. (v) tp,(lNI n B,+ , ,A uC,) does not fork over IN1 n (A uC,) [by

monotonicity of forking].

goodness of (Bf+lY C,)I.

210 PRXME MODELS [OH. my 6

(w) By (i), B,, ,, C, a m dosed in the F;-construction d, hence [by 3.31 B,,, is K-oonstruotible over A u C,, henoe (3.2) Ff-atomic over A u C, [remember we are in 6.6(1)]. (x) IN1 n B,,, is Fg-atomic over IN1 n (A u 0,) [by (v), (w) and

4.31. (9) IN1 n B,,, is Ff-constructible over IN1 u A u C,) [by (x) and

3.16, h i s regulas]. So we f i s h the proof of 6.6(1). (2) Left to the reader.

CONCLUSION 5.8: If T is stuble, IT1 is regula;r and Over A there is an Flt,,-primary model, then the FrT,-prime model over A is unique.

THEOREM 5.9 : Let Sf = { A : IAl = h imply over A there is a prime model among the Ff-compact ones}.

(1) p 2 h = A<, 3 (TI, AES: implies (2) &o=minSt,, < a0imlplie8A&~ < ~(~T~,l)(seeVII,DeJinition5.1), (3) For x = a, a, eirnitar results hol& with 2ITI i w t d of ITI.

Proof. (l), (2).

5.9A DEFINITION: (i) A type pc=S(A) is Ff-inevitable if every F f - compact model JM, A c 1N1, realizes it. For F: = F&, we omit it.

(5) @(A) is the assertion that for every A of cardinality A, and Ff- type q over A, there is an Fg-inevitable p E 8(A) , q E p .

5.9B CLAIM: (1) Ifh = h'"+lTI, then he&': @ & : ( A ) . (2) If &t(p) hob% whenever A I p < A, = A,'* + IT1 + A+, then

A ESL.

Proof. (2) Given A, IAl = A, we define by induction on i elements a, such that tp(a,, A,) is inevitable where A, = A u {aj: j < i}, and for each i, and each 1-type q over A,, lql < K, for somej < A,, q is realized by a,. It is easy to see A,, is the universe of a K-compaot model bl, A E IMl, M is F,"-prime over A .

(1) The "if" part follows from (2) (for A, = A+). For the "only if" part, let IAI = A, q a 1-type over A, 1q1 < K. Let iU be a K-prime model over A (exist by an assumption), and U E M realizes q, then p = tp(a, A) E S ( A ) extend q and is Fk-inevitable (because if A -c W, H' E"L,-compact, then M can be elementarily embedded into bl' over A, hence bl' realizes p ) .

m. IVY 5 61 VARIOUB aEBlnTs 211

5.9C CLAIM: The following are equivalent

4 A ) , (i) q ia a 1-type over A , 1q1 < K , q has no Fi-inevitable completion in

(ii) there are Fi-mpact d l . 9 M y Ma (a E N) sucla that: A E M, A c Ma, and f o r every a E N, if a realim q, then Ma omit tp(a, A).

Proof. (i) * (ii): Let M be any Fi-compact model, A E M. If a E M , a realizes q, then by (i), tp(a, A) is not inevitable, so there is a E",- compact model Ma, A E Ma, Ma omit tp(a, A). If a E M does not realize q, let Ma = H.

(ii) == (i): If q has an Fi-inevitable extension p €&A), then p is realized in M (by the definition of Fi-inevitable), so let a realize it, but then Ma omit p , contradiction.

5.9 D COROLLARY : Suppose &:(A) fail, or even A $S:, then for every p = p e K + I TI s A , QL(p) fail and 80 p $8;.

Proof. If A $ 8;, then by Claim B, &:(A') fail for some A' 2 A, so w.1.o.g. &;(A) fail. So for some A, IAI = A and 1-type q over A, 1q1 < K, q has no inevitable extension in #(A). So by Claim C, there are M, Ma (a E A ) as mentioned there. By the downward LowenheimSkolem theorem to the appropriate model, we get that Claim C(ii) holds for some A' E A, lA'l = p, so Claim C(i) holds so Qbb) fail, henoe by Claim By p $8:.

5.9 E COROLLARY : There is a sentence t,h E LIT,+," which has a model of power A i# not Qbo(A).

Pro@ Formalize Claim B(ii).

PROBLEM 5.2: What is the &st cardinal A, such that if F& satisfies Ax(X.l), IT1 = No, and over some A the F&,-prirne model is not unique, then there is such A, IAI < A (similar problems exist for other Fs, and for uncountable T).

THEOBEM 6.10: Let A > IT], A = AeA. Then the following conditim are equivalent.

(1) et, 8ath$a Axiom (xI.l), (x.1) a d F~-saturatiOn imqdies A- mpa~tne88.

( 2 ) Over any A there is a A-prime model. ( 3 ) Over any A , IAl < A , there is a A-prime model.

(4) If 1p1 < A, p an m-type over A , IAI 5 A, p hua an Fi-ieolated

(6) T ~ M 8am a8 (4) for arbitrary A . & d o n in #“‘(A).

Proof. (1) * (2) By 3.1(2), 3.10(2). (2) * (3) Trivial. (4) =- (6) Suppose A is a counter-example to (6). As A = A e h , A is regular. Defbe by induction on i 5 A, A, E A ,

lA,l 5 A. Let A, = Domp, and A, = UICd A,. If A, is defined, for every m-type q over A,, p s q, 141 < A, q haa at leaat two extensions in P ( A ) , so for some Q,(J; B,) (8, E A) . q u {Y,(E; is consistent for t E {o,~}. Let 8, = A,+l u {a,: p E q, 141 < A, q an m-type over A,}. Clearly lA,+ll s X a8 I TI < A = Aeh. Nowp and Ah provide a counterexample to (4). (6) =t- ( 1 ) Clear. (3) - (4) Letp be an m-type over A , IAI s A, 1p1 < A. By (3) there

is an A-prime model M over A. So some E lM[ realizes p. If q = tp(8, A) is Fi-isolated we finish. Otherwise by 6.3 there is an A- compact model N, A E INI, N omits q. But aa M is A-prime there is an elementary mapping f from M into N, f t A = the identity, 80

f(a) E IN1 realizes q, contradiotion.

PROBLEM 5.3: Find a similar theorem when A < ACh.

PROBLEM 5.4: What can {A: Ff satisfies Ax(XI.1)) be, z = t , 8, a?

THEOREM 6.11 : If T is countable F& does not sati8fy Ax(X. l), them FZ does not satiefy Ax(x.1) for x = t , 8, a and any p and Ft does not 8atiefy Ax(XI.1) for z = 8, a and any p or z = t , p > 12‘1.

Proof. So there a m A, a E A and $(x, a), p such that (3x)$(z; a), but there are no p €&A), q E p , such that 1q1 < A, $(z; a) ~ p , and q kp I+ 9. Let P = A, and extend (Q, P) to a E-saturated model (Sl,pl). We can assume Q, = Q, and let P, = A,. As (Q, P,) , (Q, P ) are elementarily equivalent, for every hite 1-type q, over A, such tht (3z)[$(x, a) A A q] there is 5 E A,, such that {$(E; a)} u q u {p(x, 5)t} is consistent for t E (0, 1). By the p-saturation of (Sly P1), this holds for any q, lDom qI < p. The conclusions me immediate.

DEFINITION 6.1: The set I E 1M1 is absoluteiy indi8cernible in M y if every permutation of I is induced by an automorphism of M.

a. IV, 5 61 VABIOUS RESULTS 213

LEMMA 6.12 : ( 1 ) An absolutely indiscernible set I in M , is indiscernible. (2) I f M i s strongly (I(+-saturated, (see Definition I, 1.4(2)) I c lbl(

I an indiscernible set, then I is an absolutely indiscernible set in M .

Proof. Immediate.

Remark. So the real problem is to find M and I absolutely indiscernible in M, such that ( I ( = (IM((.

THEOREM 6.1 3: Let I be an indiscernible set, T stable in ( I ] . Phen I is absolutely indiiwrnible in s m M , ((M(( = ( I ) . Moreover, I is a d d indiscernible set in M.

Proof. Let u = ur(T); so as T is stable in ( I ( , l I ( < s = 11) (by 111, 3.7) so K s I I I and even K s of 1 I ( . Let bl be q-primary over (J I . M exists by 3.1, 2.18(2); clearly (lMl( = ] I ( (as T is stable in I I ) ) . By 4.10 I is a maximal indiscernible set in M , and by 3.8(1) I is absolutely indiscernible in M .

THEOREM 5.14 :Suppose Tisstable, h = A<", and Tisstableinsomep, u s p s h .

( 1 ) Over any A, lAl s h there M a Fi-cawtr~c4ibl.e model M over A, 11M11 = A m h that: if B s JMI , IBI c A, p e s r n ( A u B ) doe8 not fork over C c_ A u B, IC( < u then p is realized in M.

( 2 ) The rrwdel M in ( 1 ) i s unique u p to isomwphim over A.

Proof. ( 1 ) The proof is similar to the proof of 3.1(3), (4). (2 ) The proof is similar to the proof of 3.9 (in fact, by 3.1, 3.8(2)

reap. applied to T h ( M , a),,,).

CONCLUSION 6.16: If I i s a (non-trivial) indbcemrible set of s e q w , and T i s stable in some p s I I1 then ) I [ i s an absolutely indiscernible set in a nzodel M, 11M11 = III.

Proof. B y 6.14(1) and 6.14(2).

THEOREM 5.16: Suppose (TI = A < X , A G L = L ( T ) , ( A ( = A, and one of the following coltditions holde.

(A) A is regular, x = 2". ( B ) There i s a tree I c A22 (i.e., 7 Q V E I 4 7 E I ) , so that II n a2(

< h fora < h but ( I n%2( = x (for x = 2", I n A > 2 i a a Kurepatree).

214 PRIME MODELS [CH. IV, 5 5

(C) x = A + , A regular. (D) There i8 a tree Z c hr2, x = IZ n "21 > IZ n A > 2 ) + . Then (1) there are models Mi (i < x) of T , each of cardinalityh, such

that: i(1) # i(2) < x, a, E IMi(DI, tp&, 0 ) = tp&, 8 ) implies that there are pt c tp(a,, O ) , lpll < A, pl k tpd(iji,, 0) (1 = 1,2).

(2) If A'" = A, we can aseume each Mi it? ~-wmpmt .

Remark. Case B applies, e.g., for singular A when (Vp < A ) @<'IA < A ) x = AofA; so for strong limit A, we get x = 2A, and for A of cofinality No, We get X = ANQ.

We can conclude:

CONCLUSION 5.17: If IT1 = A < x, A, x SatiSfy (A) OT (B) OT (C) OT (D) from 6.16 each?, E D( T)( i < io < x ) i s not F,iaoZated then T ?ma a d e l of cardidity X which Omit8 every pi.

Proof of 5.16. We can w u m e w.1.o.g. that d is closed under Boolean combinations. We leave (2) to the reader and concentrate on (1).

Case C . We define the models M , (i < x = A + ) by induction on i. If we have defined M, for j < i let dji = {tp,(a, 0): si E IN,!, j < i} (so lQi1 S A) and it suffices to find a model Hi of T of cardinality A, 8UCh that if tpd(Ti,O) E Qi, IN,( then for some q E tp,(a, 0), Iql < A, q I- tp& 0) and a little more. This follows by

CLAIM 5.18: If IT1 = A, pi (i < io I A) types (in L(T) , over 0 ) then T ha% a. nzodel M of cardinality A, such that:

(i) 7i E M , realimp, implies the &stence of q E tp(si, O ) , lql < A fur which q kpi.

(ii) If si E "1M1 a d fur 8 m e m-type q (over 8), Iql < A, q I- tp,(si, 8 ) then this hl&S fur 8Orne q E tp(8, 0) .

Proof. The proof is like that of 5.3(2), but we also do an assignment like (iii) of the proof of Case A. If A = No the set of types forbidden by (ii) of 5.18 is countable, 5.3 applies directly.

Caee B. Let {p?,(z, &): i < A} be a list of all formulas cp(z, 8) EL p E {z,: j < A} and wsume jji E {z,: j < 2i}. We define by induction on u < A, for each r) E Z n "2 a consistent set r,, of formulas in the variables {z,: i < 2a}, for A = KO, r,, is finite, and for A > KO, s No +

CH. IV, 5 61 VARJOUS RESULTS 215

[ I n '21 (note that w.1.o.g. IZ n '21 is non-decreaeing). For a = 0 let r,, = 0, and for limit a, q E ' 2 let r,, = UB<= r,,,. Suppose we have defined for a, and let us prove for a + 1. Let {(q:, qa, Z:, Z;): i < i(O)} be a list of all quadrupals (ql, q,, Z,, Z,) where ql, q, E ' 2 n I are distinct, Z,, 2, are finite sequences of variables from {zj: j < 2a} of equal length. So for A = No, i(0) < No, and for h > No, i(0) < No + IZ n a 2 ) + . We now define by induction on i s i(O), consistent set of formulm (9 E ' 2 ) in the variables {zj: j < 2a} such that II'il < No + II n '21+, and j < i implies r! G r,l. Let r,O = r,,, and for limit ri = UfC, Pi. If we have defined for i, and there is a formula Q@) E A such that ri, u { Q P ( Z ~ ) ~ ( ~ ~ ~ ) } 1 = 1 , 2 , are consistent, choose = r:, U {tp(E)}JZ1 = rip U {lp(z)}. For all other r]'s, and if there is not such Q for 7 = ql, q, too, = r;. Now let, for q E ' 2 , l E (0, 1 )

rn-<l> = ricO) u {@z)qa(~ iia) QtAzsa, ga)}

u {Zj # z,a+1:j < 2a).

For each r] E A2 n I , q = u, < A qla is consistent so some assignment qwuaf, eatisfiw it. By Tasaki-Vaught test there is a model iWn with universe {ut: i < A}. It is easy to check that the models me as required.

Owe A. As Case B wae proved, we can assume h > No, and ae Caee C was proved we can assume 2" > A + .

Let {Q,(z, &): i < A, i a successor} be a list of all formulae ~ ( z , g ) E L, gE{zj:j< A}sothat&tz{zj:j< 2i}.Let{Z,:a< A}bealistofall 5 E {z j : j < A}, such that for each such 8, {a: Za = Z } is a stationary subset of h (possible by 1.3 of the Appendix). We aesume Za E {zj: j < 1 + a}, and let L = Uuch La, 1L.I < A, La (a < A) is increasing and continuous.

We shall define by induction on a < A, for each r) E ' 2 a consistent set of formulas I',, in the variables {z,: i < 2a) such that Ir,,l < A.

8&e (i) (a = 0). r<, = 0.

A d w e (ii) (a limit). For each q E ' 2 , F,, = UB<cr r,,,,. 8 d w e (iii) (a = successor). Let (~~(2): i < i(O)} be a list of the formulas from LB with 2 E {z,: i < 2a}. For each q E B2 we define by induction on i < i(0) formulae q(2) such that F,, U {8 { (9 ) : j 5 i} is consistent, and 8; E A. If we have defined for j < i, and there is 8 E A such that r,, u {O{(Zj): j < i} u {8(2)} is consistent, and T, ~ ~ ( 2 ) k

+ 1,

-,8(2), choose such 8 = e f , , otherwise choose 6; E A such that

216 PRIBXE MODELS [m. IV, 8 6

r, u {%(2): j I i} is consistent (note that consistency is preserved for limit i’s).

NOW let for q E 82, r,l be r, u {8#9): j < i(0)) u {-,(3z)tpB(z, g8)} if consistent and r, U { q ( I j ) : j < i(0)) U {tp&ag, g8)} otherwise. For I = 0, 1 let r,-(,) = r; u {z, # za8+1: i < 2/31.

&&a&e (iv) (a = /3 + 1, /3 limit: and for no y < 8, I , = 28). For q E 82 letp, = {(38)’p(2~4,8):cpisafiniteconjunctionofformultlsof r,, ;z, n8=0}.

If there is a type q = q(If i ) , 19.1 < A, such that p , U q is consistent, and for every 8(z) E A, Z(Z) = @), for some t E (0, 1) p,, u q k O(Qt, then choose such q,; and if there is no such p, let q, = 0. Let for 1 = 0, 1

~ V < O = r, U qn-

S*e (v) (u = /3 + 1, /3 limit, and for some y < /3, I, = I,& Let us define pn (q E 82) as in the previous case, and let Ls, = {q e 62: there is 8 E A such that p , u {0(28)t} is consistent for each t E (0, 1)). By the previous case, m Ipnl < A, for each q = q(If l ) , 1q1 < A, q E S ~ , if I), u q is consistent, then for some 8 E A , pn u q u {0(18)t} are consistent (t = 0, 1). So we can find 0; E A such that for every v €82, I),, u {8i918)””9: y < /3} is consistent. We can assume, that i fq l , 11 E S ~ and for every 8(z) E A , pv1 k 8(Q e p , , I- 8(zB), then for every Y E 8’2,

Oil = 0;, (remember that we assume A is closed under Boolean combintttions).

Now for every q E 5,l = 0 , l let

r,-(l) = r, u {qY(I~)“”; < pi. For E 82 - S, let r,,..,,, = r,.

80 clearly it is consistent and let zt H a!, be an msignment satisfying it. So by the Tmki-Vaught test (and Subcase (iii)) there is a model M, of mrdinality A whose universe is {a:: i < A}. We now prove

(*)

:. Let for each q E ”, r, = Uaeh

Suppose Z = ( x ~ ( ~ ) , . . . , ztCk)), ql # qa E ’2, and let Ti, = (a::o), . . . ,Ti$:)). If tpd(Til, tp) = tp&, tp) then for at least one 1, there ie q E tp(Til, ‘p) 1q1 < A, such that Q I- tPd@l, t p h

For suppose (*) fails, choose a. such that q1 1 a. # qa 1 ao, and for = I , then qll/3, some y < ao, I , = 2. So if a. < /3 < A, /3 limit

7alP

OH. Ivy 9 51 VARIOUS RESULTS 217

Clearly there is a closed unbounded set W E A, such that for p E W, I = 1, 2, P,,,l@ is s L, and, for it, it is complete (La, for each formula #(8), z E {q: i < 2p}, # E Lg, for some t E (0, l}, I',,,,, t- #(Z)t)). We can also assume that (for each

As {b: 28 = Z} is a stationary subset of A there is a p E W , 28 = 8. So p,,rlB E flfl (as a,, Ti3 contradicts (*)) and for every 8 E A , t- 8(Z8) o pnal, k 8(Z,). @?or otherwise we can suppose l)nl18 k 8(Z,) but pml , It 8(Z8), hence for some #(zJ EI)nllB, #(Z8) k 8(Z8) but 1)va18 u {10(8B)} is consistent. For some y , a. < y < 8, # E L,; so by subcase (iii) for some

E W ) is limit, b > ao.

81 $(ZB) 81(2fl), 1e1(88) E p " l l ( y + l ) hence l O l ( Z 8 ) Epn118. But as #(z,) Epnllfl, e(E,) E Z ) ~ l l B contradiction (to tPd(al, = tpd@2~ v))].

So, by Subcase (v) Oil = O i 2 for every Y €6'2 , and $:IY(Z8)"W E

r,,rl(B+l) E r,,, where y = min{y: q l [ y ] # qa[y]}, and we get a contradiction (to tpd(Til, 0) = tpd(Sia,0)). SO we proved (*). NOW for each q E "2 we define g(q) as the set of v E "2 such that for some Ti &Ifv, 8 E H,,, tpd(ij;, 0) = tpd(6, 0), but for no q s tp(Tiv,O), 1q1 < h doea Q k tpA(Ti9 0). NOW,

(**I Is(.l>l Otherwise there are distinct v, E g(q), (a < A + ) and for each of them

there are witnessee Tia, 6,, and let a, = (?i$y*o), . . . , aty*k(a))). The number of possible k(a), i(a, 0), . . . , i(a, k(a)), 6, is A, hence there are a # /3 for which they are equal. Clemly tpA(a,, 0) = tpA(aB, 0) and we get e d y a contradiction to (*).

Now as we have assumed 2" > A+ (by c w (C)) by 2.8 of the Appendix there is U G "2, IUl = 2" such that q # V E U implies q $g(v) . It is easy to check that {Ma: a E U} is a family of models exemplifsing our assertion.

Case D. Use the proof of Theorem 5.19, Case I (with I instead "2, omitting condition (4) there). In the end use AP 2.8.

We can improve Theorem 5.16 (hence 5.18) by:

THEOREM 5.19: Suppose [TI = A, = 2A, d E L = L(T), Id1 = A, then :

(1) There are models Mi (i < X ) each of cardinality A, m h that: i( 1) > i(2), al €A&), tp,(&,, $3) = tp,(a,, $3) implies there i sp , 5 tp(a,, p)), lpll < A, pl t- tpd(iil, 0). If A is regular or 2 < A < 2 A there is also Pa E t~&2,0), l ~ 2 l < A, Pa t- t p ( G 0).

(2) If A<" = A, we can waume each Mi is K-OMn'paCt.

PRIlKE MODELS [ax. IV, 8 6 218

Remurk. So if IT1 = A, Pi E 8$(0) (i < a < 29 , then for some M, l l J f l l = A, and 8 E M, pi = tpd (a, 0) implies q p , for some q C tp (a, g), IqI < A. We can have A pairs ( A , m)’s.

Proof. (1) By Theorem 6.16 we can assume w.1.o.g. A is singular. Caee I: 2‘A < 2A. Clearly (2”: p < A) is not eventually constant so

we can choose A, (a < K = cf A), a < < S A D , A = z,,, A,. Let {vf(z; Bf): i < A} be a list of all formulas ~ ( z ; jj) E L, jj E {z,: j < A}, 80

that g1 E {z,: j < Pi}. We define by hduction on i < A, for each q E a set r,, of formulas

in the variables {z,: j < 2i}, consistent with T, such that (1) lrnl s lZ(q)l + KO, and when 6 = I(q) is limit, r,, = ua<d Il,,,,, (2) Y 4 q implies r,, c r,,, (3) if i + 1 < Z(q), then (3%) vf(z; &) -+ v,(za,; &) E r,, and

s

{za,+l # z,:j s 2i} c r,,,

T(Z)ES~~(PJ), but not rJ-r(Z), then for some O ( z ) E r , ~ , E + ( z ) , for any O E ~ ’ ~ ) .

(4) if 5 i < A,+,, 7 E f2, Y E (&a)2, Z E {z,: j < A,}, and rv 1 r(Z)

If such r exists, we denote it by r,(Z). (6) If q E f2, Z E {z,: j < 2i}, Z E (0, l}, r,-o>(Z) does not exist, then for

some 0 €4, e(z)t E r,,-&). (6) IfqEf2,ZE{z,:j < i},andforsomeq(Z),r~Sfi(~)(O), Iq(Z)I s lil,

q ( ~ ) t- r , r,, u q ( ~ ) is consistent, then r,, I- r. Let for q E ”, zf I+ af, be an aasignment satisfying r,, and M,, a model

with universe {a:: i < A}, and let g ( q ) = {vcA2: For some ~ E E , , , 6 EM,,, tpd(a, 0) = tpd(6, 0) but for no q -c tp(iiv, 0), 19.1 <. A, does q t- tpd(a, 0)). Now Ig(q)l s 2<” by (6) above. So there are qC E ”2 ( 5 < 2A)

Henoe {iK,,<: 5 < 2”) ie as required.

define inductively qe E A2 (5 < 2”), rare iffor some y = y(A) < A

such that f < 6 = rlr; $ S(TJ. But by (4) above, [Ta E S(rlS)l* 1716 E dra)l.

Case 11: 2 c A = 2A. Like Case I, omitting condition (4), and try to < 6 =- qa 4 g(qC). We call A s A2

(vq,Y)(qEA A Y E A A 7 17 = Y rr + q = v ) ,

and clearly g(q) is the union of A ram sets. It ia enough to prove A2 is not the union of <2” rare sets; and by assumption for some p < A, 2’ - 2”, and w.1.o.g. p < A,,. If A, E A2 is rare, i < io < 2Ay let y1 = y(Af), and choose induotively q, E <*J2 (a < of A) increasing by 4, such that i < io, ‘yc < Aa, Y E A, implies Y 1 A,,, # q,+, (for a zero or limit

OH. IV, 5 61 VARXOULJ RESULTS 219

there is no problem, for successor-by cardinality consideration). The limit of the qcr is q E ", q $ Uiei0 A,.

QUEBTION 5.5: Can we assume in 6.16 only that T is stable I TI s I I I ?

EXERCIBE 5.6: Let A s ICI. Define L(A) aa L u A, i.e., we add the elements of A to L aa individual constanta,

T ( A ) = T U { ~ ( i z ) : iz E A , Crp[iz]}.

(1) Investigate the connections between the fulfillment of the axioms

(2) Use (1) to generalize the uniquenew Theorem 6.6. by Fg for T and T(A) .

EXERCIBE 5.7: Let T be a theory in L, ILI = A, L = U,<hL,, L, increasing, pi is an m(i)-type for i < a s A, and for each i and q, m(i)- type q @ < A: T U (q 1 L4) Y p, 1 &#+I} has cardinality A. Show that T has a model omitting each p,. Check generalization to the situation in 5.16-5.19.

EXERCIBE 5.8: Show the consistency with ZFC of the existence of a countable T , and types pi (i < wl) as in 6.3(1) such that no model of T omits every p,. (Hint: See Hechler [He 731.)

EXERCIBE 5.9: Assume M A , and generalize 6.3 for a countable T , and <2no types.

QUEBTION 5.10: Can we in Exercise 6.8 demand the types are pairwise contradictory.

EXERCI8E 5.11: Generalize 6.16 for h A's at once.

EXERCIBE 5.12: Show that if in 6.16 we mume A = L, then we can aimplify the proof (e.g., omitting Sub- (iii)).

EXERCIBE 5.13: (1) Give an example of a theory 21, IT1 = N, s ID( T)I such that T has a Fn,-atomic model of cardinality No but not

(2) In 6.6(3) we can assume the model haa cardinality rA and is K-compact, when K < A, A'" 5 A+. [Hint: Let T = Th(M), M = ( m ' 2 , . . . , P,,, . . .)*a1 where P,, = {q 1 n: n < w}.]

> No.

PRIME MODELS [m. IV, 8 6 220

DEFINITION 5.2: M is a full model if for every formula V(Z) in L(M) I{iz E IMI: M C cp[7i]}l < KO or = 11M11.

EXERCIBE 5.14: Let T be a counfable complete theory. Give a necleesary and sufticient condition for the existence of a full model of T cardinality K, which omits the types p, i < w. Generalize 6.17. [Hint: The condition ie that for no p@) and cp = ~(3, yo, Z0,. . . , y,, 9) the following hold :

(i) TI- (3""yO)(3~,)(3""y,)(3q), . . . , (3"3,)(3~,)(3~) p, for every

(ii) For every $(Z) EP,(Z), for some n < w n < w .

TI- ~ ( ~ " " y O ) ( ~ ~ O ) ( ~ ~ " ~ l ) ( ~ ~ l ) * * *

EXERCIBE 5.15: Show that if 6.1(3) holds, any EY,-atomic, FA- saturated model is A-compact.

EXERCIBE 5.16: Concerning Theorem 6.2(1), show that for some T, it has a A-compact model, but 5.1(3) fails. [Hint: See Exercise 2.2, hint (l) , with A the universe of a non h-compact model.]

EXERCIBE 5.17: Suppose IT1 = A, singular but not strong limit, and T has a A-compact model M, of cardinality A. Choose p < A, 2' > A.

(1) *(T) s cf A. (2) There are no formulaa cp,(Z, a,) (q E "'2, iZq E lMol) such that for

any q E fl2 {cp,,&, i~,)*t"l: a < p} is consistent. (3) If A c IMol, m < w , q an m-type over A, then for some m-type

p over A, lpl KO, any Py,-atOmi~ (over 0) A-compact model is A-

(6) The A-prime model of T is unique.

(2) As llAfoII = A, Mo is A-compact. (3) By (2) (aa in the proof of 2.16(1). (4) By (3) (M is PY,-constructible) and 4.7. (6) By (l), (3) and &B 4.9(4), any FtA-atomic over 0, A-compact model

homogeneous and the conclusion of 5.2(2), (3) holds.

Hints: (1) See VIII , 4.7 for suitable reducts of M,.

22 1 m. IV, 4 61 VARIOUS RESULTS

is A-homogeneous. The uniqueness and characterization of the A-prime model, are trivial now. (6) Proved as 4.18.

EXERCIBE 5.18: Suppose IT1 = A, A singular and strong limit and T has a A-compact model of cardinality A, M,. Then

(1) ~(5") < cfh, and even A<K(T) = A, (2) T is stable in A, (3) The statement Exercise 6.17(3) holds for p = A, (4) Exercise 6.17(4); (6), (6) holds.

[Hint: (1) See VIII, 4.7, for suitable reducts ofM,. (2) If not, by 111, 6.16, by (l), ID(T)I > A, hence for some m,

ID,(T)I > A. Let M be a model of T of cardinality A, so IMI= U f < c l h A ( i ) , A ( i ) increasing IA(i)l < A. For each i , define an equivalence relation E, on the set of m-formulas: cpl(z)Eicp2(Z) iff for any li E A,, M C cpl[li] = 9&]. So E, has s 2lA({)l equivalence classes; so the number of p E D,(T) such that cpl(Z)E,cp2(Z) implies cpl(Z) E I ~ o

cp2(Z) E P , is ~ 2 ~ l ' ' ' " l < A (as IA(i)l < A, A strong limit). Hence for some p E D,( T), for every i , there are m-formula cp:(Z), d ( Z ) such that +(Z) ~p o cpi(Z) EP and cp:(Z)Eicp&(Z). Hence #,(Z) = cp:(Z) = - p , " z ( z ) ~ p but no n ~ A ( i ) satisfies @@). So M omits {@'(~):i < cfA} E p, hence is not A-compact.

(3) We first prove: (*) There am no formulee PO,^,^ = cp,,v,a(Z, li,,,v,(r) over iKo for i < cf A,

7 E nj<,

if v E nj<f a < X , (where A = x { < d h X, ) , such that:

then {cpnll.vl,.rtc, A --rvnl,,vl,,v[,J: i < of A}

is consistent. This is proved using Exercise 6.18(2), with similar technique. Next

we prove: (**) If p = p(Z) is an m-type over A s IiK,l, 1p1 < A, there is a

q(Z), 1q(Z)1 < cf A, such that p(Z) u q(Z) is an m-type over A, and: r € S m ( A ) , r z p ( z ) U q(z) implies r is Fi-isolated.

If (**) fail we can construct a counterexample to (*), and using (**), Exercise 5.18(3) is easy. (4) As in Exercise 6.17.1

222 PEIME MODELS [CH. Iv, 9 5

EXERCIBE 5.19: Show that Theorem 6.2 holde also for singular A, when cf A > KO, but when cf A = No some parts may fail. [Hint: See Exercises 6.17, 6.18.1

EXERCIBE 5.20: Generalize Theorem 6.2(2) and (3) and the previous exercises to the case A < I TI.

CHAPTER V

MORE ON TYPES AND SATURATED MODELS

V.O. Introduction

In this chapter we shall deal with stable theories only; note that for the concepts we investigate here, there is no need to distinguish between parallel types, nor between equivalent indiscernible seta.

In Sections 1-6 we investigate the notion “dimensions”, and also regular, orthogonal and minimal types for stable and superstable theories. In Sections 6 and 2 we use this to investigate theories with few quite saturated models. In Section 6 we deal with cmdinality quantifiers and strong transfer theorems. At last, in Section 7, we deal with a generalization of “algebraic”, to “of small cardinality by a cardinality quantifier”, with appropriate rank; and deal again with ranks.

The “classical ” example of those who dealt with categoricity was the algebraically closed fields. There what is the (transcendence) dimension is well understood. The natural first try (see Definition 111, 4.6) is aa follows: let p E S ~ ( A ) , and we define on {a E B: p = tp@, A)} a dependence relation: depends on I if tp (a, A u w I) forks over A. Unfortunately not all axioms of dependence relations which enable one to define dimension are satisfied (transitivity is lacking). When there are (in a given model or set) lmge independent sets we still get a true dimension (see 111, 4.21(2)), but when, e.g., A E M y and every independent set ( ~ p ( d l ) ) is finite, this does not help. So it is natural to try to deal with the types p for which transitivity is satisfied, and we shall call them regular (see Definition 1.2 and 1.9 for equivalent definitions, including the one above). For developing this we define when p i E #‘“(A) are weakly orthogonal (pl(Z) u pa(#) is complete) and when p , , pa are orthogonal (every stationarizations me orthogonal).

The main result is contained in 1.14 and 1.16 : among the sequences realizing over A, a regular type, the axioms of dependence (needed to define dimension) are satisfied.

223

224 MORE ON TYPES AND SATURATED MODELS [CH. v, 8 0

Naturally we want to have existence theorems for regular types. For stable T, we can prove that every non-algebraic type a n be extended to a minimal type p (h., for every p for some t, p U {pt} is algebraic). Minimal types are regular and we a n understand them better (see 1.17 and 1.18). But we want to know that there are really many regular types. For superstable T , M , < M,, M , # M , HI F;,-saturated; for some a E lMll - IM,l, tp(a, lM1]) is regular; and for T totally transcendental, F(t,-saturativity is not needed (see 3.6 and 3.19). So regular types are important mainly for superstable T. In this context it is interesting that if M is E-saturated, K 2 KJT) , p I = tp(si,, 1611) is regular, then p , is realized in the Ff-prime model over U iff p , is not orthogonal to pa.

From this we can prove non-orthogonality is an equivalence relation among stationary regular types, and it is a meaningful equivalence relation.

After this theorem it is natural to define the following orders among indiscernible sets: I so J if dim(I, M ) s dim(J, M ) (for every M ) and I s 8 J if when I u J s M I < M, are Fg-saturated ( K = K,(T)) Av(I, lN1l) is realized in Ma, then Av(J, lMll) is realized in M, (but this implies Av(I, lM1l) is more complicated than Av(J, lM1l), so maybe it would have been better to reverse the orders). Those (quasi-) orders are investigated in Section 2. Our intuition is that for “simple” theories (in an appropriate sense) the F&(,,-saturated models are characterized (up to isomorphism) by some dimensions. So the simplest theories will be the unidimensional, (i.e., I sro J for any indiscernible sets I, J), and we would like some dicotomy, saying meaningful aasertions on unidimensional and non unidimensional theories. So the climax of Section 2 is Theorem 2.10, which says:

(1) T is unidimensional, and every Fgr(,,-saturated model is a Ert,-prime model over U I, for some A, where I, is an indiscernible set of cardinality A and has a fixed type, and I, is based on 0. Or

(2) T is not undimensional and for every p 2 K,(T), T has a Fi- saturated, not e + -saturated models of arbitrarily large cardinals, which me not even p + -universal.

In Section 3 we shall show that for those orders, for superstable T , above each element there are only finitely many elements up to equivalence by the natural equivalence.

Though we get true dimension for regular types, we still want to get somehow the true dimension for the other types; and this is the aim of Section 3. Maybe the following example will clarify it. SupposeMis an

OH. v, § 01 INTRODUCTION 225

algebraically closed field, {a,, a,} is transcendentally independent, and iz = (a,, a,) and p = tp(iz, 0). Now this type is not regular; but it is clearly a matter of “notation” essentially. We should define a weighted dimension; i.e., let w(iz, A) be the transcendence dimension of acl(A u T i ) over acl(A), and the parallel to the dimension being true is

(*I for I = {4: i < a},

(We can define naturally w for infinite sets or sequences.) The first’question is how to define the weight w(iz, A). Our suggestion

is as follows: Assume A = 1H1, H Ero,-saturated, N is F&(n-prime over (M( u a. If we can find a set J E IN1 independent over IMI, of sequences realizing over Idl( regular types such that N is Qrtn-prime over IdlI u U J we define w(iz, IMI) = IJI (and for any p, w(p) is w(a, lUl) where tp(zi, lMl) is a stationarkation of p, w(zi, A) = w(tp(zi, A))). Now the main properties we succeed to prove are:

(A) If T is superstable, w(a, A) is always well defined end finite (see 3.9).

(B) If I is independent over A, for every there is J E I, 1 JI 5 w(a, A) such that I - J is independent over (A u a, A ) (see 3.16).

Note that previously we could get I JI < K(T); however we then get I - J is independent over (A u zi u U J, A), and this cannot be done here (even the vector space over a bite field c m serve as a counterexample).

Now by (B) we can show that dimensions are “almost” true, i.e.: (C) If p E Brn(A), J E p(Q), I l s J is a maximal set independent over

A ( I = 1, 2) then 1111 I; w(p)JIal. However, the weight defined in Section 3 does not satisfy (2); so we

continue our sectrch in Section 4. First let us contemplate an example. Suppose H is the disjoint union of two algebraically closed fields: P, and P,, and a l ~ P I . Now clearly tp((al,a,),O) has two dimensions, and the natural way is to try to decompose it. So (already in 3) for a stationary regular typep, we define lowp(zi, A) just as w(zi, A), however it is not I JI but E J: tp(8, 1HJ) is not orthogonal to p}. Note that then w(8, A) = zn lown(zi, A), where among any family @ :pregular, stationary not orthogonal top,,} just one appears. We call


q = tp(Z, A) unidimensional if q sw p * p sW q; for supentable T this means that for some p , p sw q, p regular. We may now ask:

(a) For a and A, and stationary regulmp not orthogonal to tp(si, A), can we find equivalence relations B over A, such that tp(a/E,A) is unidimenaional not orthogonal to p ?

(b) Does the parallel of (*) to low, hold? i.e., let p be stationary and regular: I = {&: i < a}, then

(b') Does (b) hold at least when U I E q(Q q EB'"(A), q uni-

Unfortunately the answer to all those questions is negative (see

The moat important notion of Section 4 is a semi-regular type (me

(i) If p ia semi-regular, it is unidimensional, 80 all the regular types

(ii) (b') holds when q is semi-regular. (iii) (P) If a$aclA = A, tp(Ti, A) not orthogonal to a minimal

type q, then for some equivalence relationE over A , tp(iz/E, A) is semi- regular (semi-minimal, in fact) not orthogonal to q.

(iv) (Pa) If T is superstable, ti $ A = acl A, then for some equivalence relation E over A, tp(a/B, A) is semi-regular (hence not algebraic) (see 4:ll).

(v) If bl is Fg-saturated, K 2 lA I + -t K,(T), p = tp(7i, A) semi- regular, then for some indiscernible set I E M over A, based on p , Av(1, A u u I) I- Av(I, 1N1) (see 4.22).

Note that (v) would not hold for a non-unidimensional type. How- ever if we me ready to wave (i), (v), we can replace semi-regulm by cl~{p}-simple ( p some regular (and stationary) m-type) and (ii), (iv) holds, the family of such types is closed under supertypes, and a stre@he+ of (iv) holds.

(iv') (P) If B $ A = aclA, tp(a,A) is not orthogonal to the stationary regular type p , then for some equivalence relation B over A, tp(Z/B, A) is clj{p}-simple and low,(?i/E, A) > 0.

Note that the family of regular types fail to satisfy (iv), and it should be easy to find families satisfying (iv) (or (iv')) but not (ii). The distinct roles of regular, semi-regular and simple will be exemplified in the proof of IX, 2.3 and 2.4. (Simplicity is defined in fact for a regdm family of types, see Definition 4.3.)

dimensional not orthogonal to p 1

Exercise 4.11). However some weakening has positive answers.

Definition 4.1). Its important properties are:

not orthogonal to it are pairwise not orthogonal.

UH. v, 6 01 INTRODUOTION 227

Sections 3 and 4 are relevant mainly to superstable theories (as for stable theories we do not have existence theorems for regular types). So in Section 5 we return to stable theories. In Section 2 we have dealt with unidimensional theories, and in Section 5 we continue this by dealing with theories with a bound number of dimensions called non-multidimensional (this means, e.g., that {dim(I, M): I E ilf) 64 F&(n- saturated, has a bound AT). The main result is again a dichotomy (see 5.8 and 5.9): (a) If T is multidimensional, H, 1 N# 2 K,(T), T stable in K,, then

T has at lead 2 I K - B + non-isomorphism F&-saturated models of cardinality N,.

(8) If T is not multidimensional, for every F&(,-saturated model there is N < M, llNll I 2ITl, N F&,-saturated, and J G independent over INI, such that 64 is F&,-prime over IN1 u (J J .

By ( f l ) we get the number of Q,,-saturated model of T of cardinality N, quite accurately (when a - 8 is big enough, T stable in N,- accurately). In IX, 2.3 we get accurate results for superstable T.

The main lemma in the proof is as follows. We ask for an indiscernible set I over A, what can be {dim(J,M):A E m, MF:r(,,-saturated, stp,( J, A) E stp,(I, A)}. So we choose I, realizing

does not fork over A. Maybe we can choose the dimensions dim(I,, M ) at random. Otherwise we prove that for some a < K,(T), {a, :i < a} is independent over M , a, realizes Av(I,,M) and N is F:r(,,-prime over M U {ai:i < a> then N realizes Av(l,,M) (see 5.3).

In Section 6 we try to deal with cardinality quantifiers, and two- cardinals theorems. Think of a complete (stable) theory T , in a language L(3<X), x regular; and we assume that every formula is equivalent to a predicate (in T). So we can replace T, by its fbst order part, T, together with the following requirement on the model M: if (V@[(3<Xz)tp(z, 8) = #(@I E T,, tp, # first order then M C #[a] e Itp(M, 7i)l < x for each 7i E !MI.

We generalize this somewhat in order to include more cases (e.g., we want just P ( z ) to have a fixed cardinality). So T will be a (complete, stable) first order theory, R = (F, W, A, p ) where: F is as in Chapter IV; W is a set of triples (p(x, ij), &8), x ) which means that for good M, M I. #[6] * Iv(M, 7i)l < x (note we demand only e- and not 0).


We do not require anything from W , but for any formula tpfz, a) we define C(cp(z, a)), which is the minimal x (or co) such tbt our previous requirements imply Iq+lf,ti)l < x (for good bl; we just notice algebraic formulas are < x, and the union of less than x sets each of cardinality < x is < x ; we use just the rules corresponding to this). We formulate six conditions C1-6 from which we prove various results. For a function h, h(x) < x, we define an h-good model, assuming C3 it says that for zi E 11111, h(C(tp(z, a))) I Itp(H, a)l < C(tp(z, a)). The main result is 6.7 which says that: any good model can be extended to an F-saturated 8-good model. Note that the question whether ltp(Y, a)[ < x in any good Y, is not an elementary property of 8, but is equivalent to “6 does not satisfy a type pX”.

Now if F = R, K 2 KJT) , p. = K,(T), h = (2IT1)+ and h 5 x, T stable in x for each x appearing in W, then all the conditions are satisfied. But there are other cmes which interest us, where this is not the case. Hence we suggest another set of conditions C*1-8, which maybe me less nice looking. Their consequences are somewhat weaker, but they are satisfied in more cams. In 6.12 we prove the existence of k-good models. Note that in treating C*1-8, we use more good sets and less good models. In 6.14 we find when the various conditions me satisfied, and we go on to “applications” to two- cardinality quantifiers, which should exemplify the flexibility of our treatment. Those assertions fail in general for unstable theories.

THEOREM: (A) If T is (stable and) countable, M < N , M # N , P ( M ) = P(N) there is a model N* of any cardidity > lIN/l, such that N < N*, P (N) = P(N*) (see 6.14, the countability is essential; it is generalized to F-saturated: Tnodels).

(B) If IPdWl = X I , x o < * * * < x,,, and IT1 s xo I - - I x” then T b a model N, (P,(N)I = x1 (see 6.16).

(C) Suppose xo < . . - < x,, are regular, for any ~ ( z , 5) and 1 for some &@), for every E M y bl C $:[a] o lv(bl, T i ) [ < x I . Then for any xi, IT1 < xo < . . < xn T has a model N , such that for any ‘p, 2 and 8 E IMI, Itp(M, a)] < x1 o bl C @$i] (See 6.16).

then there is N < My IlNll = A, IP(M)l 5 2’*’ (see 6.16, in fact we can get IP(M)I 5 I TI ; for T stable this fails).

(D) If T is 8wpeT8kZht!ey IP(&f)I < h = 11N11, h regular for SiW&@iCitY,

In Section 7 we return to ranks. First, we note that in Section 6, the property “Itp(M, “)I < x for each good My’ is quite similar to “tp(x, a)

OH. v, 8 01 INTRODUOTION 229

is algebraic”. Part of Section 7 is devoted to this generalization, so we have K-minimal, and other concepts in the same vein, and a rank D(p, L, 9) where now D(p, L, K) 2 a + 1, for finite p , means that for any x, for some good M , p has > x pairwise contradictory extensions q over M, D(q, L, 9) 2 a. Some of the theorems are meaningful for the usual notation. In 7.3 we show that when tp sK # (i.e., for good M , Itp(M)l s I#(M)l + xOIx), then we can partition tp to “small” parts, indexed by #, and when tp (or #) is (weakIy) K-minimal, we get better results. So in 7.4 we can characterize when tp sK #. In 7.6 (and 7.7), we deal with tp(z, fj) and q such that

( *4) for any good M , {(tp(H, a) ] : algebraic} has cardinality < &,, where A. < A=.

realizes q, tp(z, a) not K -

We show that then we can replace ho by some k < w , q by a h i t e subtype. When tp(z, 8) is weakly K-minimal for a realizing q then also

ltp(M, Z)l = 1tp(M, 6)1 for any good M , or tp(z, a), tp(z, 6) are K - algebraic” is a first-order property of a-6 with finitely many equivalence classes. It follows that I{Itp(M, a)l: tp(z, a) not K-algebraic}l < A. < AK for good M, implies that “tp(z, a) is not K-algebraic” is a first order property of a. We can get (still assuming (**)) also that “tp(z, a) is weakly K-minimal” is first-order property of for the trivial R; and if ~ ( z , a) is minimal, T totally transcendental, we can get in fact a similar result (see 7.7(2)).

Later we try to prove theorems of the form: if q(Q) is divided, the parts indexed by #(a), each part has a bounded rank, then we can bound tp’s rank using #’a rank and the bound on the ranks of the parts. More accurately, suppose

L <

m, Y) 1 t p ( 4 A #(Yb t p ( 4 (W(z , Y).

#(Qh then RCtpk), L, sol I 4 8 + 1) or a = 0, R[tp(z), L, sol s B. By 7.8, if /3 = R[#(z), L, so], R[B(z, b) , L, KO] s a for every b E

For the rank R( -, L, 00) we get a similaz result when /3 < o, and a better bound (a + 8) when a,B < o. It follows (see 7.10) that when T 5 $9

R(*, L, No) < * R(% L, K O ) < a,

R(#, L, 00) < w * R(tp, L, CO) < o.

At last we deal with another rank, L, defhed for complete types, without taking finite subtypes (so we lose the finite characteristic). We let L ( p ) 2 a + 1 if L(q) 2 a for some extension q of p which forks

230 MORE ON TYPES AND SATURATED MODE= [OH. v , $ 1

over Domp, and let L(a, A) = L(tp(B, A)). This ranks satisfies more than the above: L(a-6, A) s L(a, A u 6) @ L(6, A) (@-the “natural sum” of ordinals, see Definition 7.6). The rank has other natural properties, e.g., T is superstable iff (Vp)L(p) < 00, if tp(a, A) does not fork over B c A then L(G, A) = L(6, A ) ; L ( p ) s R(p, L, 00).

In this chapter T will be stable.

V.l. Orthogonality, regularity and minimslity of types

DEFINITION 1.1: (1) If p(Z1), @a) are complete types over A, p an m-type, q an n-type, we call p weakly orthogonal to q i f f p(Zl) u @a) is complete (over A).

(2) Let p1 be complete or stationary and p a complete or stationary. Then p1 is orthogonal to pay if for every A, Dom p1 u Dompa E A, A the universe of a F$,,-saturated model, and any stationarizations q1 of ply 1 = 1, 2 over A; q1 is weakly Orthogonal to qa (see Definition 111, 4.2(2)).

(3) The infinite indiscernible sets I , J are orthogonal if Av(I, U I), Av( J, U J ) are orthogonal (see Lemma 1.1 (3)).

(4) The type p is orthogonal to the set A if p is orthogonal to every complete type over A.

LEMMA 1.1 : (1) If p is algebraic then p is orthogonal to any type q (when both are stationary or complete). If p €Sm(A), p is realized by jzcst one sequence, then p is weakly orthogonal to any complete type over A.

(2) If p 1 = pI(EI) = tp(7il, A) then p1 is weakly orthogonal to pa iff

(3) Weak orthogonality and orthogonality are symmetric relations. (4) I f tp(Sil-&,A) is [weakly] orthogonal to tp(61^6aYA) then

tp(7il, A) is [weakly] orthogonal to tp(6,, A ) (foralE1(7il), Z(aa), Z ( & ) , Z(Ea)) . ( 6 ) If p I (1 = 1, 2) are complete types over A , and they are not orthog-

onal, then for some jinite B there are statimrizations q1 of p I over A u By such that ql , qa are not weakly orthogonal.

( 6 ) When p I = tp(7i1, A,), p1 is orthogonal to pa iff for any a; realizing

@(a,, A) 1 tp(7iay A u 7il) iff tp(7i1, A) F tp(si1, A u as).

P I , stp(a;y A,) is orthogolzal to Stp(a;, Aa).

Proof. Clearly (2) is a restatement of Ax(V1) for FL (see Lemma IVY 2.9). Then (3) follows; and (4) follows from Ax(1V) for FL. Now ( l ) , (6) are trivial and (6) is easy.

CH. v, 8 11 ORTHOOONALITY, REQULARJTY AND -Y 23 1

THEOREM 1.2: (1) Let B c A, and for 1 = 1, 2, p , i8 a complete type over A, p , does not fork over B. If Pa 1 B i8 8tU4%mtZ?’y, ply pa are weakly orthogonal, then p1 1 By pa 1 B are weakly orthogonal. (2) If ply pa are complete types over A, and they are orthogonal, and

one of them is stationary then they are weakly orthogonal. (3) If bl i8 F;,,-saturated, p , are wrnplete types over hf then ply pa are

orthogonal iff ply pa are weakly orthogonal. (4) I f p , i8 parallel to qr for 1 = 1, 2 ( h e w they are stat~onay) then

ply pa are orthogonal iff ql, qa are orthogonal.

Proof. (1) This is a restatement of Claim IVY 4.16. (2) Follows by the definition and (1). (3) By 111, 2.15(2) ply pa are stationary, so if they are orthogonal,

they are weakly orthogonal by (2). So suppose they are not orthogonal. Then there is a l!‘&,-saturated

N , M c N , and stationarbations q, of p , over N such that qly are not weakly orthogonal. By 111, 3.2 and 111, 2.9 there is C EM, ICl < K(T), such that p , doea not fork over C, p , r C is stationary, hence q1 does not fork over C , hence q1 is definable over C , by 111, 4.8(2).

There is 6 E N such that ql(Z) U qa(g) U {cp(Z, g ; 6)’) is consistent, for t E (0, l}. Choose 6, E Y, tp(6,, C) = tp(6, C) (since Y is F&+&urated). By the F&-saturation of N and the definability of q, over C, ql(Z) u qa(g) u {cp(?Z, g; 6,)’) is consistent (for t E (0, 1)) hence pX(Z) u pa(@ u {cp(Z, 8; 6,)’} is consistent, hence ply pa are not weakly orthogonal.

(4) Choose a F&,-saturated model My Domp,, Domq, c Y for 1 = 1, 2. Let r, be the stationarization of p, (and q,) over Y. By the symmetry it suffices to prove rl, ra are orthogonal iff ply pa are orthogonal. The “only if” can be proved by (l), (3) and the “if” by the definition.

LEMMA 1.3: The types p , q are orthogonal (in Q) iff pea, 40 are orthogonal in Qeq.

Proof. Use 1.2(3) for Ceq, remembering lMeql = dcllMl in V a .

LEMMA 1.4: (1) fluppwe {at: i < n} is intiependent over A (see 111, Definition 4.4(1)). Then tp(6, A ) is orthogonal to tp(B,; A ) for i < n iff

(2) If p , = tp(Bty A) are stationary and p ino i se orthogonal (i < a) then Ut<apl(Zt) is cinnpiete, or equivaientiy tp(7i,, A ) I- tp(Z,, A U U,<,Z,).

tp(6, A) i8 orthogonal to tp(BonB1n- * *-Bn-1, A).

232 MO- ON TYPBS AND S A T U R A ~ D MOD- [CE. v, $ 1

Prmf. (1) If tp(6, A) ia orthogonal to tp(Sion-..n7i,-,, A) then by 1.1(4) tp(6, A) is orthogonal to tp(Zt, A). For the other direction, by 1.1(6), 1.2(4) we can assume A = w, M is F;(,,-saturated. Now mume tp(Si,, A), tp(6, A) are orthogonal for 1 < n, and prove by

induction on k < m that tp(6, A) k tp(& A u Ulck a,). For k = 0 it is immediate; for k 2 1 i t follows by 1.2(2), (4). So tp(6, A ) and tp(Sion. - -hSisil)_l, A) are weakly orthogonal, hence by 1.2(3) they am orthogonal.

(2) The proof ie immediate by (1).

CONCLUSION 1.6: (1) Let I be an indiemnible eet over A , based on A, and tp(Z,A), Av(I,A) are o r t h u g d . Then tp(Z,A), tp,(I, A) are orthogonal (i.e., for Si, E I , tp(Z, A), tp(SiOn. - A) are o r t h u g d for all n < w).

(2) If I , J are indiscernible eets over A , baaed on A, and Av(I, A ) ie orthogonal to Av( J, A ) then tp,(l, A ) ie or t7wgd to tp,( J , A) .

Remark. For a strong converse to 1.6(2) see 2.1.

Proof. (1) By 1.4(1). (2) By a double use of 1.6(1).

LEMMA 1.6: (1) If for i < a, tp(6, A ) ie o r t h o g d to

and tp( 6, A ) ie e t a t i m y then tp( 6, A ) k tp( 6, A u Ut at). f?o tp( 6, A)

(2) We can r e p h in (1) tp by stp and m i t the aBmmpth that i8 o r t h O g d t0 tP(Z0”-

tp(6, A ) i8 8kctiOnary.

.â,,, A ) (We (1.2(3)).

Proof. (1) Immediate by induction oq a, using 1.2(2). The eecond part by 1.2(3).

(2) Can be easily proved in a similar way, or through W.

LEMMA 1.7: If pt, (i < a) are p i r w h e o r t h o g d , but q ie not orthog- mucltop,, qieBtationarythena < ~(T) ,andeven <K~, ,~(T) .

Proof. By 1.2(4), 1.1(6) we can assume q, p , are complete types over a Ft(,,-saturated model M. Then use IV, 4.12(3).

CH. v, 0 13 OBTHOG)ONILLITY, REGULARITY AND 233

DEFINITION 1.2: (1) A non-algebraic type p €&"'(A) is regular if for every B 2 A and r such that p E r E Bm( B); if r forks over A then p , r a m orthogonal.

(2) A stationary type p is regular if its stationarization over Dom p is regular.

LEMMA 1.8: (1) I f p , q are parallel (and stationary) then p is regular iff q M regular.

(2) If tp(Ti, A ) is regular, then stp(Ti, A ) is rqular. ( 3 ) I f tp(B, A ) = tp(6, A), then stp(B, A ) is regular iff stp(6, A ) M

(4) tp(Ti, A) M regular in (E iff it is regular in P. regular.

Proof. (1) By the definition, we can assume p , q are complete. Let C = Dom p u Dom q, and let Y be the stationarization of p (and q) over C. By the symmetry it suffices to prove that r is regular 8 p is regular.

If p is regular, and r -c r1 EB"'(A), r1 forb over C c A, then r1 forks over Dom p, hence is orthogonal to r .

So suppose p is not regular; then there am pl E P ( B ) , Dom p E By p E p , , p , forks over Dom p but p , is not orthogonal to p. We now know p and r are stationary and parallel. By 111, 2.6(3) there is an elementary mapping f of (E such that f 1 Domp = the identity, and for every 6 E B stp(6, Dom p ) = stp( f (6), Dom p), and tp( f (6), C) does not fork over Domp. Hence pa = f ( p l ) is not orthogonal to p, so p2 is not orthogonal to r. Let p 3 be the statiomrization of pa over C u Dom pa; p 3 1 C does not fork over Dom p, by Ax(VI1) for F/, . But p C p z E p,, hence, as p is parallel to r, r E p , . Now p , , being parallel to pa, is not orthogonal to r. This shows r is not regular.

(2), (3) Immediate. (4) By 1.3.

QUEBTION 1.1: Does the converse of (2) hold?

THEOREM 1.9: Let p eBm(A). The following conditim are equivalent: (1) p is regular. (2) If A s By I v {E} a set of sequences realizing p and f o r every

Ti E I , tp(Ti, B) f o r b over A , and tp(Z, B v U I ) f o r b over A , t h n tp(E, B) f o r k s over A .

(3) If B = A u U J, J v {Ti, Z} i s a set of sequem realizing p , and tp(Ti, B) f o r b over A and tp(E, B v Ti) f o r b over A, then tp(Z, B) f o r b over A .

234 MORE ON m P E S AND SATURATED MODELS [CH. v, 5 1

(4) I f . 2 K,(T),K > IAI,A E M , M b e - 8 d U r d d , p E q=tp(ii,M), q does not fork over A , and N i s @-prime over M u ii; then for no 6 E IN1 - lMl, 6 realizes p and tp(6, M) forb over A . (If p is stationary, then for all 6 E IN1 - 1M1,6 realizes p * 6 redim q).

(6) The hypo&& and con~luSion of (4) bold for 8 m M , 6, N , K .

Remark.. If p is stationary, we can in (4) omit the condition K > IAl.

QUEBTION 1.2: Can we in (4) replace e by Fx? Can we replace K,(T) by K( T ) 2

Proof. (1) * (2) Supposep is regular, A E B, tp(?i, B) forks over A for every 7i E I, and tp(8, B) does not fork over A, and we shall prove tp(i5, B u U I) does not fork over A (clearly this is sufficient). By 111,2.6( 1) and 111,1.4 stp(8, B) has an extension q, q z T E Bm(B u U I), which does not fork over A. Hence it suffices to prove stp(E, B) k stp(8, B u U I), so it suffices to prove this for finite I. Let I = {a1: i < n}; so aa tp(Z1, B U {a,: j < i}) forks over A and extends p, it is orthogonal to p, hence to stp(E, B). Hence by 1.6(2), stp(E, B) k stp(i5, B u U I ) . (2) * (3) Immediate, as (3) is a particular case of (2); taking I = {a}. Before proving (3) + (4) we need:

CLAIM 1.10: (1) If M is F:-saturded, K 2 K,(T), p = tp(& 1M1) i8 orthogod to r ~B"(11111), T is non-algebraic and N i s Cqrime over ]MI U zi, then T ha% a unique extenaim in Sm(INI), hence i s not realized in N. (2) Moreover, for every 6 E INI, 6 $1M1, tp(6, IMI u a) forks over [MI.

Proof of 1.10. (1) As M is @-saturated, there is an infinite indiscernible set I, 111 = K , I c 1611, Av(I, [MI) = r. Hence by the hypothesis and IV, 4.10(2) (remembering K 2 K,(T)) Av(I, ]MI) = T k Av(I, IMI u a) 1 AVV, pq). (2) The same proof.

Continuation of the proof of 1.9. ( 3 ) =. (4) Suppose (4) fails, and let T = tp(6, IM]) , which is stationary; hence by the claim tp(6, lMl u a) forks over ]MI, hence tp(zi, lMl u 6) forks over lMl (by Ax(V1) for FL). Let J = {Z: 8 E IMI, 8 realizes p}, B = A u U J ; we shall show that A, B, a (for 8) and 6 (for a) form a counter-example to (3). Clearly

CH. v, 8 11 ORTHOQONALITY, REGULARITY AND -Y 235

tp(si, B) does not fork over A, and tp(si, IMl u 6) forks over A , and tp(6, !MI) forks over A. We need only to replace M by B in the last two types. By the following claim, tp(si-6, [MI) does not fork over B, thus by 111, 4.14, the result follows.

CLAIM 1.11: If x = 8, t, a, M a q-compact maEel, p , (i < n) Fg-type8 over ]MI , and si, realizes p , , and B, = U {a: 8 E M, E realize8 p,} , then tp(aon. - snsi,,, Ute,, Domp, u U,<,, B,) k tp(sion. - --sin, M ) .

Proof. W.1.o.g. p , is a l-type (if it is an m type, let

p! = ((32,. - ~ x i - l q + l - . x , , , - ~ ) A q: q a finite subset of p,}

and replace the p,’s by the pi’s and the sit’s by si,[l]). We can also aasume p , = p (replace p , by p = {V, <,, tp,: tp, E p,}, clearly an element realizes p iff it realizes some p,).

Let B = B,, si, = (a,), si = (ao,. . . , a,,-l). Suppose Ctp[si, E l , E E M ; then by 11, 2.2 and 11, 2.13 for some 6 E B and #, Ctp[si‘, E ] o !+[a’, 61 for any 8’ E B. As M < 6, M F:-compact, CJI[si, 61.

Suppose tp(si, B U Domp) lftp(Z, E), so somesi* realizes tp(8, B uDomp) (hence a*[l] realizes p) but C-,tp(si*, 17). So,

q = {tp(xi; 6): Z < n, tp@; 6) ~ p }

u {#(xo,. . .,%t-1; 61, 1 d x 0 , * * - 9 %-l, a,} is realized by a*. Clearly q is a Fz-type, hence q is realized in M, by a**. So a** E B, t#[a**, 61, C-,tp[a**, E ] contradiction to the definition of #.

Continuation of the proof of 1.9. (4) 3 (6) Trivial. (6) - (1) Suppose p is not regular (i.e., (1) fails), K 2 K,(T) + IAI +,

A G 1M1, M q-saturated, and tp(si, IMI) is a stationarization of p and N a FE-prime model over 1M1 u si. As p is not regular there is B, A E B, and q E P ( B ) , p E q, q forks over A, q is not orthogonal to p . We can assume that IB - A1 < min{x(T), Hl}, and q is stationary.

As M is Fl-saturated, we can msume B c M , and let q1 E P ( ~ M I ) be a stationarization of q. So ql , tp(& IMI) are not orthogonal, (by 1.2(4)). By 1.2(3) there is a 6realizing q1 such that tp(6, wl U a) forks over A (as q1 is stationary, being over a model). So (by 111, 1.1(5)), for some $ , c E M , l=$(6,6,@ and $(Z,6,@ forks over wI. Clearly q U {$@, aLq} is a consistent F,“-type over N , hence realized by some EN. This b contradicts (5 ) , so not (1) implies not (5 ) . This completes the proof of Theorem 1.9.

236 MORE ON TYPES AND SATURATED MODELS [CH. v, f 1

THEOREM 1.12 : 8 u p e p , q are complete typea Over I M I , 6 reulizes q, M ia F39aturatedY K 2 K,(T), N is Fg-prim over lMl U 6. If p , q are not orthogonal and p i.9 regukcr, then p is realized in N .

Proof. By 1.2(3), asp, q am not orthogonal, they am not weakly orthogonal. Hence p has an extension p1 E P ( ~ M ~ u 6), which forks over 1611; let Ti realize p1 (Ti is not n e d y in N ) . Choose C s lMl, ICl < K(T) such that p = tp(& IMJ) does not fork over C and tp(Ti, C) is stationary (by the saturation of bl). Let B = u {Z: Z E N , E realizes p 1 C). Then tp(Ti, C u (B n M)) does not fork over C. If tp(a, C u B) does not fork over C then by 1.11 also tp(Ti, N ) does not fork over C, contradiction as 6 E N . So tp(Ti, C u B) forks over C, hence over C u (B n M) (by 111, 4.4). So for some soy.. . , a,, E N realizing p , and E E C, and 9, cp(E, a,, . . . , d,, Z) E tp(si, N ) forks over C u (B n M). If for every i the type tp(&, C u (B n M) u u,<, 3,) fork over C then it would be orthogonal to p (as p is regular) hence by 1.6

tp(Ti, C u ( B n M)) k tp a, C u ( B n H) u U 2, 198

But the second type forks over C u (B n M) contradiction. So the a, for which the above tspe does not fork over C realizes p and is, of course, in N .

CONCLUSION 1.13: (1) The r&ion of non-orthogdity among tire stationary regular typea, i.9 an equivalence relation.

( 2 ) .If p , q are ~tationary, regular, and not orthogonal, then r i.9 orthogonul t o p iff r ia orthogonal to q.

Proof. (1) Reflexivity is trivial (using non-algebraity). Symmetry was proved in 1.1 ; and transitivity is immediate by 1.12 (going to parallel types) -

(2) Also easy. Remember that Ti depends on I over A if tp(Ti, A u u I ) forks over A.

THEOREM 1.14: FW a set A tlte relation of &pendew Over A among sequences F for which stp(F,A) is regular satis$es the axioms of dependency relation (Def. AP 3.3) . If p , is regular, on {F:stp(c,A) regular not orthogonal to p , ) it is a nice dependence relation.

Procf. (1) The exchange principle. It suffices to show that if {a,: j < a} is not independent over A, then some Ti, depends on {al: i < j} over A. This is 111, 4.19(6).

OH. v, 4 13 ORTHOQOWLUIITY, REQULlWTY AND M I N U T Y 237

(2) The hite character. That is, if a depends on I over A then iz depends on some finite J c I over A. Easy, see 111, 1.1(6).

(3) Transitivity: Here we use regularity. Suppose iz depends on I over A , each 6 E I depends on J over A and I, J m independent over A. We mn replace I by a finite subset, and then J by a finite subset, without changing our assumptions. W e can work in F, and replace A by acl A, (see 1.8(4) and 111, 6.3(3)), so every complete type over A is stationary. Let I , = (6: 6 E I, tp(6, A) is not orthogonal to tp(iz, A)}, J , = {Z: 3 E J, tp(E, A) is not orthogonal to tp(iz, A)}. Now iz depends on I,, for otherwise let I - I, = {st: 1 < n}, and as tp(iz, A u u I , ) does not fork over A and is stationary 1.6 shows tp(iz, A u U I,) k tp(8, A u U I , U u {6[: 1 < n}). Hence the latter does not fork over A, contradiction. Similarly, every 6 E I , depends on J1 over A. Let I, = (6C i < k), and assume iz does not depend on J , over A. As tp(st, A u U J,) forks over A, tp(6{, A u u J , u u (6': j < i}) forks over A, hence (by the regulazity of tp(8, A)) is orthogonal to tp(Ziy A), hence to tp(iz, A), hence to tp(8, A u UJ,) (using 1.13, 1.8(1) and 1.2(4)). So by 1.6 tp(iz, A u u J, ) t- tp(iz, A u u J , u u I,), so the latter does not fork over A, so tp(7iy A u U I , ) does not fork over A, contradiction.

(4) Full transitivity (for {F: stp(c, A ) regular not orthogonal to pol): Repeat the last part of the proof of (3).

CONCLUSION 1.15: (1) If p E Sm(A) , A E M , I, regular (or euery e t a t i o n a ~ z a t h of 7i ie regular) then dim@, M ) is true.

(2) If i5 E I * atp(c', A ) i s regular, J,, J , are mu&mal subsets of I independen4 over A , then I J,I = I J,I.

Proof. By the previous theorem and AP 3.10.

LEMMA 1.16: Let A G IMl,p E S ~ ( A ) , I r lMl a d d i n r E e p e n d e n t 8ei over A of 8t?quences realizing p in M, p regular, M ie c-saturated, IAl+ + K(T) s K , K regular.

(1) For every ~tatiorucrizution q = tp(i3, 1M1) of p over M y stp(B, A u U I ) t- q, anal if p i s stationary, q (A u U I ) t- q.

(2 ) For euery i5 E M there is J E I , IJI < K ( T ) 8w:h that tp(c', A u U J ) k tp(c', A u U I ) .

(3) Let p be r@r, Mi (i 5 a) c - s a t u r d modeb, i < j == M , ( M,; Md ie F:-pimuy over UiSd lMil, M, =; M , and I , G lMi+,l ie a mmxthal independent set over IMiI of sequence8 realizing 8ome stationarization of p over (Mil. Then U,,,I, u I i s a dd &ei of M , i d - over A of sequence% realizing p.

238 MORE ON TYPES AND SATURATED MODELS [CH. v, f 1

Proof. Easy, using 1.11 and 1.14.

DEFINITION 1.3: (1) An m-type p is minimal ifp is not algebraic but for no cp(Z, 8) are p u {cp(E,

(2) An infinite indiscernible set I is minimal if Av(I, U I ) is minimal. (3) A formula cp = 'p(Z, a) is weakly minimal if 'p is not algebraic but

it has a bounded ( < R ) number of non-algebraic pairwise contradictory extensions.

t = 0, 1, both non-algebraic.

THEOREM 1.17: (1) The m-type p ie minimal iff there is aJinite A , euch that for every Jinite A 2 A,, P ( p , A , No) = 1, Mlt(p, A, KO) = 1 iff for every cp, P ( p , 'p, No) s 1, and Bm(p, cp, KO) = 1 * l W p , 9, No) = 1, andfor some 9, R"@, 9, KO) = 1. I f p is complete, MIt(p,q, KO) = 1 for every p.

( 2 ) The m-type p ie algebraic iff for every A , Bm(py A , KO) = 0 iff for every 'p, lP(p,cp, KO) = 0 iff P ( p , L , A) = 0 (any h L KO) iff Dm(p,L,m) = 0.

(3) The fornzuZa cp(Z; 8) ie weakly minimal iff Dm['p(Z; a), L, a33 = 1. If p ie etatimucry, 'p(Z, 8) E p ie weukly minimal, p i s not algebraic, then p ie minimal.

(4) If Bm(p, L, KO) = 1, Mlt(p, L, KO) = 1, then p w minimal; and the converse holde for finite p.

Proof. (1) Suppose p is minimal, then let A, = {E = PI. As p is not algebraic, there are distinct 8, (n < w ) which realizes it, hence p u {Z = a,} is consistent and pairwise contradictory. So for every A 2 A,, R"\(p, A , KO) 2 1. Now it is easy to prove (by (2)) that if d is finite, Rm(p,d, KO) > 1 or Rm(p,A, 8,) = 1, Mlt(p,A, KO) > 1 then for some cp(2, @) E A , and 8, p u {cp(f, a)'} is not algebraic for t = 0, 1, contradicting the minimality of p. This proves that the first condition implies the semnd and third. The other direction is easy.

(2) If p is algebraic, some finite q c p is realized only by n < No sequences; hence q does not have n + 1 pairwise contradictory extensions, and no kn + 1, k-contradictory extensions. On the other hand if p is not algebraic, for every h there are distinct (i < A ) which realizes p , so p u {Z = 8{} are consistent and the formulas Z = are pairwise contradictory. Observing this, (2) is easy.

(3), (4) Easy too.

THEOREM 1.18: (1) if p is a minimud m-type over A , t h it hae a unique nowalgebraic & d o n q E Sm(A).

OH. v, 5 11 O R T E O Q O N ~ Y , REGuLhIlITY AND -Y 239

(2) If p is a m i n i d type, then there is a unique (up to equivalence)

( 3 ) If p is minimal, p c q, q forks over Dom p then q is algebraic. (4) If p E Sm(A) is minimal, then p is stationary and regular. ( 6 ) A complete type parallel to a m~nimal type is minimal. ( 6 ) If tp((al, . . ., a,,), A) is minimal, then for some 1 ii is algebraic

over A u {a,} (where ti = <al, . . . , a,,)). (7) If M is ~,-wmpaCt, A3(T) 5 A, p an m-type over bl, t h n p has an

extenaim q, IqI < A, even lq - p1 c A3(T), q a type over M , q is m i n i d .

Remark. In view of (l), (4) and (€9, we shall call minimal types stationary, and extend the definition of parallel accordingly; (and also accompanying concepts, “ I based on p ”).

non-trivial injnite indiscernible set of sequences realizing p.

Proof. (1) A s p is not algebraic, we can find > 21-4 + elements realizing it, so infinitely many of them realizes some q E Sm(A), which is the right extension. Clearly it is unique. (2) As p is not algebraic, there is at least one such I . If I , J are two

such sets, let Domp u I u J E M ; then Av(I, M ) , Av(J, M ) are non- algebraic extensions of p in S*(M), hence they are equal, hence (by 111, 4,17(1)), I , J are equivalent.

(3) Follows by 1.17(1) and 111, 1.2(4) and 1.17(2). (4) By 111, 4.2, and 1.17(1) p is stationary. By 1.1 every type is

(5) Easy, using 111, 4.2. (6) By 111, 6.2(6) for some 1, a, # acl A . If a, 4 acl ( A U {al}), then

tp(6, A u {a,}) forks over A (as a, # acl A , q = a, forks over A ) but it is not algebraic, contradicting (3).

(7) Easy, like IV, 2.16(2), remembering the finite character of being an algebraic type.

orthogonal to an algebraic type hence p is regular.

EXERCISE 1.3: In 1.16(1), we can get q Si, E I realizes stp(8, A).

( A u U I) I- q when some

EXERCISE 1.4: Suppose tp(6-6, A ) is regular, ti 4 acl A. Prove tp(6, A)isregular.[Hint:LetA c IMI,MFR-saturated,A > IT1 + IAI, and w.1.o.g. tp(6-6, [MI) does not fork over A. Suppose tp(sil, 1611) extend tp(si, A) and fork over A; and it suffices to prove that tp(si, IMI), tp&, 1611) are weakly orthogonal. Choose 6, such that tp(ii-6, A) = tp(iilnb,, A). Clearly also tp(Si1”gl, 1M1) forks over A. Hence


tp(ZlnEl, [MI) is weakly orthogonal to tp(a-6, 1M1). So by 1.1(4) tp(Z,, IMI), tp(?Z, 1611) are weakly orthogonal as desired.]

EXERCI8E 1.5: Suppose M is Fg-aaturated, K 2 K,(T), tp(4, IMI) are orthogonal and tp(6,, 1x1 u a,) are e-isolated (for 1 = 1,2). Prove tp(iT1-6,, 1M1) ( I = 1,2) are Orthogonal; and generalize by replacing the sequences by sets. (Hint: Use 3.2.)

V.2. Dimensions and orders between indiscernible sets

DEFINITION 2.1: (1) Let I, J be infinite indiscernible sets. I I, J if for every K 2 K,(T) and Fg-saturated model M and I,, J, E M (infinite) indiscernible sefs equivalent to I, J respectively, dim(I, M) s dim( J,, M) (w stands for “weakly”). (2) Let I, J be infinite indiscernible sets, I S, J when for every

K 2 K,( T) and e-saturated model M and I,, J1 s M equivalent to I, J respectively and Fg-saturated model N, M < N if Av(I,, M ) is realized in N then also Av(J,, M) is realized in N (8 stands for “strongly”). (3) For stationary types p, q, p I , q (p sa q) if the corresponding

relations hold between indiscernible sets based on them.

LEMMA 2.1: (1) FOP any in,nite indiscernible eet I , I I, I , I I , I . (2) I,, sa are tradtiwe. (3) I I,J*I s,J. (4) If I,, I , are equivalent and J, , J, are equivalent then for x = 8, w,

I , 5, J, 0 I , 5, J,.

Proof. (1) Immediate. (2) Suppose I,, 1 2 , I, are infinite indiscernible sets, M a Fg-saturated

model, 1; c M, Ij c M and I ; , Ij are equivalent to I,, I,, respectively (where K 2 KJT) ) , w.1.o.g. I, = I ; , I, = Ij.

We can find, for 1 = 1, 3 sets A, E U I,, such that I, is based on A,, Av(I,, A,) is stationary and lAll < min{K(T), 24,) (see 111, 4.17(4)). So we can find an elementary mapping f, f 1 (A, u A,) = the identity, and f maps A, into M ( B exisfs as M is R-saturated).

So we can find in M an infhite indiscernible set I; based on f(Av(I2, As)). Clearly, I ; is not necessarily equivalent to I , . But for x = W, 8, I , 5,. I, iff I ; 5, Is ; and I, I, I, iff I , sZ I; .

So if I, I, I , s, I,, dim(I,, M ) I dim(&, M) and dim(I;, M) I dim(I,, M). As M, I ; , I ; were arbitrary, it follows that I, I , I,.

CH. v, 8 21 DIMENSIONS AND ORDERS BETWEEN SETS 24 1

Also if I, I, I , 5, I,, zi, realizes Av(I,, M), N is FE-saturated, IN1 u zi, E 1x1 then Av(l;, M) is realized by some E N (aa I, 5, I ; ) , hence some EN realizes Av(I,,M) (as 1~1,). Again I, s , I , follows.

( 3 ) Suppose I SW J, so for some Fi-saturated model N, K 2 K J T ) end I,, J , E M equivalent to I, J respectively, but dim(I,, M ) > dim(J,, M ) 2 K ; w.1.o.g. I,, J , are maximal indiscernible sets EM, lJll = dim(J,, M), II,l = dim(I,, M ) . Hence ]Ill > lJll 2 K . So by 111, 3.6 there is I , E I,, ] Ia [ = I Jll such that I, - I , is indiscernible over lJ I, u lJ J,. By IV, 3.1 there is a F,4-prime model N over UZ, U UJ,, N G M . By IV, 4.10 N omits p = Av(J,,UZ, U UJ,). Choose si E I, - I,, so si realizesp; but M omits (by the definition of J , ) Av (J,, N). This situation shows I J.

(4) Trivial.

LEMMA 2.2: Assume K 2 K , ( T ) , M is F3saturated I, J c M inJinite

p i m e model N over N u a, Av( J, M ) is. realized. indi8~et~ibk 8d8, and zi realizes Av(I, M ) . T h I I, J iff in the’F,O-

Proof. The “only if” part follows by the definition. So suppose Av( J, N) is realized by 6 E N , and we shall prove I I, J.

So let K( 1 ) 2 K,( T ) , M, is F&saturated, I,, J, E H, are equivalent to I, J respectively, zi, realizes Av(I,, M,), N , be a F,”-saturated model, lM1l U si E N,. We must prove N, realizes Av(J,, M ) . As there is a Fi,,,-prime model N , c N over 1M,1 u a,, we can assume N, is F,O,,,-prime over lMll u a,. Choose A 2 ~ ( 1 ) + K , A regular A > IT1 + llMll + l[Mlll, = A, and let M* be a saturatad model of power A, 1M1 U 1M11 E 1M*1. We can assume that a, = si realizes Av(I, M*) = Av(I,, M*). Let N* be e F:-prime model over IM*I u si.

By the symmetry in assumptions, i t suffices to prove that Av( J, M ) is realized in N iff Av(J, M*) is realized in N*.

Suppose Av( J , M ) is realized in N by 8 E N. Let

d = (IMI U a, {bf: i < a}, {Bf: i < a})

be a Fi-construction of N. We prove by induction on p that dz = (IM*I uzi,{b,:i < /3})isa~-construction,andtp,[si~{b,:i < p}, 1M*1] does not fork over IMI. So d; is a FE-construction, hence a Fi- construction, hence tp(8, 1M*1 u zi) is F;f-isolated, hence tp(E, 1M*1) is realized in N*, does not fork over M , and extends Av(J, M), so

242 MORE ON TYPES AND SATTJRATED MODELS [CH. v, 5 2

tp(E,IM*I) = Av(J, M*) so this will finish one direction. We prove by induction on 8; for fl = 0 and @ a limit ordinal it is trivial. For + 1

show q I- tp(b,, 1M*1 u iz u {bi: i < 18)) (the non-forking of we know q = stp(b8, BB) stp(b,, 1M1 u a u {b{: i < p}). It suffices to

tp*@ u {b,: i s p}, p*1> over

So let Zl E a U {b,: i < p}, Za E lM*l, and ~(z, 8, Z) be a formula; and we shall prove q I- ~(z, a,, Za)t for some t. Choose C E 1M1 such that tp(B8 U 17, U iz, 1M*1) does not fork over 0, ICl < K (C exists by the induction hypothesis, and aa K J T ) s K ) . Choose EL E 1M1 such that stp(E;, C) = Stp(E2, C) (possible as M is q-saturated). By the definition of C, q I- ~ ( z , El, Ea) e Q t- ~ ( z , Z,, $). As for some t, ~(z, El, EL)t E

tp(b8, lJfl u a u {biz i < P}), Q I- ~ ( z , Ei, E d t - Now we prove the other direction, i.e., suppose that 5 E N* realizes

Av(J, 1M1*). As h is regular ~ K J T ) , tp(E, IM*I u a) is q-isolated. So there are C E B E lM*l, ICl < K(T) , IBI < A, tp(E, B u a) t- tp(i3, IN*\ u iz), tp(3-5, 1M*1) does not fork over C, and tp(anE, C) is stationary. As M* is a saturated model, there is an automorphism f of M , f 1 ( I V J) = the identify, f (C) c M , (w.1.o.g. I, J are countable) and for K = No, we can demand just that f (I), f (J) are equivalent to I, J resp. ; so we can assume C G M. Now claim III,4.22(2) shows tp(a,Cu iz) I- tp(E, lM*l) (1ettingC u = A, B u a = B, 1M*1 u a = C).

u iz will then follow).

THEOREM 2.3: The following crmditiona on the infinite indi8cernible set8 I , J , and infinite cardinals p > A 2 K J T ) are equivalent:

(1) I s,J. ( 2 ) If M is Fi-saturated, I , J E M , then dim(I, M ) I dim(J, M ) . ( 3 ) There i s ru) q-saturated ncodel M , with I , , J1 c M equivalent to

I , J rapectively, dim(I,, M) = p, dim( J,, M ) = A. (4) Por every I , , J1 equivalent to,I, J reqectively, 1111, lJll 2 K,(T),

I1(J1) indiscernible ower U J , (UI,), Av(J,, U J , ) has a F&,-isoZated extension in P(U I , u U J,).

Remark. So if there is a ql-saturated model M , with I,, J, c M , equivalent to I, J respectively such that dim(I,,iK,) = p1 > dim(J,,M,) 2 A , 2 KJT) then for all ,uLz > A, >, K,(T) there is a F:,-saturated Ma with I , , J, E Ma equivalent to I, J respectively such that dim(I,, Ma) = pa, dim(J,, Ma) = ha.

CH. v, f 21 DIMENSIONS AND ORDERS BETWEEN SETS 243

Proof of Theorem 2.3: (1) * (2) by the definition. (1) =F (3) by the definition. (2) or (3) =- (4) Suppose I,, J1 contradict (4). Without changing the

hypothesis on I,, J , , we can assume 1111 = ,!A, IJ,1 = A (by 111, 4.22). Let N be Fi-primary over U I, u U J,. By IV, 4.10 N omits Av(J,, U I, u U J , ) and Av(I,, U I, u U J,) . From the first omitting it follows that dim(J,, N) I II,l + lJll = p; but dim(J,, N) 2 lJll = p, so dim(J,, N) = p. (See 111,3.9). On the other hand dim(I,, N) 2 II,l = A; suppose dim(I,, N) > A, then there is I* E N equivalent to I,, II*l = A + . As A 2 K,(T), some E € I * realizes Av(I,, U I,). Let d = (I, u J, , {af: i < ao}) be a Fi-construction of N. Going through q,n,(T) (see IV, 4.6-4.7) we can aasume E E {af: i < a}, a < K,(T) (see IVY 1.4(2)) so there are I, s I,, J , E J,, 11, u Jal < KJT) such that tp*({af: i < a}, U I, u U J , ) does not fork over U (I, u J,) . We can also aasume Av(I,, U I,) does not fork over U I,, Av(I,, U I,) is stationary, and similarly for J,, J, (see 111, 4.17(4)) and for every E E U (I, u J , ) , tp(E, U (I, u J,) is stationary.

Now we prove by induction on i that

91 = tP(.fJ u (In u J a ) ” b,:j < i})

tP(% u (I1 u J , ) u {a,:j < i}).

This holds by 111, 4.22(2) as p1 is Ff-isolated. So clearly

tp*({.,: e: < 4 u ( I , u J, ) ) I- t p* (h : i < 4, u (I1 u J, ) )

hence tp*(E, U (I, U J, ) ) k tp(8, U (I, U J, ) ) . So we could choose a < w, so we could choose I,, J, such that 11, u Jal < K(T) , and remember tp(E, I, u J,) I- tp(E, I, u J,) . So tp(Z, I, u J , ) is F&)- isolated, and it extends Av(I,, U I,), contradiction to the assumption on I,, J,.

So dim(I,, N) = A, dim(J,, N) = p, and (3) fails. If we choose A > III + IJI, we can assume I , J E [MI, hence getting a contradiction to (2). So we proved that (2), or (3) implies (4).

(4) =- (1) Suppose K 2 K,(T), M is F:-saturated, I , , J , E M are equivalent to I, J respectively; we shall prove dim(I,, N) s dim(J,, N). Suppose not, and w.1.o.g. I,, J, are maximal indiscernible sets in N. By 111, 3.9 !Ill = dim(I,,M) > dim(J,, M) = lJ1l, and of course lJll 2 KJT). By 111, 3.5 there is I, E I,, II,l 5 lJll + K,(T) = lJ1l, such that I, - I, is indiscernible over J,. By 111, 4.17(4) there is J , E J , , lJ81 < K,(T), Av(J,, UJ,) does not fork over U J,,

244 MORE ON TYPES AND SATURA!CED MODELS [a. v, 5 2

Av(J,, u J,) is stationary; and similarly I, E I , - I,, and we can assume tp,(I,, u J,) does not fork over u J,. Then tp*(I, - 'I, - 12, 1, U J1) does not fork over I, U J,, hence tp*( J, - J,, (I, - I , ) u J,) does not fork over I, U J,. So I ; = I , - I, - I,, and J; = J, - 5, scltisfies the hypothesis of (4) and

Now we define by induction on p < I J, I + sequences E,, E IblI and setsI: E I ; such that J: = J; u {Zt: i < p}isindiscernible, II:l I I Jll and I ; - I;, JZ satisfy the hypothesis of (4). For p = 0, p limit this is clear. For p + 1, by (4) there is a Fk,,,-isolated type q EP( J; u (I; - I:)) which extend Av(J:, u JZ), so some E,, E M realizes it, and for some

= 11, 1I;+11 I; IJ11, I ; E I:+, and tp(E,,, (I:+, - I:) u J;) k tp(E,,, ( I ; - I:) U J*). Clearly the induction hypothesis are satisfied, and for /3 = I Jll + , JZ contradicts dim( J,, M) = I Jll.

= 1111, IJ;I = I Jll.

LEMMA 2.4 : (1) If p , q are stationary, not orthogonal, and q is regular then p G 8 q (hence p <,a).

(2) If p , q are stationary and orthogonal then p $, q (hence p & q) .

Proof. (1) Immediate by 1.12. (2) Immediate, by IV, 4.10, and 1.4, 1.10.

THEOREM 2.6: Let I be a (non-trivial) indiscernible set, I I I = p > A 2 K,(T), M a Ft-prime model over I , J E M a (non-trivial) injnite indiscernible set. 'Then dim(J, M ) = A iff dim(J, M ) # p iff I $, J.

Proof; As bl is R-saturated, dim( J, M) 2 A. If J' E M is equivalent to J , IJ'I > p, then for some J " E J', IJ"I I p, J' - J " is indiscernible over (J I, and its cardinality is, of course >p, contradiction to IV, 4.9. Hence A 5 dim(J, bl) I p, dim(1, M ) = p.

If I I, J then dim(J, M) 2 dim(1, M ) = p hence dim(J, M) = p. If dim(J, M) = p clearly dim(J, M ) # A. To complete the proof we assume I $ w J and prove dim(J, f l ) = A. By 2.3(3) there is a Fi- saturated model N, and I,, J, E N equivalent to I, J respectively, dim(I,, N) = p, dim(J,, N) = A. Choose J p E J , lJal < K(T), such that Av(J, u Ja) is stationary, Av(J, u J) does not fork over u J, (by 111, 4.17(4)). By IV, 3.12(3) M is Ft-prime over I u J,. W.1.o.g. we can find an elementary mapping f of I u J, into N, such that f (I) is equivalent to I,, and f ( J a ) c J,. By the primeness we can extend f to an elementary mapping f ' of M into N. So A I dim(J, M) I dimlf'(J),f'(M)] I dim(J,, N) = A. So dim(J, M) = A, and we finish.

OH. v, 5 21 DIMENSIONS AND ORDER8 BETWEEN SETS 245

THEOREM 2.6: 8uppoee I is a non-trivial indiscernible set, IIl z p > A z +(T), M &I q+me over I, and J c M an infinite ind&Imible set.

Then I is orthogonal to J iff Av(J, M) i s F,”+-isolated iff Av(J, M) i s q-h&t?d fOr 8- x 5 p.

Proof. We can w u m e I JI = A, I, E I , II,l = A, I - I, is indiscernible over I , U J, I, U J c N c M, N is Fi-primary over I, u J, and is ~-conetrudible over I u J, and M is 5-constructible over I u N (see IV, 3.17 and IV, 3.9). Clearly I - I, is indiscernible over N.

If I, J ctre orthogonal then by 1.4 Av( J, N) I- Av( J, N u I) and by IV, 4.10 Av(J, N u I) t- Av(J, M) and Av(J, J u I , ) F Av(J, N). As (J( + (I,( = A, i t follows that Av(J, N ) is F;+-isolated, hence Av(J, M) is F;+-isolated, hence it is Fi-isolated for some x < p.

Now wume Av(J, M) is q-isolated, x s p, and we prove I, J am orthogonal, and so we shall finish. Let A c M, IAI < x 5 p be such that Av( J, A) I- Av( J, bl). By IV, 3.17 and IV, 3.9 there is a model N, A E IN1 E 1611, N Fi-prime over I, for some I, c 1(1111 < p) and Fi-constructible over I, bl is R-constructible over I u N, IN n I1 2 No, IN n JI 2 X,. So Av(I, N) is realized in bl (by any c‘ E I - I , ) and Av( J, N) I- Av( J, H). Hence Av( J , N), Av(I, N) are weakly orthogonal, hence by 1.2(3) they are orthogonal.

CONCLUSION 2.7 : If 1, J are (non-trivial, inJinite) indiscernible sets Over A, I, J are not orthogonal, t k n tp,(J, A), tp,(I, A) are not weakly orthogonal.

Remark. See 1.5(2) for a converse.

Proof. Suppose tp*(I, A), tp*( J, A) are weakly orthogonal. As I, J are infinite, we can wume IIl > I JI > K = ~ ~ ( 2 1 ) + IAI +, I indiscernible over A U J, and J indiscernible over A u I. By Ax(V.2) for Fb, for any c’ E J (letting J‘ = J - { E } ) tp(E, A u J’) F tp(E, A u J’ u I). So if N is F:-primary model over A u J’ u I, then Av(J’, A u J’) = tp(E, A U J’) I- Av(J’, N). So Av(J’, N) is q-isolated, ( A = I JI) hence by the previous theorem I, J me orthogonal.

CONCLUSION2.8: I f p E P ( A ) , B E A,porthogonaltoBortotp,(I,A). I an (infinite) hdi8Cernibk. set over A, tp,(I, A ) does not fork Over B, then p is ort7cogOnal to AV(I, U I).

246 MORE ON TYPES AND SATURATED MODELS [OH. v, 5 2

Proof. We can assume A = "1, bl F&,,,-Saturated. Let J be an indiscernible set baaed on p. By 1.4 tp*(I, M), tp,(J, M) are weakly orthogonal, hence by the previous conclusion p, Av(l, Ul) are orthogonal.

THEOREM 2.9: Let I be a (non-trivial) indiscernible set, IIl = p > A 2 K J T ) , M a q-prime &l over U I , A c M, IAI < A, p E P ( A ) , A1 the $rst cardinal 2 A in which T is stable, p > hl, and

J = (6 E M : 6 realizee p}.

Then IJI 2 p i f I JI > A1 iff p has an d o n q such t7rat Av(I, u I ) It0 Q.

Proof. If p has an extension q, Av(I, U I ) 5, q; then let q E Sm(A u B) IBI < K(T), so we can assume B c M , (of course, q is stationary). So p = dim(I, M) 5 dim(q, M) 5 IJI. If I JI 2 p then I JI > Al, and if I JI > A, then by I, 2.8 there is an indiscernible J, s J , I J,I = A:. So dim(J,, M) > A1 2 A, hence by 2.5 I 5, J, so letting q = Av(J,, A u u J,), Av(I, U I) I, q and q extends p (of course J, is not trivial, q not algebraic).

Notation. (for Theorem 2.10). (1) Let IN, be a non-trivial indiscernible set, IIKol = KO. For every A let IKo E I,, I , indiscernible, lIAl = A, and let M(I, ) be a F&,- prime model over u I,.

(2) Let x 2 K J T ) , T stable in x. THEOREM 2.10: The following 8tatenwnt.3 on T are equivalent (remember

(1) For every A 2 K ~ ( T ) , T has nzaximally-~-saturated rnodels of

(2) As (l), but the models are in addition A-universal but not A+-

T i8 Stable).

arbitrarily high cardinality.

universal. (3),,, Not every F & ( ~ ) - 8 d U T d e d model i8 i8onzOrphiC to M(Ij,) for 8Ome h. (4), Not every two F&(~)-8aturated. d e l e of T of cardinulity x+ are

( 6 ) For some (non-trivial infinite) indiscernible sets I , J, I $, J. (6) There are (non-trivial infinite) indiscernible sets I , J such that J is

(7) There are (non-trivial infinite) indiscernible sets I , J , I $, J, J is

(A) I is minimal, or (B) J * I, J s, J * for some minimal J*, and I , J are orthogonal.

iscnniwph&%

minimal and I is orthogonal to J.

based on 0 such thd

CH. v, 8 21 DIMENSIONS AND ORDERS BETWEEN SETS 247

DEFINITION 2.2: If T fails to satisfy the above conditions, i t is called uni-dimensional.

Proof of 2.10. Trivially (2) =- (1); (6) + (6) (by 2.4(2)). (6) * (1) By 2.3. (1) - (a), By 111, 3.12 T hae a saturated model of cardinality x + ;

and by (1) and w T is stable in x + , T has a maximally-~ro,-satturated model of cardinality x+ . As A L K,( T) they are not isomorphic.

(4), =- (3)rHO Clearly p 5 II,l I llM(I,,)\l; and if p I x, then as T is stable in x, IlH(I,)I( s x. So IIM(I,)II = x+ iff p = x + ; hence one of the F&,,,-saturated models of T of cardinality x+ , is not isomorphic to any

(3)IH0 3 (6) Let J be a countable (non-trivial) minimal indiscernible set (exists by 1.18( 7)). Let M be a F;=,,,-saturated model not isomorphic to any M(I,) . We can wsume J E M and let J , E H be a maximal indiscernible set in M extending J. By 1.16 and 1.18(4) for every E E I M I tp(8, u J, ) is l?&,-isolated. If I* E 1M1 is a (non-trivial) infinite indiscernible set over lJ J,, then we can assume that for some J , E J,, I J21, I J , - J21

are orthogonal, so (6) holds. So suppose there is no such I*, then by IV, 4.14 (the characterization theorem) M is FEr,,,-prime over lJ J,.

Let p = I J , I, then in M(1,) we can find a maximal indiscernible set J,, such that for distinct a!, . . . ,a; E J, , 1 = 1, 2, tp(7iin. - . 7i;) = tp(sifn. . .-a:). As before we can assume M(1,) is F&,-prime over J,. If I Jal = p, there is an elementary mapping f from u J, onto u J2. By IV, 4.14 we can extend it to an isomorphism from M onto M J I ) , contradicting the choice of M. If I J21 # p then as dim ( J 2 , M) = I J,I, by 2.6 J, Sw I,, hence J, , I , are orthogonal (by 1.12) so (6) holds.

(6) 5 (7) Let I, J be as in (6), and J, be a non-trivial infinite indiscernible set, based on9 (exists by 111,1.12). If J $,J1 then taking J, J , for I, J shows that (7)(A) holds. So assume J sur J,. If J, is orthogonal to I then I $, J , so taking I, J,, J for I, J, J* shows that (7)(B) holds. ( J , sw J holds as they are not orthogonal, and J is minimal hence regular). But J, .is orthogonal to I by the following claim :

W,).

2 KO, I J2I < Kr(T) + HI, tP*(I*, U J 2 ) t. fp*(l*, J 1 ) SO by 2.7 I*, J ,

CLAIM 2.11: If J sW J , , I is orthogonal to J then I is orthogonal to J , .

Proof of 2.11. Let M be A-saturated, IlMll = A > K ~ ( T ) (by stability.) and w.1.o.g. J,, J, I be countable and cM. Define J* so that I J*l = A+,


J* U J is indiacernible and J* is indiscernible over 1M1; and let N be E-prime over IMI u U J*. Clearly dim(J, N ) 2 IJ*I = A + , hence dim( J,, N) 2 A + > A, so some 2 E N realizes Av( J, , 1Ml). On the other hand, as I is orthogonal to J , Av(I, IMI) F Av(I, 1M1 u U J*) and by IV, 4.10(2) Av(I, lMl u U J * ) FAv(I, IN]) , hence Av(I, ]MI) k Av(I, [MI U a), so by 1.2(3) J, , I am orthogonal.

Continuation of the proof. ( 7 ) - ( 2 ) Let I, J be as in (7) and if poaaible, as in (7)(A) ; and ,u > h 2 K,(T). We can assume J, I are indiscernible over U I, U J reap. and 111 = p, I JI = A. Let N be a F:-primary model over U I u U J ; so by 2.3 we know N is maximally FZ-saturated, and of come lldlll 2 111 = p . As Af is E-saturated, it is A-saturated, hence A-universal.

Choose an indiscernible set J 1 over U I , J G J1, I JII = A + , and chooae If E I, 11'1 = A + . Suppoae f is an elementary mapping from U I' u U J1 into N; we try to get a contradiction, and thus show that N is not A+-universrtl. Clearly dim(f(Il), N)) is > A thus aa in 2.6 I s , f ( I 1 ) ; and similmly I 5, f ( J 1 ) . If J* 5, J I , J*, J* minimal then for some minimal J** E IMI, f ( J 1 ) 5, J**, hence I 2, J**. As dim( J , H) = A, dim( J**, M) = p, clearly J** sW J , so we could have started with J , J** instead of J, I, and then (7)(A) holds, so I is minimal.

If I is minimal, thenf(ll) is minimal too, hence by 2.4, 1.18(4) f(P) I, I, thus (as I 5, f ( J 1 ) and by 2.1(2))f(11) I, f ( J 1 ) hence I' I, J 1 contradiction.

As we proved (6) =- (6) =$ (1) =- (4), * (3),,0 =- (6) =- (7) - (2) =- ( l ) , we finish.

PROBLEM 2.1: Is there a uni-dimensional, stable, but not superstable theory? (see Exercise 6.4).

QUESTION 2.2: In which places in this section can we replace K J T ) by K( T) ? Or should we find a natural K"( T), K( T) I K"( T) 5 KJ T) for those theorems ?

PROBLEM 2.3: (1) For every stable IT and A < K J T ) does T have a A-saturated model which is not A + -universal ?

(2) For every atable theory T with ID(T)I > No does T have a model which is not N,-universal, or at least not I TI -universal ?

EXERCISE 2.4: (1) Prove (1) above when A 2 KJT).

UH. v, $31 WEIGHTED DIMENSIONS AND SUPERSTABILITY 249

(2) Show that if K,( T ) < I TI then T has a model which is not I TI- universal.

EXERCISE 2.5: Suppose p 5 , q, r I, q, and show that p , r axe not orthogonal, except when they are algebraic.

V.3. Weighted dimensions and superstability

THEOREM 3.1: Suppose {a, 6} is independent over A and stp(C’, A) is regular. Then it is impossible that both tp@, A u E), and tp(6, A u 5 ) fork over A.

Proof. W.1.o.g. we work in Beq, A = acl A. Let M be X-saturated, IAl + 12’1 < X = llMll,A UauEuE E M,andJ = {a:aEM,$realizes stp(C,A)}. By 11, 2.2, 13 there is J , c J , lJll < 12’1 such that tp(& A U U J ) is (A u u J,)-definable (see Definition 11, 2.1(4)). Clearly if tp,(Ji, A U 7i) = tp,(J,, A u a), J; E J (with suitable indexing) then there is an automorphism f of N, f A = the identity, f (a) = a, f maps Jl onto J ; ; hence we can replace J , by J; . So we can assume that tp(J,,A u Z U 6) does not fork over A ua. Hence tp(6, A u u U J , ) does not fork over A u a (by Ax(V1) for FI,) so it does not fork over A (by 111, 4.4). Similarly we can define Ja E J so that tp(6, A u U J) is (A u U J,)-definable, and

t p * ( J , , A u a u 6 u U J , )

does not fork over A U 6. Hence (by Ax(VI1) for FI,))

tp,(6 u J,, A u a u u J , )

does not fork over A, so tp,(J2, A u J , ) does not fork over A, in particular J , n Ja = 0. Choose maximal I, c J , independent over A. So I, u I , is independent over A, I, n I , = 0.

We can find & E M , i? realizes stp(c,A), but

tp(E’, A u U J , u U J , u ti u 6)

does not fork over A. So (3, a’} is independent over A. But {a, E} is not independent over A, hence tp(anE, A) # tp(a-E’, A). As Z, E’ e J, and tp(8, A u U J ) is (A u U J,)-definable, it must be that tp(E, A u J , ) # tp(E‘, A u J1). As stp(Z, A) = stp(E’, A ) is s ta t ionq tp(8, A u J , ) forks over A. So E depends on I , over A (by 1.9). Similarly E depends on I, over A . Contradiction by 1.14.

250 MORE ON TYPES AND SATURATED MODELS [CH. V, $ 3

LEMMA 3.2: Suppose M is Fi-saturated, A r K~(T) , tp,(Al, [MI u A,) doesnotforkover lM[ andN,ie Fiqrimeover [MI u A, (1 = 1,2). Then:

(1) (i) tp*([N1[, INaI) does not fork over [MI. (ii) stp*(IN,I, IMl u A,) btp*(lN,I, (MI u A1 u IN,[).

(2) If tp(6, )MI u A,) ie Fi-ieokted then: (i) tp(6,l MI u A,) does not fork: over I MI.

(ii) stp(6, I MI u A,) k tp(6, [ MI u A, u A,).

Proof. (1) We can in the lemma replace [N,1 by sets B, Fg-constructible over [MI U A,, and then prove i t by induction on the length of the constructions. So it suffices to prove (2).

(2) If this is true for every finite subset of A,, it is true for A,; so we can msume A, = 8. Similarly we can assume IA,I < A.

AS Kr(T) S A, there is B G ]MI, JBI < A such that tp,(A, u 8, ]MI) does not fork over B, tp,(A, u 8, B) is stationary and stp(6, B u A,) k stp(6, [MI u A,). As tp(8, 1M1) is stationary by 111, 2.15(2) and tp(8, [ M I U A,) does not fork over [ M [ (by assumption) tp(8, 1 M 1 u A,) is stationary. If (ii) fails, then there is Si E (M 1 such that stp(6, B u A,) br tp(6, B U A, u Ti u 8) and w.1.o.g. Ti E B. Now as M is Fi-saturated there is 8' E I MI, stp(8', B) 9 stp(8, B), so

atp(a', B u A,) E stp(8, B u A,)

(remember tp,(A,, lM[ u 8) does not fork over 1M1, hence over B (by 111, 4.4)). Hence stp@, B u A,) br tp(6, B u A, u 8'), contradicting stp(6, B u A,) b stp(& [ M 1 u A,).

Remark. Really 3.2(2)(ii) implies tp(6, M U Al) I- tp(6,M U A, U As), as tp,(Al U 6, M), tp,(A,, M) are stationary.

CLAIM 3.3: If M ie Fi-euturated, A r K,(T) regular, tp(ii, [MI) regular, N is FR-primary over [M 1 u Ti, and 6 E IN[ , 6 $ 1 M 1 then N is F",primary over IMI u 6.

Proof. Choose B G [M 1, I BI < A so that tp(5-6, M) does not fork over B, tp(~i-6, B) is stationary, and stp(5, B u 5) k stp(6, (MI u a) (such B exists by III,3.2,111,2.9(1) and IV, 3.2). By III,6.14 tp(6, B u a) I- tp(6, lMl u a)(B, IM[, a correspond to B, C, A.) If stp(z, B u 6) br tp(Si, 1 M 1 u 6) then some 3' E N realizes stp(Si, B u 6), but tp(zn5, I M I ) # tp(a'- 6, m). As tp(anb, B) = tp(n '6, B), tp(6, B u a') forks

OH. v, $31 WEIQHTED DIMENSIONS AND SUPERSTABILITY 251

over B, so &#M, hence by 1.9 tp(&, wl) = tp(a, WI); tp(an&,B) =

a contradiction.

and IV, 3.6.

tp(a’^b,B), tp(a^b,M) # tp(n’^ b,M), tp(6,B u a) I- tp(6,M u a),

Hence tp(&, wI U 6) is Fi-isolated, and the rest is easy by IV, 3.16

THEOREM 3.4: Suppose A G B, f an elementary mzpping whse domain is €3, f A is the identity, and p cSm(B) is stationary, and stp,(B, A ) E stp*(f (B), A), stp*(f (B), B) does not fork over A . Then p is orthogonal to A iff p is Orthogonal to f ( p ) .

Proof. Define elementary mappings fi(i < IT1 +) so that stp,(B, A) EE stp,(f,(B), A), and tp*(f,(B), u,<f f ,(B)) does not fork over A, and we let fo = the identity over B, fi = f.

Suppose p is orthogonal to f ( p ) , so clearly the f i ( p ) ( i < I T ] + ) are pairwise orthogonal. So if p is not orthogonal to A, for some a, p is not orthQgona1 to stp(& A), hence stp(a, A) is not orthogonal to every ff(p), contradicting 1.7. So we have proved: if p is orthogonal to f (p) then p is orthogonal to A.

Now suppose p is orthogonal to A. W.1.o.g. there is I E B, p = Av(I, B), I indiscernible over A. So p is orthogonal to tp,(f(l), A), hence by 2.8 p is orthogonal to Av(f(I), u f ( I ) ) , which is parallel to f (PI = Av(f (0,f (B)).

DEFINITION 3.1: Then pair (p, tp) is called regular, if tp = p(E, a) E p and (clearly the definition does not depend on M)

(*I Let p be over [MI , M Fg-saturated ( K = K ~ ( T)), tp(Si, 1M1) a stationarization ofp, N F;-primary over u si. Then for every 6~ INI, 6 4 IN], such that 6 realizes tp;

tp(6, [MI) does not fork over E .

Remark. Clearly if ( p , tp) is regular, then p is regular. Also if tp is weakly minimal, p ~ p , p is minimal then (p,p) is regular.

CLAIM 3.6: Suppose M, N are Fgr,,,-saturated, IMI c IN] , si E IN] , si 4 [MI, u = R[tp(Z, \MI) , L, m] c m and u = R[tp(iE, E ) , L, 001 where p@, 5 ) E tp(Z, INI). Suppose also (i) for every 6 E INI, 8 # /HI, u I R[tp(6, /MI) , L, 001, or (ii) for every 6 E tp(N, a), 6 # !MI;

a I R[tp(8, pq), L, ml. Then p is regular and even (tp(t3, ]MI) , tp(3, E)) is regular.

252 MORE ON TYPES AND SATURATED MODEL8 [CH. v, 8 3

Proof. Choose A s ]MI, IAl < K(T) so that tp(ii, IMI) does not fork over.A, tp(a, A) is stationary and E E A and a = R[tp(ii, A), L, a33 (A exists by 111, 3.2,111, 2.9 and 11, 1.2). If q E S” (IMI) is not algebraic (where rn = I@)), tp(& A) s q, and q # tp(a, M ) then p forks over A, hence by 111, 1.2(4) R(q, L, oo) < a, so N omits q by hypothesis. Thus we can conclude, by 1.9, that tp(si, 1M1) is regular. Similarly (tp(ii, IMI), ‘p) is regular.

DEFINITION 3.2: The weight of p, w(p) ( p stationary or complete) is the minimal cardinal p 2 0 (possibly finite) such that for some regular h 2 K ~ ( T ) , some Flf-saturated model M , and a stationarization tp(8, IMI) of p , there is an independent set {af: i < p} over IN[, sif E N where N is q-primary over 1M1 u si, such that N is Flf-primary over 1M1 v {a1: i < p} and tp(sif, [MI) is regular. If there are no such B, ii, iil, N w(p) is said to be a3 or not defined. We let w(a, A) = w(tp(3, A)).

LEMMA 3.6: (1) If w(p) is defined, (i.e., <a), i t is < K ( T ) (and even < K ~ ~ ( ! I ’ ) ) and if 8, M, af are in the definition, tp($, IMI v a) forks over IN\.

(2) If h 2 KJT) is regular, ikf is Flf-saturated, tp(8, IMI) does not fork over A, A E 1M1, N is Flf-prime over V 8, p = w(a, A) < a3 then we can$nd iif (i < p) a8 in Definition 3.2.

(3) If M is Fg-saturated, h = K,(T), tp(iZl, M ) regular, {a,: i < p} independent over M, and tp@, IN1 v {af: i < p}), tp,({a,: i < p}, lMl u a) are Q-isolated then w(8, ]MI) = p.

Proof. (1) By 111, 4.21 it suffices to prove the second part. If tp(ai, WI U a) does not fork over M , N omits it by IV, 4.10(2).

(2) Left to the reader as an exercise. (3) Easy.

LEMMA 3.7: When T is superstable, h 2 KO regular, M is Fg-saturated, ii a sequence then we can Jind n < o, a, (1 < n), M , (1 I n) such that No = M , Mz+l is F,”-prinzary over lMzl V a,, tp(Z,, lMzl) is regular; and M,, is Flf-primary over 1MI V 3.

Proof. We define by induction on 1 < o, 3, and M,, M , being Fg- saturated.

Let M o = M, and if M , is defined let N , be K-primary over lM,l v a. If a E M,, we finish, otherwise among a, E lN,l - lM,l there is one with minimal CL = R[tp(Ti,, IM,l), L, a]. Thus by 3.6 tp(a,, lM,l)

CH. v, 5 31 WEIGHTED DIMENSIONS AND SUPERSTABILITY 253

is regular. By 1.10(2) tp(Ti,, lM,l u a) forks over IM,1, hence (by Ax(V1) for FL) tp(Ti, IM,I u Ti,) forks over IM,I. Let M I + , be Fa-primary over 1M,1 u Ti,, F,Q-constructible over IM,I u a (exists by IV, 3.17). So clearly tp(zi, IMz+ll) forks over IM,I.

Clearly M , c M , E M , c + - - As T is superstable, K ( T ) = X, so for some n < w , M , + , is not

defined, so Ti E M,,. As M , + , is Fa-constructible over 1M,( u Ti, M , is F:-primary over WI U a (by IV, 1.3(2)).

LEMMA 3.8: &uppose that A 2 K,(T) is regular, M , (i s a) is FZ- saturated, Mf+l is Fg-prime over lMrl U Tif ( i < a), Ma is Fg-prime over Ui<aMi for 6 ( s a limit) and tp(Ti,, / M i l ) is regular. Then for every zi E Ma, we can j nd a set {6,: i < a} c IMaI independent over IN,], tp(& IMol) is regular, tp[G lMol u {gf: i < p}] is Fg-isolated and p I a. I?& f d p < K ( T ) .

Proof. We first prove that we can assume that for every i, tp(Ti,, lM,l) either is Orthogonal to 1M,I or does not fork over IMol. For each i , choose B c /Mi l , IBI < K ( T ) , and A c IMol, IAl < K,(T) such that: tp(Ti,, ]Mi l ) does not fork over B, tp,(B u Z,, IM,l) does not fork over A , tp(ai,B), tp,(B U ai,A) are stationary, and w.1.o.g. A c B: and if tp(Tif, / M i l ) is not orthogonal to lMol, it is not orthogonal to A. As M , is F,Q-saturated, we can choose B‘ G lMol, 7i; E lMol tp,(B u Z,, A) = tp,(B’ U a:, A). By 3.4 tp(Ti,, B) is orthogonal to A iff it is orthogonal to tp(Ti;, B’). So if tp(Ti,, IM,l) is not orthogonal to lMol, it is not orthogonal to A, hence not orthogonal to tp(Ti;, I?’). As tp(Ti,, B) is regular, also tp(Tii, B’) is regular, hence for for some Ti: E M i + , , tp@, lMil) extends tp(Tii, B‘) and it does not fork over B‘ (by 1.12). By Claim 3.3 M i + , is Fi-primary over ]Mil u Ti:, and clearly tp(si:, IM,]) does not fork over lMol. So we can replace Ti, by Ti:. So we can assume that tp(Z,, / M i l ) is either orthogonal to lM,l or does not fork over 1M01. So [ M a [ is lMol u {E,: i < p} where for each i one of the following cases occurs :

(i) tp(E,, A,) is Fa-isolated, where A, = lMol U {E,: j < i}. (ii) tp(E,, A,) is regular, orthogonal to lMol and stationary; more-

over, it does not fork over some B, c A,, lBil < K,(T), where tp(Et, Bi) is stationary.

(iii) tp(E,, A,) is regular and does not fork over IMol. As in IV, 1.4(2), 4.4-7 we can assume that for some y -c K,(T) , y < p

~ E A , , . By 111, 3.2 and 111, 2.9, we can choose B C W,J, IB( -= K,(T)

254 MORE ON TYPES AND SATURATED MODELS [CH. v , 8 3

suchthattp*[{Ej:j < y}, IMoI]doesnotforkoverB,andtp,[{Ej:j < y} , B] is stationary and if case (ii) occurs for E,, tp(E,, B u {Z,: j < i}) is stationary. Let C = U {Z,: tp(Z,, A,) does not fork over lMol, and is regular}. We now prove that for i < y , tp(E,, A, u C) is F;f-isolated. If cue (iii) holds, E, E C E A, u C, so this is trivial. If case (i) holds, then tp,[C - A,, A,] does not fork over IMol, so our conclusion follows by IV, 4.10(2). If case (ii) holds, as tp(Zf, A,) isorthogonal to liKol, also tp(Ef, B u {Zj: j < a}) is orthogonal to B. As tp*[{Zj: j < i}, IMol] does not fork over B, for every 6 E lMol tp(6, B u { E j : j < i }) does not fork over B, hence it is orthogonal to tp(Ef, B u { E j : j < i}); but the latter is s h t i o n q , so they are weakly orthogonal by 1.2(2). Hence

tp(Z,, B U {Z,:j < i}) k tp(E,, B u {Ej: j < i} u 6).

As 6 E lMol was arbitrary,

tp(Zf, B U { E j : j < i)} I- tp(Ef, lMol u {Ej: j < i}) = tp(E,, A,).

As i < y < K,(T), IBI < K J T ) I A, tp(E,, A,) is F:-isol&d hence q-isolated; so u before tp(Z,, A, u C) is q-isolated. So we finish the three cases hence A, is R-constructible over lMol u C, so by IV, 3.2 tp(8, lMol u C) is F!-isolated so we finish.

THEOREM 3.9: (1) If T is euperstable, for every a, A w(a, A) is well- d e j i d and jinite.

(2) Assumi?&g the conditions of Lemma 3.8, w(8, IMol) exist8 and is

4,el 14. Procf. (1) By 3.7 i t suffices to prove part (2).

Before proceeding to (2) we make two observations:

Observation 3.9A: Suppose IAl + IBI + ICl < K , C c C*, F = FE for x = t , 8, a and K 2 K,(T) is regular. If

(i) tp,(A, C* U B) is F-isolated over C u B, (ii) tp*(B, C U A ) I- tp*(B, C*), (iii) tp,(B, C u A) is F-isolated,

then tp,(B, C* u A) is F-isolated over C u A.

Proof of 3.9A. The idea appears in the proof of 3.3, so we leave it to the reader.

Observation 3.9B: Suppose q E Sm( INI) is regular; I, J are sets of sequen-

OH. v, 0 31 WEIGHTED DIMENSIONS AND SUPERSTABILITY 255

ces realizing q, independent over [MI, 111 + I JI < K, M is FE-saturated where K 2 K(T) is regular, and A c liKl

u u I) forks over A then there is B c IMI, IBI < K such that tp,(U J, B U U I) t- tp*(U J, ]MI) .

If for every EEJ, tp(E,

Proof of 3.9B. Choose B c 1M1, 1B1 < K such that tp*(u (I u J), IMI) does not fork over B, and q B is stationary, and let I* E ]MI be a maximal independent set over B of sequences realizing q B. Clearly tp*(u J, B u u I) is orthogonal to q, hence to tp*(U I*, B) hence

Now for any ~ , E I + = ( 6 ~ 1 M ) : 6 realizes q B} tp(60n..-n6,,, B u U I*) is orthogonal to q, hence

(ii) t p ( u J, B u u I*) I- tp(U J, B u IJ I+). By claim 1.1 1

Combining (i), (ii), and (iii) we are finiahed.

6) tp*(U J , B u u 4 t tP*(U J , u u I*) .

(W tP(U J , B u u I + ) tP(U J , 1q.

Proof of 3.9(2). We can easily add to the conclusion of 3.8 that (letting IMol), tp(6,, IMol) are orthogonal or equal. So let

I = {6,: i < rs) = U,<y I,, the 1,'s are pairwise disjoint, and for every E I j , tp(E, 1611) = p j , and for j(1) # j(2), pj(,), p,,,) are orthogonal.

Let N be E-prime over 1M1 u {6,: i c a} such that it EN. Now choose N , E N Fg-prime over ]MI u it and choose a maximal

J = U,<7 J, c N,, independent over ]MI, suoh that eaoh sequence in J, realizes p,. Let N , c N , be Fg-prime over 1MI u lJ J.

By the maximality of J, no sequence in N , realizes over N , a complete type which is a stationarization of some p , ; hence by 1.12 tp(a, w21) is orthogonal to each p j . By 1.14, there are 11 E I j such that for every E E If, tp(Z, IN1 u J, u (I, - I,')) forks over 1611, hence is orthogonal to p, , and J, u (I, - If) is independent over M. Aa clearly IJI I 111 < K ( T ) , by observation 3.9B there is B c [HI, IBI < A such that letting I? = J j u ( I j - I ; ) , I* = (J f<7 I?:

As tp*(U I,, [MI) (j < y ) are pairwise orthogonal

Hence

As N is &-prime over lMl u U I , tp*(UI*, ldll U U I ) is &- isolated (w.1.o.g. over B U UI), and tp,(UI,B U u I * ) is F,"-isolated (as (B U UI*( < A ) ; so by Observation 3.9A (with B,M,UI ,UI*

= No)

(9 tP*(U IjS B u u I f* ) t- tP*(U It , IMI).

(W U,<Y tP*(U I,,IMl) t- tP*(U I , IMI,.

(Ci) tp*(U I , B u u I* ) I- tp*(U 1, IJq.


standing for C, C*,B,A, respectively) tp,(Ul, /MI U uI*) is Fi- isolated.

By IV, 3.6 as N is FT-prime over WI U UI, it is also F;-prime over 1M1 U u I U u I*, but as t p ( u I, u (J I*) is F~-isolated, N is Pi-prime over [MI u U I*, hence tp(a, !MI u u I*) is FR-isolated.

By 1.4(1) I* is independent over WI, so as N , is F,”-prime over 1M1 u J, by 3.2, I* - J is independent over (INal, !MI). As tp@, lN21) is orthogonal to each p j , it is orthogonal to tp*(U (I* - J), IN,[) , hence tp*(U (I* - J), lN,l U a) does not fork over IN,l, hence over IMl, hence tp(7i, 12111 u U I*) does not fork over IMI u U J. So by IV, 4.3, as tp@, 1M1 u I*) is l$-isolllted also tp@, IMl u U J ) is q- isolated.

On the other hand, as J E N,, N , c-prime over 1H1 u a, also tp,(J, U a) is F$-isolated, so by 3.6(3) w@, 1x1) is well-defined and equalto I J I s IIl = 1/31 s 1.1. LEMMA 3.10: Let p be etationary or complete.

(1) The type p is algebraic iff w(p) = 0. (2) If p is regular then w(p) = 1.

EXERCAYE 3.1: Prove that the converse of (2) fails. (Hint: Exeroise 4.11.)

Proof of 3.10. (1) Trivial by Definition 3.1. (2) Immediate by 3.3 and Definition 3.1.

LEMMA 3.11: (1) If {it, 6} ie independent over A then w@-6, A ) =

w(a, A ) + w(6, A ) (i.e., if the right side is de$ned, then 80 is the left side and they are eqd . )

Remurk. Notice that if {a, 6} is independent over A then w(6, A) = w(6, A u a).

Proqf. (1) We can msume A = IMl, Y is q-mturated, A = x,(T). Let N , be q-primary over [MI U and also over IMl u { E i : i < w(a, A)}, and N , be E-primary over 1M1 u 6 and also over

where tp(E,, M ) is regulax and {Et: i < w(8, A)}, {Et: w(3, A ) I; i < w(a, A ) + w(6, A)} are independent sets over A. By 3.2 tp,(lN,I, IN,l) does not fork over IMI, so { E t : i < w(a, A ) + w(6, A)} is independent

(2) w(a, A ) I; W(a-6, A ) s w(E, A ) + w(6, A u a).

lMl u {E*: w(a, A ) s i < w(a, A ) + w(6, A)},

OH. v, 8 31 WEIQHTED DIMENSIONS AND SUPERSTABILITY 257

over ]MI. By 3.2 INl] U IN2[ is Q-constructible over IM1 u B u 6, so tp*[{ci: i < w(Z, A ) + w(6, A)} , 1M1 u ii; U 63 is F,Q-isolated. In a similar way tp(Z-6, [MI U { E f : i < w(ii;, A ) + w(6, A)} is F!-isolated. So we finish.

(2) Left to the reader.

LEMMA 3.12: (1) If { E f : i < a} is independent over A , (and Ef 4 acl A) tp(Ef, A u a) fork8 mw A (for i < a) then 1.1 s w(a, A).

{El: i < a} independent over 1611, E, E IN1 - 1611 t 7 m 1.1 5 w(a, A). (2) If N i 8 CF'??W over IMI U a, M it? F;-8durded, h = K ~ ( T ) ,

Proof. (1) We can clearly assume w(a, A) < m, i.e., defined and that A = 1611, M is R-saturated, A = ~~(2'). Let N be Fff-pfimctry over

U a, and {a,: i < p = w(a, A)} be aa in Definition 3.1. Now dehe by induction on i < p sets u, E a such that tp*[{Z,: j E a{}, 1611 u {aj: j < i}] does not fork over 1611, and la - ufl s li l,j < i u, c u,.

narningletitbe{j:j < Iufl}.Ifforevery5 < Iufl,tp[Zc, ]MI u{E,:j < t} u {a,: j I i}] does not fork over IMI u {a,: j < i}, then by Ax(VI1) for F', tp,[{E,: j < luil}, )MI u {a,: j I i}] does not fork over IN) u {a,: j <: i}, hence by 111, 4.4 and the induction hypothesis it does not forkover1611.Sowecanletu,,, = uf.Sosupposetp[Zc,INIu{E,:j < 5) u {B,: j I i)] forks over 1611 u {aj: j < i}, and this is the first such 5. For every 6 E u {E,: 5 < j < lufl}, {Ec, 6} is independent over 1611 u {Ef : i < 5) u {a,: j < i } (because { E , : j < a} is independent over 1M1 and the induction hypothesis). Now tp(Zf, U {Z,: j < t } u {Z,: j < i}) is regular (it does not fork over [MI U {a,: j < i} by the choice of 5, and over 1MI as {aj: j < p} is independent over IMI using 111, 4.4). By the choice of 5, tp[Zf, IMI U { E j : j < 5 ) U {a,: j < i} U Ec] forks over /MI u {E,: j < 5 ) u {a,: j < i}. Combining those three facts and remembering 3.1 we get that

For i = 0, u, = a, and for fi&t 6, = nf<d U,. If U, iS defined, by m-

tp*[{E,: 5 < j < lUf l } , (MI u {E,:j < 4) u {a,:j s i}]

does not fork over 1H1 U {E, : j < t} U {Z,: j < i}, hence it does not fork over 1611. So clearly tp,[{Z,:j < lufl,j # 5}, IMI u {7ij:j I i}] doesnot fork over 1M1. So let uf+l = uf - (5). For any j E u,, we get tp[E,, ldll u {ii;,:j < p}] does not fork over ]MI. So by 3.2, tp(E,, ]MI u a) does not fork over ]HI, contradiction. So u,, = 0 hence 1.1 s p.

(2) Follows by (1) and l.lO(2).

258 MORE ON TYPES AND SATURATED MODEL8 [OH. V, $ 3

CONCLUSION 3.13: Let I , A be given. ( 1 ) If J,, J , are muximal subsets of I independent over A , then I J,I s

2 {w(7i, A ) : 7i E J,}. ( 2 ) If p E Sm(A), J a maximal subset of (5: 5 E I , 5 realizes p} inde-

pendent over A then dim(p, A , I ) s I JI s w(p) dim@, A , I ) . (3) If p E Sm(A), w (p ) = 1 then dim(p, A , I ) h true.

Remark. See Definition 111, 4.5 for dim(p, A , I ) .

Proof. ( 1 ) C l w l y i f for some B E J,, w(a, A ) = ao, the conclusion is trivial, eo we can asaume w(Zi, A ) < ao. E'iret asaume J , ia finite and let

del J = { c l : l < n ) . Let F C = E , , " . . . - - cn-,, so by 3.11(1) k = w(F*,A) = 2 1 , n ~ ( F l , ~ ) . SO if k < IJ.1, by 3.12(1) for some FEJ., tp(c ,A U F*) does not fork over A, hence {Zi: i < n} u {Z} is independent over A , contradiction to the maximality of J,. Henoe

So we finish for finite J , and for the U t e J,, i f A s 1 J,I is regular, IJ,I > IJ,I, then for some finite J ; _c J,, for A many ZEJ,, tp(5, A u U J i ) forks over A , and we get a contradiction as above.

(2 ) is immediate from (1) and the definition o f dim, (3) follows from (2).

LEMMA 3.14: If 1M1 c IN], M , N are F&,,-durated then there is a cardinal p > 0 8w:h that for every d m a l set {a,: i < a} independent over lbl), of 8 ~ ' l M W e 8 at € A??, w(%, 161)) < 00, this e q ' l d h hQEds; p =

2 i c a ~ ( $ 9 1H1).

Proof. Through the regular types and is left to the reader.

THEOREM 3.16: ( 1 ) If T is muperstable, I independent over A , a 8q'&WK% t h for 80TPl.4 J E I , IJI 5 w(G, A ) , ( I - J ) u {Ti) i8 independent over A .

( 2 ) POT eulperetablt? T ,

dim(p, A u Zi, I ) s dim(p, A , I ) I; dim(p, A u a, I ) + w(Z, A) .

Remark. Work for stable T too.

Proof. (1 ) W e can assume w.1.o.g. that A = 1611, M is E-saturated A = K ~ ( T ) . Let I = {d : O < i < a), n = d' ; by 3.6(2) there is a family

OH. v, f 31 WEIGHTED DIMENSIONS AND SUPEBBTABILPTY 259

{Z,!,: k < w(Z,,A)} independent over A , tp(Z,!,,A) regular, and tp[Z,, 1M1 u {Z,k: k < w(2, A)}], tp,[{a:: k < w(a, A)} , A u @] are Fff- isolated. By 3.2 {Z,!,:O < i < a, k < w(2 , A ) } is independent over A. By 1.14 there is u c {i: 0 < i < a}, 1.1 s w(a0, A ) , such that {ak: i $ u, k < w(Z,A) } is independent over (A u {a:: i = 0 and k < w(Z, ,A)} ,A) . By 3.2 the results &re immediate.

Now we define upper and lower weights. (2) Left to the reader.

DEFINITION 3.3: Letp be stationary or complete. Define the upper weight p , upw(p), aa the supremum of the cardinals p such that there exists My N , I satisfying the following: (1) M is q-saturated, K = K ~ ( T). (2) p has a stationmization which is a complete type over 1x1

(3) N is (4) I is a set of sequences realizing non-algebraic types over 1x1 and

realized in N by a. @-primary model over 1x1 u a.

I is independent over 1M1. (5) Ill = p. Define U P W ( ~ ) = upw(8, A) = Upw(tp(tZ, A)).

DEFINITION 3.4: For p as above define the lower weight, low (p), as the maximal cardinal p such that there me M y N , I , satisfying: (1) M is E-saturated, K = K,(T). (2) tp@, 1M1) is a stationarization of p. (3) N is @-primary over [MI u 8. (4) I E IN1 is a set of sequences realizing regular types over 1H1, I

independent over ]MI, and is a maximal such set.

If in addition, q is regular and stationary define low&) to be the maximal p satisfying the above conditions plus B E 1 tp(c', lMl) is not orthogonal to q.

(6) III = p.

LEMMA 3.16: (1) In Dejnitiona 3.3 and 3.4, if th appropriate weigh4 is p then for any My N , I , satisfying (1)-(4), aho 111 = p.

(2) upw(p) = upw(r), low(p) = low@), low&) = low&) i f p = tp(a, A), r = stp(a, A), or p , q are parallel.

( 3 ) If w ( p ) is dejned, then upw(p) = w(p) = low(p) (so this alway8 blh for superstable T ) . Also low@) s upw(p) < K(T) .

(4) low&) = 0 iff p is orthogonal to q (q reguhr, of course).

260 MORE ON TYPES AND SATURATED MODIDLS [UH. v, 5 3

(6 ) If q, r are non-orthogonal, and regular then low&) = low,@). ( 6 ) If {q,: i < a} is a maximal family of stationary, regular, p a i r h e

( 7 ) upw(7in6, A ) 2 upw(7iy A) + upw(6, A u ii) when {ii, b} is inde-

(8) lowq(Gn6, A) s lowq(7i, A) + lowq(6, A u 8). (9) low(Gn6, A) 5 low(7iy A) + low(6, A u 3).

o r t h q d types not o r t h d to p thm Z,low,,(p) = low(p).-

pendent over A .

(10) If {a, 6) is independent over A, thm equality lrolds in (8) and (9). (1 1) lowq(8, A) s lowq(Gn6, A) and similarly for low and upw.

Proof. We leave to the reader (1)-(7) and (lo), (11). (8) W.1.o.g. A = 1611, M is F,O-saturated, K = KJT), q is over 1611.

Let N , be q-primary over IMI u ii, let 6' be such that tp(iZn6,1M1) = tp(an6', 1M1) and tp(6', N,) does not fork over 1M1 U 8. Let Na be F,O-primary over lNll u 6'; and let q,, Pa be the stationarization of q over IMI, lNll respectively (SO clearly q E q1 c qa). Let I , be B maximal set of sequences from INl] realizing qI which is independent over 1M1 when 1 = 1, and over lNll when 1 = 2. Clearly I , U I , is a set of sequences from IN21 realizing q1 which is independent over 1611. It is a maximal such set. For suppose E E "21, E realizes 42. By the maximality of I,, tp(c', lNll u l a ) forks over lNll hence over 1M1. By 1.11 letting J = {d: d EN,, a realizes q,} tp(i3, IMI u J u l a ) forks over IMI. But for every a E J , tp(& IMI u I , ) forks over IMI, hence by 1.9 tp@, 1M1 u I , u l a ) forks over 1M1. If E realizes Q, but not qa this holds trivially so I , U I , is indeed maximal. Now let N* E N , be F,O- primary over 1611 u u 6'; and I be a maximal set of sequences from N* realizing qr which is independent over w. Then by 1.14 111 = 10~,,@~6, A), and by 1.14, [ I ] I; ] I , U l a 1 = Il,l + IIal; and this proves our atmrtion.

CLAIM 3.16A : ( I ) In DeJinition 3.4 there is a maximum.

regular (and over M ) I is a maximal family satisfying:

- -

(9) Follows by (6) and (8).

(2) Assume M is Ff-saturated, K = c f ( K ) 2 K,(T), IAI < K , p is

(i) for each @ € I , tp(6,M) is regular not orthogonal to p , (ii) for each @E I , tp(& M U A ) fork over M,

(iii) I is independent over M.

(b) tp(U I, M U A ) is Ff-isolated, (c) if N is Ff-constructible over M U A Ff-primary over M U u I ,

Then (a) 14 = low,(fp*(A,M)),

then tp*(A,N) is orthogonal to p .

OH. v, 8 31 WEIUHTED DIMENSIONS AND SUPEBEITABILITY 261

( 3 ) Suppose for p ~ { l , 2 } , K , M , p , A,, I,, N, are as in ( 2 ) and {A, ,A,} is independent over M and N is Ff-primary over Nl U N,. Then K , M , p , A , U A,, I , U I,, N are as in (2).

Remark. We can replace p by a regular family 9 (see Definition 4.5:

Proof. Straightforward.

QUESTION 3.2: Is the independence aasumption necessary in 3.16(7) ? When equality holds ?

QUESTION 3.3: In Definition 3.3 is the mpremum obtained?

QUESTION 3.4; In Question 3.3, prove the answer is positive when upw(8,A) = KO.

PROBLEM 3.5: Try to generalize the results of this section on superstable theories to stable T for which K ~ ~ ( T ) = KO.

DEFINITION 3.5 : (A) Define lgw(p) just as low ( p ) but p is minimal and (4) for B E I , in the F,”-primary model over ]MI u B there is no

independent set with more than one element, and (6) I is a maximal set satisfying (a), of sequences 5 E IN I, 5 # 1M1. (B) Define ugw(p) just as lgw(p), omitting (6), and (3) N is c-primary over u I .

If there are no such M , N , I , ugw(p) = 00.

LEMMA 3.l7: (1) If tchd(T) = No, t h Igw(p) < .KO. ( 2 ) We get tlce aame IIl for eveq M, 8, N in De$nitim 3.6. ( 3 ) If {a, 9 ie independent over A then:

ugw(8-6, A) s ugw(B, A) + ugw(6, A). (4) If T ie -~&bZe t h ~ ( p ) = ugw(p) = Igw(p). (6) lgW(P) 5 Ugw(P).

Prmf. We leave it as an exerck to the reader.

QUESTION 3.6: Try to &pt the theorems and definitions of Seotion 2 (e.g., s ~ , sJ, Section 3 (e.g., w(p)) and also Section 1 to other kinds of F, mainly F& (replace, if necessary “ T superstable” by “ T totally transcendental” or add it) and also FL,, (for countable T ) . (See below.)

262 MORE ON TYPES AND SATURATED MODELS [OH. v, f 3

Remark. By III,4.2, for all stationarizationsp, ofp, *(ply A, , 2) is the same number.

Procf. (2) == (4) Let A , M, N , b be aa in (4). By 11,2.12(2) the properties of zi, (i c k) me expresaable by hitely many formulas, with parameters from El. Hence we oan assume a, E IMl; El E IMI as A c 1M1.

As q = tp(6, 1M1) is a stationariaation of p , P (q ,d , , 2) = a, (for

THEOREM 3.18: Suppose tp = tp(Z,E) E p E Bm(A); then the following & $ ~ ~ r t a 8di8fy: ( 1 ) * (2 ) 0 (3) 0 (4 ) 0 (6 ) and if IAI + IT1 5 so then &o (6 ) 0 (6); and if T is totdly tram- then also (7 ) o (6).

( 1 ) The pair ( p , t p ) is not regular. (2 ) There are tpl(E; 4), #,(Z; at) (for i < k) , B,(Z; go,. . .,&i), a,) (i <

1 5 k ) and$nite A, (i < k) , such that, letting a, = Rm(pl , A,, 2 ) (i < k ) for any stationarimtion p1 of p, the fol2owing conditions hold:

(i) tpl(Z; E l ) E p, n ( Z , E l ) I- ~ ( 3 ; El . (ii) at) I- tp(Z; a ) for i < k.

(iii) tpl(Z; a,) 1 Vi < I (3g0, . . , gn(i,)CAjsn(i) #i.,(gj; iZ,) A e:(z; go, . ., &(ip 411 v V l r t e k $i(Z; Zi().

( iv) P [ $ , ( Z ; Ti,), A,, 21 < a, when i < k. (v) For any i < I , and Eo, . . . ,

(3 ) The same aa (2) but n(i) = 0, #i = $, 8, = 8, A, = A for i < 1; and for 1 5 i < k , A, = AO, $, = $; and 8, i s a amcatenation of sequence8 satisfying t p ( f ; E ) V3 = E for i < k.

(4 ) I f A G !MI, tp(6, 1 ~ 1 ) i s a statiolucm'mtion of p, 1M1 u 6 c IN1 then for some 6' E INI, 6' 4 I M I , 6' ~a t i s$e~ tp(3; E ) but tp(6', 1M 1 ) i s not a stationarization of p.

(6) There are K > IAl + ITI, A c IMI, M a c-saturated model, N E-primary over IM) u 6, tp(6, )MI) is a stationarization of p, and 6' E INI, 6' $1 M I , 6' satisfies t p ( ~ ; z), but tp(6', ! M I ) i s not a stationarization of p.

(6) There are a countabie M and 6 such tluct tp(6, I M I ) is a idatimariza- tion of p, A r IMI, and for every N , for whid lMl u 6 c INl,'smne 61 E IN(, 25' $ lMl eatis* q ( Z ; a) but tp(6', ( M I ) i e nd a stationarizcction of P.

(7) like (6), for not nmeawri2y countable M .

OH. v, 5 31 WEIGHTED DIMENSIONS AND SUPERSTABILITY 263

j c k), hence by (2)(iv) +(Z; 4) E p when I I i < k; and clearly by (2)(i) cpl(Z; a,) E q. So by (2)(iii) necessarily for some i < I,

N C (380, - . - gn(d [A $&& a,) A el@; go, . . . , tin(,), al)]. jsn

Hence there are a,, . . . , an(,) E IN1 such that N t $@,; a,] (forj 5 n( i ) )

c a,, hence fI,(s; a,, . . . , d,,(,), a,) 4 p; but p = tp(6, 1611) so for some j I n ( i ) zj 4 [MI. But as N C $,[a,; a,], by (2)(ii) N C cp[Z,; a]. So a, satisfies cp(3, a), and as $,@, 4) E tp(Z,, [MI) , clearly Rm[tp(Zj, [MI) , A,, 21 I Rm[$l(Z; a,), A,, 21 < al hence tp(z,, IMI) is not a stationarization of p. So a, satisfies the requirements on ii', hence (4) holds.

- and N C 4(6; a,, . . . , a,,(,);%). By (2)(v)Rm [UZ; a,,. . . , a,), A,, 21

(4) + (6) Trivial. (6) + (3) Let {p, : i c I 2ITI} be a list of the stationarizations of p

inSm(lYI) andp, = tp(6, 1611). Choose $,@; a,) E tp(61, 1611) such that +,(E, Ti,) E po and we can assume that Go E u I u a (by 1.1 1) where I = {a E !MI: Ccp[a; Z]}, and $,(E, a,) I- cp(Z; a). SO by 1.11 tp(6', A u U I U 6) is F$-isolated, hence it forks over A u u I hence tp(6, A u U I u 61) forks over A u u I. So we can find Zo E A u U I c IMI and 8 such that C8[6, 6', a,] and 8(3; 61, a,) forks over A u U I.

Hence by 111, 4.2 we can find a finite d such that

m p , u {em ii', go)}, A , 21 < RW,, A , 2).

As by 1.11 tp(6,A u U I) = p,, we can replacep, by tp(6, A u U I). By replacing 8 ( E ; 61, ao) by 8(Z; ii', a,) A $ l ( ~ , al) where $ l ( E ; al) E

tP@, A u u I ) ,

~ y p , u {e(Z; 61, a o ) } ~ , 21 = ~ m [ { q ( ~ ; iq, e(Z; P, a)), A , 21

we can msume that Rm[8(Z; 6', a,), A , 21 < Rm(po, A, 2). Similarly by 11,2.12,3 we can assume that for every 6", a",R"[B(z; 6", &), A , 21 <

So clearly for every 6* realizing p , there is 6** satisfying $,@; a,) such that !+9[6*; 6**, a,]. Clearly for every i < there are a,, 7i' E [MI such that p,, $,(Z; a,), 8(Z; 9, a') satisfy the parallel assertion.

Hence for every 6* satisfying cp(Z; 13) 6* satisfies some formula $, -+h E n,<,p,, or for some i c a C (3g)[$,(g; 8,) A 8(6*; 8, a,)]. By a compactness argument we get i(n) < a, (n c I ) and - ,$ , , (P;@)E 0, <8 p , ( I s n c k) such that:

Rrn(p,, d, 2).

Q(Z, a) I- V $n(Z, a:) A V (~Y)[$O(@, zf(n)) A W ; 8, @?I* tSn<k n<t

264 MORE ON TYPES AND SATURATED MODELS [CH. V, 5 3

Clearly for each n, 1 s n < k, p u {#,,(Z; a:)) forks over A, hence there is rp1(3; El) ~p such that pl(Z; c,) t p(Z; 8) k d pl(Z, El) A Sn(Z; a:) forks over A. By 111, 4.2 there is a finite AO such tht for every 6" realizing p, and n < 1 P[stp(6", A) u {pl(%, 4) A #,,(f; a:)}, AO, 21 < ~ [ s t ~ ( 6 " , A), AO, 21. By 111, Definition 2.1 there is E(z, #, a*) E PE(A) such that for any 6" realizing p Rm[stp(6", A), do, 21 = P[E(3, p, a*), do, 21.

Now by notational changes we can assume B, = Gi, and

Now (3) is clear (by 1.11 it is easy to get iZi E U I u E). (3) 3 (2) Trivial. (1) 3 (6) Immediate by the definition of a regular pair. (4) 3 (6) (when !A( + ( T ( I KO) Immediate. (6) 3 (5) (when I A I + I TI I No) For each +b = #(Z, 3) such that

p u {#(Z; a)} forks over A, E JM1 (e.g., is inconsistent) and k(33)(rp A #) let r* = {rp(3; E ) , #(Z; a)} u {Z # a: a E I M I} and r be the set of such r,'s. So r is countable and there is no model N, IMI u 6 c IN1 omitting all types in r. Hence by IVY 6.3(1) there is r, E r, 8, E 1 MI and a formula rpl(Z; a,, 6) such that C(~Z)~,(Z; a,, 6) and p , ( ~ ; a,, 6) t- r,. So choose a I TI +-saturated model M,, M E M,, such that tp(6, 1 M, 1 ) does not fork over (M 1.

Ae clearly pl(Z; iZl, 6) I- I # J for every Z E IM,I, in the FAT)-prime model over I MII u 6, some b' 1 Mll will satisfs p(l, 5) but tp(6, '1 MI]) will not be a stcttionmhation of p. Hence (6) holds.

(4) =- (7), (7) e= (6) (for T totally traneoendenfal) Left to the reader.

CONCLUSION 3.19: (1) If T is totally-tramoendental, M E N, M # N then for 80W b E IN1 - 1611, tp(b, 1611) is regular.

(2) If M = N, b E IN1 - IMI, V(X, 5) ~ t p ( b , IMI), R[97(~, E), L, KO1 = R[tp(b, I MI), L, Ho] < ao and for no b, E IN1 - (MI which adis* F(x ,~ ) R[tp(bl, lMI), No1 < R[tp(b, IMI), Ly KO] t h [tp(by IMl), cp(x, E)] is regular.

Proof. (1) By (2). (2) Use 3.18 for A = IMl, p = tp(b, ]MI).

Clearly condition (4) fails, by our hypothesis, hence (1) fails, but not (1) is our conclusion.

EXERCIBE 3.7: Suppose a,, ti, are sequences, M is K-saturated.

UE. v, 8 31 WEIGHTED DIMENSIONS AND SUPEBBTABILITY 265

(1) If tp(7il, 1M1 U Zz) is c-isolated then tp(Z1, IN1 u a,) is F&- isolated [i.e., for every p there is &(Z; EJ E tp(zi,, Ill21 u 4) such that #*(% C@) tP@@lY 14 u aZ)l*

(2) The converse of (1) holds when K 2 K( T). (3) Suppose there are N, a,, a,, #@(Z; ZQ) as above, and tp(ai;, INI) is

Then we can find a:, Ei such that tp(Zi;, I N [ ) = tp(si;, IN] ) , and parallel to tp(Z,, [MI) , N is F&,-saturated.

#@@, z;) E tP(Zi;, IN1 u a;), #@(% Z;) I- tP&;, 1" u a;).

EXERCIf3E 3.8: (1) Suppose I so J ; then

w(Av(& u I ) ) 2 w(Av(J, u J ) ) ,

and for superstable T if they are finite and equal then J < , I . (Hint: See 3.9(2), and its proof.)

(2) For superstable T I I J iff for every regular q, low,(Av(I, u I ) ) 2low,(Av(J, u J ) ) , and I S w J iff for every regular q,

low,(Av(I, U I)) < 1 =- low,(Av( J, U J ) ) < 1.

(3) The relation I = J = d d I s8 J A J s o I is an equivalence relation, and for superstable T, for each I, the number of J/ = , I s8 J is finite (in faot s 2 w ( A v ( ' 9 . A similar result holds for sw.

In (I series of exercises, we now check whether Section 3 Exercise 3.8(2) generalizes to P"?-primeneee.

EXERCIf3E 3.9: If M is K-compact, 1M1 5 A, B is Ff,-constructible over A, 6 E B then tp(6, A) forks over ]MI. (Compare with l.lO(2). What about P i ? ) [Hint: It suffices to prove that if tp(6, A) does not fork over IN], tp(c, A) is Ft,-isolated, then tp(6, A u {c}) does not fork over ]MI; for this it suffices to prove tp(c, A) k tp(c, A u 6). Suppose

. not, then for some p c tp(c, A) lpl < K, p k tp(c, A), p is closed under conjunotions and for some 8 and Z E A, p u {O(z, 6, a)'} is consistent for t = 0, 1. Let Y = {(3z)(p A e(z, g, a)) A (3z)(p A 4 ( z , g, a): 'PEP} so Y E tp(6, A), 1.1 .c K . As tp(6, A) does not fork over IMI, it is finitely satisfiable in it. Hence by the definability lemma and the K-

compactness of My some 6' E IN] realizes it. So p u {e(z, F, 7i)t} is consistent for t = 0, 1, oontradiction.]

DEFINITION 3.6: The pair (p, p) (p ~p €Brn(A) for some A) is strongly regular if it fails to satisfy conditions (2)-(6) of 3.18. We dl the complete type p strongly regular if for some Q (p, p) is strongly regular.

266 MORE ON TYPES AND S A T m T E D MODELS [CH. v, 9 3

EXERCIBE 3.10: Show that the pair (p , ’p) is strongly regular, iff q~ ~p €Brn(A) for some m, A, and every stationary and complete q containing ‘p is orthogonal to p or is a stationarization of p .

EXERCIBE 3.11: Suppose T is totally transcendental M c N, M # N. Show that for some c E IN1 - IMI, tp(c, 1M1) is strongly regular. .

EXERCIBE 3.12: If (~(33; a), p) ( p E Sm(A)) is strongly regular, the dependence relation on ‘p(C, a) “tp(ii, A u iil u . - - u an) does fork over A, or do not extend p ” satisfies the axioms of a nice dependency relation (Def. A P 3.4). Prove a similar assertion for several (cpa,p,)p, E

Sm(a)(A) strongly regular, pairwise not orthogonal ; together.

EXERCIBE 3.13: Suppose N is F&,-primary over [MI U ii, tp(iZ, lMl) is strongly regular, 6 E INl, 6 4 1M1. Then N is F&o-primary over

u 6. (Hint: See 3.3 and 3.18(7).)

EXERCIBE 3.14: For totally transcendental T, show that in 3.1 we can assume tp(iZi,, lMll) is strongly regular; and does not fork over or is orthogonal to ]MI . We can replace Fft by Fco. (Hint: Use Exercise 3.11 and Exercise 3.9. Or see XI, $3.)

EXERCIBE 3.15: Suppose T is totally transcendental, tp(iZ, 1M1) is strongly regular, and tp(6, 1x1 u iz) is orthogonal to lMl. Show that not necessarily tp(6, u 7i) is FM,-isolated.

Remark. It is Fi,-isolated by the proof of 3.9. In the example one equivdenoe relations and No disjoint one place predicates suffice.

EXERCIBE 3.16: Suppose M c N, ii E IN!, iz $Ibil, p E B ~ ( ~ M ~ ) is strongly regular and not orthogonal to tp(ii, 1M1). Then p is realized in N . (Hint: Use Exercise 3.12. This is similar to 1.12 and see.)

EXERCI8E 3.17: Show that in 3.15(1) we cannot demand I - J is independent over (A u U J u a, A). (Hint: Use vector spaces over a finite field, I a linearly independent set, A = 0, J E I finite, a =

ZmJ x-)

EXERCI8E 3.18: If 6 E acl(A u i i) then stp(iz, A) s8 stp(6, A) hence w(6, A) s w(iz, A). Check the parallel wertions for other weights.

OH. v, 8 41 SEMI-REGULAR AND SEMI-MINIMAL TYPES 267

EXERCI8E 3.19: Suppose K 2 K J T ) , M is e-saturated, N e-prime over IMI u U I , I a set of sequences realizing over IMI regular types, and J E IN1 a maximal set independent over !MI, of sequences realizing over regular types. Then N is FE-prime over 1611 u U J . (Hint: See the proof of 3.9(2).)

THEOREM 4.1 : Assunte: (i) K 2 K ( T ) is re@hr.

(ii) M E N are F,O-saturatecE, A c 1611, IAl < K .

(iii) p ie a stationary regular type. (iv) El E N (1 < k < No) are ewh tlrat p l = tp(El, 1MI) is regular and

(v) p , does not fork over A ( 1 < k) . (Vi) p l r A &? stationary (I < k) . (vii) E v e y 6~ IN1 - IMI which r d i z e a p , r A r d i z e a p , ( 1 < k) . (viii) I G is a maximal independent (over !MI) set of sequences

not orthogonal to p .

which realize &er [MI regular t ypa not ortirogrmal to p .

Then tp@, 1M1 U u I) is F$isolated, and even Q-bolated. (ix) E ie algebraic over IMI u U l < k 8,.

DEFINITION 4.1 : (1) A stationary type q is sem-regular if it is not algebraic and there me K , M , N , A , k, 5, 5, &B in Theorem 4.1 and tp(i5, lMl) is parallel to q. Clearly we can aasume there is an n s k such that El depends on {Z,: i < I ) over [MI iff 1 2 n ; and 1 < n =. p , = po, and I =

(2) If in addition the types tp(E,, IMI) are minimal, q is called semi- minimal, and then we can wsume n = k. Clearly every stationary regular (minimal) type is semi-regular (minimal).

{E,: i < n}.

Proof of Theorem 4.1. By 111, 3.6 there is J E I, I JI < K ( T ) such that tp(E’,IMI UUI)doesnotforkoverlNl u UJ,whereE’ =Eon...-- c k - l ;

let J = {a,: i < 18). Choose B E 1611, IBI < K , A c B such that tp,(E U U l c k El u U J, IMI) does not fork over B and its restriction to B is stationary (possible by 111, 2.9(1) and 111, 3.2), and E is algebraic over B u 5’. By the maximality of I , for every I < k , tp(El, ]MI u I u U,<, E,)

forks over 1611, hence tp(El, lMl u U J u UfC1 a,) forks over 1611, hence

268 MOEE ON TYPES AND SATITBATED MOD= [OH. v, 8 4

it is orthogonal to p t (as pt is regular). As non-orthogonality is an equivalence relation among the stationary regular types (by 1.13(1)) it is orthogonal top. So by 1.6(1) tp(Eon. - IMI u U J) is orthogonal to p , so tp(E‘, M u U J) is orthogonal to p, hence by 1.4(1) to

(1) tp(8‘, IMI u U J) t tp(a‘, ]MI u U I). Let I, = {&: i < y } be a maximal set of sequences in 1M1, each realizing some p t t B, which is independent over B. By the Definition of B, tp(Z’, ]MI u U J) does not fork over B u U J, I, is independent over B u U J and tp(8’, B u U J) is also orthogonal to p t 1 B. As the latter is stationary:

(2) tp(E’, B u U J) I- tp(6, B u U J u U I,) . Now suppose a E C =

U {a: B E 1M1, si realizes p t B, 1 < k} and we shall prove that tp(E’, B u U J u U I , ) , tp(& B U U J u U I,) are weakly orthogonal. Let J , c I,, IJ,I < K(T) be such that tp(& B u U I,) does not fork over B u U J,, and we can assume a = Jon,. . . ,-a,, a, E [MI, a, realizes pipts 1 B. If the above mentioned types are not weakly orthogonal there are JI,, J , E JI, c I,, IJI,I < K(T) and a sequence a‘ =

tp,(I - J, 1x1 u u J). SO by 1.2(1):

. in N such that

tp(a’, B u u J u u JI,) = tp(Z, B U u J U u JI,) but tp(c’-d,B u U J U UJ;) # tp(c’-d’,B U U J U UJ;) (because K > IBI + I JI + IJ,I and N is FE-saturated). By the maximality of I,, tp(& B u U I,) forks over B, hence tp(& B u U J,) = tp(4, B u U J,) forks over B, hence tp(&, B u u J&) forks over B, hence tp(&, 12111) # pi(1,. By hypothesis this implies a; E [MI, hence 2’ E 1M1. Now by the choice of 3, for some 6 E u J, tp( 6-i?, B U (J I, u a u a) forks over B U u J,, remember tp(6^c’, B) is stationary, hence tp(bÊ’, WI) forks over B. But by the choice of B, tp(6“ F’, WI) does not fork over B, contradiction. So for every 2~ C,

tp(Z’, B u u I , u U J ) t tp(Z’, B u u I, u u J u 2) hence

(3) tp(E‘, B u U I , u U J) t tp(8, B U U J u C). Now let C, = U (7i: B E N , ii realizes pt 1 B, 1 < k), so by 1.11 for every 6 E 1x1, tp(b, B u C) k tp(6, B u C,) hence (noting 8’ u J E C,)

tp&’ u J, B u C) t tp*(d u J, 1H1)

(as this holds for every 6). Thus (4) tp($, B u U J u C) t tp(Z’, [MI u U J ) .

CH. v, Q 41 SEMI-REGULAR AND SEMI-MINIMAL TYPES 269

Combining (2), (3), (4) and (1) we see that

tp(", B u (J J) I- tp(Z', 1" u u I ) .

As E is algebraic over B u Z, and IBI, IJI < K , we finish.

LEMMA 4.2: For every semi-regular type q, there is a stationary regular type p such thal:

(1) p , q are not orthogonal. (2) For any type r, r is orthogonal to p iff r is orthogonal to q. (3) If I , J are in$nite indiscernible sets based on q, p rap . , then

I s,J & I .

Proof. Easy. (For (3) use Theorem 4.1 .)

CONCLUSION 4.3: The relation of being non-orthogonal, b an equivalence reWion among the semi-regular types.

Proof. By the preceding lemma and the fact that non-orthogonality among the stationary regular types is an equivalence relation.

Notation. Hence if p is stationary and regular, q is semi-regular and they axe not orthogonal, we shall define low&) as low#).

LEMMA 4.4: If tp(Z,, A) are minim1 and pipwise nd orthogod, then tp(Eon. . ^Zk- A) is semi-minimal.

Prmf. Easy (as here Condition (vii) in Theorem 4.1 holds trivially.)

Proof. We use the notation of 4.1, end assume in addition that N ie q-primary over IMI U U J . Then by 1.14, IJI 5 k < No. Hence by 3.9 w(Z, lMl) s k < X,.

LEMMA 4.6 : (1) Let 6~ acl(A U c), 6$ acl.4. Ifstp(r,A) is semi-regular [semi-minimal] then stp(6, A ) is semi-regular [semi-minimal] too, etp(6, A ) ie not ort?mgonal to stp(5, A); and w(6, A ) s W(Z, A ) , and lowJ6, A ) s low,@, A) for any p .

270 MORE ON TYPES AND SATURATED MODELLI [OH. v, 5 4

(2) If {a, 6) ie indepdd over A and stp(Z, A), stp(8, A) are eemi- regular and not orthogonal then stp(6-Z, A) ie semi-regular, not orthogonal to either of them, and w(6-Z, A ) = w(E, A ) + w(6, A). Bimilarly for eemi-minirnal.

Proof. W.1.o.g. we can assume, in both parts, that A = IMl, Y is E- saturated, where K = K,(T) and then the proof is easy (using Exercise 4.8.).

THEOREM 4.7: Buppoee p = stp(Z, A) i s eemi-regular and A E B. If stp@, B) f o r b over A then IowJa, B ) < w(Z, A ) = w(p) .

Proof. W.1.o.g. we can assume A = 1M(, Y is R-saturated where K = K,( T). By 4.6, as p is semi-regular there am El (1 < k) and N such that: for no 1 < k, there is 5; E INI, Z; + IMI, realizing tp(Z,, Cb(tp(Zon. - .n Zk-,, /MI)) but not tp(Z,, IMI); tp(El, IMl) is regular, not orthogonal to p , {E l : 1 < ko = w(ii, 12111)) is independent over 1Y1, N is %-primary over 1M1 u ii and also over 12111 u Ulcko Z,; E IN1 for 1 < k, and ii is algebraic over 12111 u Ulek Z,. We can assume that tp(Z,, B u Ute, Zt) forks over IMI iff 1 2 k,; so 0 I k, 5 k, I k. Notice that for 1 2 k,, tp(Z,, B u U1<, 4) is orthogonal to p , as tp(Zl, 12111) is regular, not orthogonal top. As ii is algebraic over B u u, ck Z,, clearly low,(i~, A) = w(ii, A) = ko and low,,(Z, B) I low,(Eon- - .nEk-l, B); and by 3.16(8) it is easy to check that the latter is k,. Hence it suffices to prove k, < k,; soasaumek, = ko.By4.1forsomeC E 1M1,Icl < K,Z~E~C~(CUU,,,Z,),

can assume that tp(Z,O . ~ ~ n E k - , y 12111) does not fork over C. As tp(Z,O . 'hZko-l, B) does not fork over 1M1, it does not fork over C. Thus clearly (by the q-eaturativity of Y) stp(Zon. - -hZkk-l, C u Ui <ko ZJ b tp(Zon.. -AZkk-l, B u Ulek0 at). So tp(5,O". - a n Z k - , , B) doeg not fork over Y, thus tp(8, B) doee not fork over 1x1, OontFBdiOfion.

stp(80n' "hZkk_ly u U { < k o 4) btp(Zon* * ' h Z k - l , u u { < k o Z{). w e

CONCLUSION 4.8: A type p ie regular iff it ia eemi-replar, and

w(a ) - 1.

THEOREM 4.9: (1) 8-e for e v q b E By stp(b, A) ie eemireplar, not 0rthogon.d to the &a&nmy regular type p . Then there ie a cardinal p = low,,(B, A) euCi, th& for every I = {5{: i < a}, U I = B, the

foUozoins hda8.- I( = G low,(h, A u U,<' 5,).

OH. v, 5 41 SEMI-REUULILB AND SEMI-- TYPlDS 271

Remark. We define low,(B, A) by the natural extension of Definition 3.4.

(2) If p i 8 r@r a d 8 ~ ~ r y tp*(B, A) h orthogod to p , t h for every ii, lowp(si, A) s low,(ii, B).

(3) If for every 6 E I, stp(6, A) i8 regular, not orthogonal to p , I = {6,: i < a} then z,<ulowp(6i, A u ujei 6,) = max{lJI: J c I, J i d - pendent Over A}.

Prmf. (1)Choosearegular~ 2 ~,(Z’),let M be F:-etlturated, tp,(B, IMI) does not fork over A E 1611, and let N be e-prime over !MI u B. W.1.o.g. p ~ f l ~ ( I M l ) , and let J E IN1 be a maximal set of sequences realizing p , independent over 1M1. So IJI = lowp(B, A) by the natural extension of Definition 3.4, and it clearly depends on tp,(B, A) alone (and not on My N, J). Let p = z,<u lowP(6{, A u Uj<, E j ) .

Now we prove IJI s p (this does not depend on the assumption “stp(b, A) is semi-regular for every b E By’). We define by induction on i s a models N,: Nd = M , N , is e-prime over ujcL (INjI U 6j), but we choose it such that tp,(B, N,) does not fork over A u Ujcf 6j. Now let J , c lN,+ll be a maximal set, independent over INil, of sequences realizing the stationwization of p over N,. Clearly IJ,I = lowp(6f, A u Uj<, 5j), and U j e u J j E 1N.I is a maximal set, independent over lNol = 1611, of sequences realizing p (see 1.16(3)). Clearly there is an elementary embedding f of N into Nu over M y hence

Now let us prove p s IJI, thus finishing. As for each b E B, tp(b ,]MI) is semi-regular, there is a finite set Jb c N of elements realizing over lMl regular (complete) types not orthogonal to p , such that b~

we define by induction on i < a, J: c Jb,, such that J: is a minimal set (by E), independent over (1M1 u UJP, 1M1) satisfying 6, E

acl((M1 U U J:+J where Jf = {E E J * : tp(E, ldll u u J:) fork over IMI}, J: = u c e j J : E J * .

I J I 5 Z < U IJd = P.

ad(lbll U u Jb). SO B C aCl(lbll U u J*) where J* = UkBJb . Now

Now IJI 2 IJtI = zl<u IJ:l, so it suffices to prove


third relation by (2) below, aa for eachj, and c E Jf+l - Jf, tp(c, u u Ji U UEG1bt) forks over WI, in fact over /MI U u Ji U (for c E J r as Jf E Jb, (and we could choose such J,,) otherwise by 1.9, hence is orthogonal to p, and use 1.6.)

(2) We can mume A = lMl, M F$saturatd, K = K,(T), and B s N, N F:-prime over M U B and tp(n, m) does not fork over B. Let Nl be Fg-saturated, Q-constructible over 1M1 u 8 and also over IN1 u 7i (see IV, 3.17) and N2 Q-prime over INl] u INI, hence also over IN1 u 8. We can msume p ~ f P ( l M l ) and J s INl! is a maximal set, independent over 1M1 of sequences realizing p. Clearly 1 JI = low,(Zi, A). Now by 1.4 tp*(U J , IMI), tp,(B, IMI) are orthogonal and so by 3.2 tp,(u J, ]MI) , tp,(lNI, 1M1) are orthogonal. Hence J is independent over (INI, IMI) hence low,(8, B) = low,(7i, N) 2 IJI = low,(8, A). Note also Av(J, U J ) 1 Av(J, IN1 u J), and Av(J, IMl u J ) t- Av(J, IN&.

(3) Eaey by 1.14.

LEMMA 4.10: If p = stp(8, A) i8 semi-minimal, A s B, t h stp(8, B) is either algebraic, or semi-minimal and not orthqonal to p .

Proof. W.1.o.g. A = 1M1, M is Q-saturated, K = KJT) . So there are Zl ( 1 < k ) such that tp(Zl, IMI) is minimal not orthogonal to p, and 8 E

acl(lM1 u u I < k 4). By renaming we can aasume that tp(zl, B u U1<l Zi) forks over A iff 1 1 k,. Hence Zen. - -nZk-l iS algebraic over B U

u l < k o Zly thus also 8 E acl(B U UlSko Zl). {Zl: I < k,} is independent over (B, A), tp(ZI, B) is minimal, so tp(EooA- - 'nZko-l, B) is semi- minimal, hence tp(i5, B) is semi-minimal; except when k, = 0 and both rn algebraic.

THEOREM 4.11 (in P) : (1) If 8 $ ad A, T is superstable t h n there is an e q u i w reladion 1 which is almost Over A such thud stp(8/E, A ) is semi-regular (so 7i/E $ acl A).

(2) If tp@, A ) is not o r t h g d to some minimal type q, then for 8 m

equialence relation E which is almoet Over A , stp(a/E,A) is semi- minimal, and not orthogonal to q.

Remark. E is a formula E ( f , @, 6), 1(f) = I ( @ ) , 6 E A .

Proof. Choose a regular K > IAI + K(T), and a E-saturated model M such that A E M , tp(8, lM() does not fork over A, and in part (2) the

CH. v, 6 41 SEMI-REGULAR AND SEMI-MINIMILL TYPES 273

minimal type q is over lN1. Lk N be q-primary over IN1 u a. For part (1) choose c E IN1 - IMI such that R(p, L, 00) is minimal where p = tp(c, IMI) and for part (2) so that p is a minimal type not orthogonal to stp(i3,A). In both cases p is regular (by 3.6, 1.18(4)). Now choose B E 1M1, A c B, 1B1 < KJT) + IAI + such that tp(7in(c), lMl) does not fork over B, tp(Si-(c), B) is stationary, R[tp(c, B), L, co] =

Now we can find elementary mappingsf,(i < K) such that Domf, =

U,<,f , (B)] does not fork over A, andf, = the identity. Let B, = f r (B) andp, be the stationarization off,(p 1 B) over 1M1, sopo = p. Clearly aa stp(8, A) is not orthogonal to p 1 B, it is not orthogonal to each pi, hence by 3.4 the p,’s are pairwise not orthogonal. If b E IN1 - IN1 realizes p , r B,, it necessarily realizes p, (in part (1) because of the minimality of the rank, in part (2) by the minimality of p , B,). Clearly it suffices to prove:

R[tp(c, p q , L, 001.

B, Range(f,) = IMI and StP,(h(B), A ) = stp*(B, A ) and StP,Lfi(B),

CLAIM 4.12 (in Eeq): Let a , c , A , M , N , f i , B , , p , , ~ be as above (the minimality and superstability conditions are not necessary). Then there is an equivalence relation E almost over A so that s tp(a /E,A) is semi-regular and not orthogonal to p , and if the p0s are minimal, it is semi-minimal.

Proof. By the choice of B = B,, tp(c, B, u a) forks over B, so there is a 6 E B,, such that btp[c, a, 61, and {tp(z, a, 6)) forks over B. Let C, = {d: d E N, d realizes p, 1 B,}; clearly Cb[stp(Si, B u C,)] is algebraic over B U zi (see 1.11, Cb-canonical base, see 111, 6) and there are 0 and B E Cb[stp(a, B, u C,)] E acl(B, u C,) n acl(B, u a) such that for d E C,, btp[d, a, 61 iff be[& d, 61. Let I@?, if, 6,) (6, E B,) be an algebraic formula such that C+[E, a, 6,] and $(Z, a, 6,) k tp(5, B, u a). Now let E(Z, 8) say that for infinitely many i < K, (e(E)[$(%, Z,f,(6,)) = $(Z, g,f,(E,))]. By 111, 2.6 B(Z, g) is almost over A. If #(Z, a,f,(g0)) is realizedbyE!(Z < k < Ko),then8/1iaalgebraicoverU,,,(B,u ( J I < k B ! ) , because E(z, a) iffforinfinitelymanyi’s(W)[$(Z, ~,f,(6,)) = V,J = B!]. Hence tp(a/E, IMI) is semi-regular (and if the p,’s are minimal, semi- minimal) provided we show it is not algebraic (as clearly 13: E IN1 as it is algebraic over B, u C,). But for big enough n, Z/E is algebraic over U,<,, (B, u {a!: Z < k}). Clearly tp,({Zi: 1 < k, i < n}, 1M1) doe8 not fork over U,<,, B, (as 17: is algebraic over B, u a), and it is not algebraic by the choice of ‘p. But clearly tp({i$: Z < k, i < n}, IMl u a / q ) forks over

274 MORE ON TYPES AND SATUBATED MODELS [CH. v, 3 4

IMI, thus 8/E $ M. Clearly 8/E E dcl[(clcl A) u 7i] hence 8/E E acl(A u 8). Of course, when p is minimal, stp(@/E, A ) is semi-minimal. More fully, tp(a/E, M) is semi-regular and @ / E c dcl (M U ui < z, < I$) E dcl(M U ul<k C,. Hence, tp(a/E, A ) is semi-regular as tp(a/E, M ) does not fork over A and tp(@/E, M) is semi-regular.

CONCLUSION 4.13 (P): 8wppo8e F eatieJka Axifma (X.l) a d (XI.19. (1) If T ie &wpr8tabk, for every COnetmCCtiOll d there ie a conetruetion

d' such thd d, d' have the earn dumain and range, and tp(a;, di) ie algebraic or eemi-regular.

( 2 ) Forunidimensionul T , thesameconclusionwithtp(a,', A; ) algebraic or eemi-minimal.

Proqf. Immediate.

DEFINITION 4.2: The m-formula ~ ( b , 7i) is semi-weakly-minimal if for some weakly-minimal +(q, ,?), every 6~cp(6, a) belongs to acl(F U

+VL f3).

THEOREM 4.14 (in P): 8uppose tp(8, A) ie not o r t h a g d to e r n e minimd type q to which e m weakly-minimd form& Q b e 4 8 . Then for e m equ~valence reh4bn E which ie over A , 7i/E 8ati8Jka 8ome semi-weakly- minimal formukt # which is over A , but 8 / E $ acl A. Also every etationav type which conhim # i8 nd orthogod to e m etationary type to which Q b-8.

Procf. The proof is the same as that of 4.11(2) with the following additions. Ueing the notation of 4.11(2), let Q(Z; 8:) E tp(c, B,) be weakly minimal, and let ~ ~ ( 3 ) "say" w.1.o.g. that for inhitely many i < K ,

(3 < '"Z)#(z; % h(60)) A (vZ)[$(z; f~ h(bo)) * QO@Y fi(6:))l-

Let Qa(y) "sayyy that y = Z/E for some f satisfying ~~(3). Then Qa(y) is eemi-weakly minimal, but it and J me only almost over A. This can be oorrected (see 111, 2.4). For the last phrase, see 4.6.

THEOREM 4.16: 8-e p , q are wnaplete type% over A, and are station- M?/. If p , q are eemi-minid and not o r t h a g d then there are 1, < w, and BI (1 < 1,) rdiz ing p such thud every eequi?m? rdizing p i8 algebraic over UlCl,, 8, u A u U (6: 6 realizes q}.

OH. V, 9 41 SE~I-BE~ULAR AND SEBCI-MINIMAL TYPES 275

Pro$ Let h > IAI + IT1 be regular, and it2 a R-saturated model, A c 1611, and r ~ B ~ ( l M 1 ) a minimal type not orthogonal top (nor q). Let I = {ii E Q: 7i realizes p } , J = (6 E Q: 6 realizes q}. Let pl, q1 be stationarizatione of p, q (reap.) over IN). Clearly for every ii realizing p , there are c2(Z < n) such that r2 = tp(c2,M) is minimal, not orthogonal to r, @Eacl(wI U UlGncl). Also for any F realizing r2 there is an a realizing p, such that cEacl(m U I%). Similar assertions holds for q,; so for every @ realizing p , there are 6,, . . . , 6l rea,lizing q1 such that ii E ad( ]MI u Ufdl &); hence for some 5 E lit21 we can have ii E acl(5 u Uiel 6{). Clearly there are C1 c (U I ) n !MI, Cg E (UJ) n lit21 where ICll + ICgl s IT1 such that tp(5, (A u U I u U J)) is definable over Cl U Ca (as M is Fi-eclturated, 1.41 + IT1 < A) . By 1.11 for each nrealizingp, there are 6,,, . . . , 62 realizing q1 such that 7i E ad(UlsI 6 U Cl U C, U A). We can replace Cl, Ca by finite subsets without changing the last conclusion, by a compaotneae argument.

SO let Cl = U(<I(o) $, C, = U{<l(l) E$', and choose 4, 6f SO that

stp*({$: i < Z(O)} u {q: i < 2(1)}, A ) = Stp*({a:: i < Z(O)} u {6f: i < Z(l)}, A) ,

does not fork over A. Then for every 7i realizing p there is j < K(T), such that tp(ii, A U Ut 7i{ u U1 @) does not fork over A. As p is stationary for every a' realizing ply and ii' realize the same type over A u U1 7i{ u U1 6{; hence for every 7i realizing g there me EOy . . . , 6l realizing q such that ii E aol[(A u Ui,, a{) u (U:,, 6{ u U1 &)I; so we finish.

QUEBTION 4.1: Try to replace in 4.9 semi-regular by a weaker con- mpt, and investigate this concept. (See Definition 4.2 and 4.4 for a suggestion).

EXERCHE 4.2: Generalize 4.9 to semi-weakly-minimal formulas.

QUEBTION 4.3: It is true that in 4.11 we amnot replace semi-regular (semi-minimal) by regular (minimal) even if we replace ii/E by any b E ad(A u a) - acl A ? What if T is N,-ategoriWl ?

276 MORE ON !l'YPHS AND SATUBATED MODELB [a. v, 8 4

EXERCISE 4.4 (Cq): Suppose p E S(A) is minimal, I and J am sets of elemente realizing p , which are independent over A, but I u J is not independent over A. Show acl(A u I), acl(A u J) not nectxsady have an element in common, which is not in aclA [e.g. T = theory of the complex field].

QUESTION 4.5: Clemeralize 4.16 to semi-regular types.

EXERCISE 4.6: Prove that if &?,a) is semi-weakly minimal, then D"[cp(Z; a), L, co] < w . [Hint: See Section 7.1

QUESTION 4.7: Prove 4.11 does not hold for stable T in general; and try to find a substitute.

Remark. Let T be defined by the following axioms: (i) P, (r] E @'2) a m pajrwise disjoint, (ii) P; (1 = 0,1) are partial one-place functions, from P, onto

(iii) U, is a partial two-place function, pn-<t>,

T exemplifies the non-exiatence of regular types.

EXERCISE 4.8: SUPP Mi E Ha E M3, p ES"(A), A E MI, and for I = 1, 2, every 6~ lMt+l l , 6# 1M,1 which mlizes p, realizes a stationariaation of p over M,. Show that every 6 E 1M31 which realizes p, realizes a stationarimtion of p over lMl I.

EXERCISE 4.9: Show that 4.4 does not hold for regularity, even for Ko-stable T . [Hint: Let M = (1611, P , Q , E , P , +), is the disjoint union of P and Q which me W t e , E an equivalence relation on Q, + is a two place function from P to P , ( P , +) is an abelian group, each element of exponent two and P is a two-place (partial) function, P(a, b) is defined iff aEb, a, b E Q , and P(a, b) + P(b, c) = P(a, c).

for every c E M. Then tp(a,, ]MI) is regular but tp((al, ag), IM1) is not Choose a,, a, €&(a) - 1611, t ~ B ( a 1 , C) A E(a1, a,) A P(a1, as) # c

semi-regulw . ]

DEFINITION 4.3: (1) A formula (~(3, a) is (strongly) semi-minimal if for every Z satisfying it, stp(Z, a) is semi-minimal not orthogonal to some (fixed) minimal type oontaining a minimal formula, or is algebraic.

(2) A formula ( ~ ( 5 , a) is almost minimal iff it is equivalent to a disjunction of minimal formulas.

EXERCI8E 4.10: (1) (P@, a) is semi-minimal iff there is an almost minimal I,$@, 6) and Z suoh that p(C, a) E ad(E u $(C, 6)).

(2) The formula p(Z; a) is almost-minimal ifF R”’(p, L, No) = 1. (3) In 4.14, if in the hypothesis we replaoe weakly minimal by

minimal, then in the oonolusion we get a semi-minimal formula. For A = a01 A, even strongly semi-minimal.

(4) The following families of formulas me olosed under disjunotion: almost minimal, semi-minimal, weakly minimal and semi-weakly minimall.

EXERCI8E 4.11: Let V be an infinite veotor woe over the field with two-elements. Let us define for each n, iK( = M,,) :

lMl = V u V x V x - - - x V (n times), P = V (a one plaoe predioate), 4 = (<(%’ - .s ai, a1+1, * * *, a,,), (a13 * * - 9 ai, 4+1, * * ., 4)):

a1,. . .,a,, 4 + 1 # . . ., 4 E V},

RI = (~~,~4,...,~l,CCI+1,...,CCn~,~~,,...,~i,~l+l + c # ~ ; + 2 , - # 4 ) :

C, 01, - an, 4 + a s -. - 9 4 E v),

R = ((a, b, c): a, b, c E V, a + b = c},

80

= (pfl, P, El, R, m<,, T = ThW). Choose N X1-satmated.

(A) Prove T is totally transoendental and unidimensional, and oate-

(B) Show that here it is possible that (when n 2 3) w(6,A) =

(C) Show that 4.16 (UuLllot be substasltially improved.

gorical in every A.

w(a-6,A) = w(ii,A u 6) = 1.

DEFINITION 4.4: (1) Let B denote a subset of B* 5 (stp(Z, A): A, a}. B is oalled regular if:

(i) Whenever A E B, stp(si, A) E 9, stp(ii, B) fork over A, then stp(a, B) is orthogonal to B (see 2).

(ii) If A G B, stp(zi, B) does not fork over A, stp(a, A) E B then Stp(a, B) E 9.

278 MORE ON TYPES AND SATURATED MODELS CUE. v, 6 4

(2) A type q is orthogonal to B if it is Orthogonal to every p E d? If 9 = {p} , we write p instead of {p} .

(3) An m-type q is strongly Orthogonal to B if q E ql, ql complete implieg q1 orthogonal to 9.

(4) For a regular set B lowg(q) = zpOBc1) lowp(q), where B(1) E 9 is a maximal subset of pairwise orthogonal types (it is well defined) (see 4.16( 13)). Similarly low,(a, A) etc.

olO(9) = {p : p = stp(ii, B) is a sbtionariea;tion of q E # q E p } n B*, oll(9) = {p : p pardel to some q E LP) n 9*, olj(9) = {P(p ) : p E P an automorphism of CE and Pw 1 a01 A = the

0l3(9) = { p : p regular, stationary not orthogonal to LP) n 9*. Now oPJ(9) = 01‘ ol’(9) and we let ol’(9) = oli(9).

(6) For a set of types 9 let

identity} n 9*,

DEKUCITION 4.6: b t 9 be (L r e g ” f 8 d y . (1) An m-type q is B-simple when for some A, every ii realizing q is

algebraio over

A U U{b:stp(b,A) is 9-regular (see (4) below)} U {b:stp(6,A) is strongly orthogonal to B}.

(2) A stationary nz-type q is strongly 9-simple if there am E- mturated mod& bl, N ( K = K~(!P)) , M < N, and 8,B, E N(Z < n) suoh that

(i) tp(ii, [MI) is a stationarization of q, (ii) tp(c,, WI) is 8-regular (see (4) below),

(3) An nc-type q is 9%emi-regular, if (2) holds with the addition (iv) if B E IN1 realizes tp(Bl, Cb(tp(B,, [MI)), B + lM(, then B realizes

(4) An m-type q is 8-regular if every extension of q which forks over Domq is strongly orthogonal to 9, but q is not strongly orthogonal to 9.

(6) W e say si is 9-simple over A if stp(ii, A) is 9-simple. Simil&ly for 9-semi-regular, strongly @-simple and 9-regulm.

(iii) si € aol( lMl u u,<n B*).

tP(% “1).

LEMMA 4.16: (1) I f p i8 a regular etdionary type tlren: q ie 8e~ni-regukcr not orthogonal to p iJ q ie oP{p)-senti-rq&r, not algebraic (eo oP{p} ie regular).

(2) If q is @-semi-regular, then it is strongly 8-simple, and if 9 = c12(9) any complete strongly 9-simple q, is 9’-simple. Also if q is

H. v, 8 41 SEMI-REQULLSB AND SEMI-MINIMAL TYPES 279

9-regular, it is 9-simple and i f all stationarizations of a complete p are in 9, p is 9-regular. A n y stationary 9-regular type is 9-semi-regular (9 is a regular family, of course).

(3) If p = tp(si, A ) ie r@r, p E q, q forb over A, p E e B regular, then q is strongly orthogonal to 9. Also p i s 9-regular.

(4) If r k q, q $8 9%mple, ie regular then r is B-&mple. 80 if A E B, si 9-eimple Over A then li is B-eimlple over B. If r k q, q ie @-regular then r is 9-regular or is strongly orthogonal to 9. If a i s 9-simple, then w,(E,A) < w . (6) If q ie strongly orthogonu1 to e r k q then r ie etrongly orthogonal to 2 (6) If si ie 93-eemi-regular over A thn low&, A ) = w(si, A) . (7) If q ie B-regular, q over A , q E ql , q1 forks over A then. q1 ie

(8 ) 8uppxe q, r are parallel, then q ie etrongly B-eivle if r ie etrongly P-eimple, a d q ie B-eemi-regular i# r ie B-eemi-regular.

(9) If B e d i e f i (i) of Definition 4 4 1 ) then cl0(B) is regulur. If 9 c

(10) If p i s minimal, 9 = el3@}, q complete and stationary then : q is semi-minimal not orthogonal to p iff q is strongly 9-simple not algebraic.

(1 1) If p , q are parallel and complete (9 = cI2(9), 9 regular), then p is 9-simple iff q is 9-simple, and p is 9-regular iff q is 9-regular.

(12) Suppose 9, = el2(g1) c g2 are regular families: i f q i s g1- regular and complete [91-semi-regular] [strongly P1-simple], then q i s g2-regular [92-semi-regular] [strongly P2-sirnple].

etrongly orthogonu1 to B.

then: P1 is regular, implies clO(9) is regular.

Proof. Easy. In (2), note that even a stationary q may have a stationarization in 9 but not be in 9 (instead 9 = c12(9) also 9 = cll(9) suffice). In (4)) use (13) below.

(13) Let T be superstable. Every complete stationary 9-regular type p is regular.

Remark. Part (13) sounds, a t first, a truism, but an extension may be orthogonal to 9 while it is not orthogonal to the type.

Proof of 4.16(13). W.1.o.g. p e S m ( M ) , M is Fgo-saturated, C E M , r is over M and r e 9 is regular, not orthogonal to p and d realizes p . If p has weight > 1, there are n < w , n > 1, and d$(i < n) such that tp(di,M) is regular, {d i : i < n} is independent over M and tp(d,,^dln*m.dnE,_l,M U d), tp(d,M U u,,,di) are Fgo-isolated (by V,

280 MORE ON TYPES AND SATTJBATED MODELS [UH. v, 8 4

3.9(2) and see Def. 3.2). Since we have assumed p is not orthogonal to r , w.1.o.g. (by 3.9(1)'s proof) tp(do, M) is not orthogonal to r . Now p , = tp(d, M U dl) forks over M, whereas tp(do, M U dl) does not fork over M, so tp(do, M U z U dl) forks overM, and hence should be orthogonal to r . But tp(80,M U d u dl) forks over M, whereas tp(zo,M U dl) does not fork over M so tp(d,,,M U d U dl) forks over M U dl: hence, tp(d,M U do u dl) forks over M u dl. So stp(d, M U dl ) , stp(do, M U dl) are not orthogonal. But the latter is regular and parallel to tp(&,M) which is not orthogonal to r , so stp(d, M U dl) is not orthogonal to r, contradicting the 9-regularity of p .

So p should have weight 1. If it is not regular it has an extension q, forking over M , not orthogonal to a regular type r' iff r' is not orthogonal to r (as w0@) = 1,p not orthogonal to r ) . We conclude that q is not orthogonal to r , but that q forks over M and hence over C, a contradiction.

So p is regular, so we have proved 4.16(13).

LEMMA 4.17: (1) If zi E d ( A u 6), 6 ia 9dmpZe over A then zi ie B- eimpk over A, and low& A) s low& A) . 8im;hrly for B-eemi- regular, 9-regular and (except BE acl A ) strongly 9-simple.

(2) I f q ie B-eirnpk, q ia over A, ti r d i w q, then low&, A ) iajinite. (3) B = {p: p a minimal type} is a r@r famiZy. Only aebraic type8

(4) tp(zi-6, A) ie &&n& iff tp(zi, A), tp(6, A) are 9-eimpk. ( 5 ) For any ordinal u , 9 : is a regular family, where 8:g' @ : p a

regular type, R @ , L , a ) = 01, and p is orthogonal to every stationary q for which R(q, L , 00) < CO} n 8*.

(6) 8: = clo(9:) = cl'(8:) = c12(9:), and any formula $(x, C) with R($(x, c), L, m ) = u is Pz-regular or is strongly orthogonal to 8:. Also, every regular p E 9: contains ( i f it is closed under finite conjunctions) a 9:-regular formula.

(7 ) For superstable T , for any stationary regular r , for a unique u the type r is not orthogonal to S:, so some S:-regular formula ~ ( x , F ) is not strongly orthogonal to r .

are e t r q l y o r t h o g d to 9.

Proof. Easy.

LEMMA 4.17(8): (aq) Suppose r is regular and stp(B,A) is not orthogonal to r , r not orthogonal to 9:. Then

OII. v, 8 41 SEMI-RE(XULAR AND SEMI-MINIMAL TYPES 281

(i) There is a formula E = E ( Z , y) an equivalence relation over A

(ii) Moreover, tp(a/E,A) contains a formula O(y,b) that is

(iii) Also, for every 6' realizing tp(5, $), O(y, 6') is P,*-simple.

such that tp(a/E,A) is @,*-simple but not orthogonal to r .

@',*-simple.

4.17 A remark : Instead of 9': we can use any regular 9' = el; '(9') if A = aclA and any ~ € 9 ' has a 9'-regular formula in it, e.g. any ~ 1 ~ ~ ~ ( 9 ' ) , 9 E 9':, is O.K. and el3@), p stationary regular.

Proof. Choose cp(x,C) as in Lemma 4.17(7). Clearly, {cp(x,C)} has a stationary completion p which is regular, not orthogonal to r , and w.1.o.g. p(x, ES(C) (as we can increase C). So q(x, @, stp(@, A) are not orthogonal, hence w.1.o.g. their stationarization over A U F are not weakly orthogonal. Let {c, : n < o} be an indiscernible set based on A,c0 = F. So each p, = &,I?,) is regular, not orthogonal, to A (as po is). Hence, by 3.4 the q n ' s are pairwise, not orthogonal ; hence, by 1.13(1) (as qo,r are not orthogonal) each q, is not orthogonal to r . Also for each n, stp(@, A) has non-orthogonal stationarizations over

Now repeating the proof of 4.11 we get E , an equivalence relation over acl A as required in (i).

So let E l , . . . , E, be the images of E under the automorphisms over A. As @',* is preserved by the automorphism, for every d and 1 the type stp(d/E,, A ) is @',*-simple. Let E*(Z, y) 'Zp A f - l E z ( ~ , y), then @/EEdcl(A U U,a/E,) so tp(a/E,A) is S',*-simple.

Now tp(@/E,A) contains a formula O(y,b) such that for some n, ~ ~ ( 1 < n) realizing tp(c, A )

A u cn.

r 1

As each q(Z, 4) is @',*-regular (by Lemma 4.17(6)) clearly O(y, 6) is @,*-simple. So (ii) holds. Also for every automorphism F of aeq,F(Fl) realizes tp@, g) , hence each cp(x,F(~?J) is 9':-regular, hence O(x, F(6)) is @',*-simple. This proves (iii).

THEOREM 4.18: 8uppOcre B is regular, B = ol%(B) and I = {6{: i < a}. 2 " h lowB(U I , A ) = Zlsrr low&,, A u U,<l 6,) if every b E ut < 6, i8 B-8inzple ouer A.

Pmof. Like 4.9.

282 MORE OW TYPE8 AND SA!lWBATmD MOD= [OH. v, 8 4,

Remurk. This really generalizes 4.9( 1).

LEMMA 4.19: (1) If A c B, 9 regular, B = c1:(9), ii 9-simple Over A then low&, B) s low& A ) .

(2) If in addition nis 9-semi-regular over A , tp(n, B ) fork over A then low,(E, B) < low,(& A ) .

Proof. We prove (2) like 4.7, and (1) like 4.9(2).

THEOREM 4.20 (in P): S p e p is stationary and regular, and stp(ii, A ) is not orthogonal to p . Then for some equivalence relation E almost over A , stp(ir/E, A ) is strongly c13 @}-simple, low,(a/E, A ) > 0. It is also cP@}-simple.

Proof. Like 4.11.

Similarly

CLAIM 4.21: flwppo8e I c ~ ( a , 6), I an infinite indhrnib le set, tp,(I', A) = tp,(I, A ) implie% I', I are not orthogrmal. Then there are n, 6,(l < n) i2 E A, and # m h that:

( A ) tP(61, A ) = tp(6, A) . (B) For s m c', #(a, a) c acl(a u Ulen q(&, 6J), in fact there ie a set

{6fni5y: 1 < n, a < K } (any K ) independent over A stp(6?, A ) = stp(6f, A ) m h that for every a(1) < K , #(a, a) G ac1(Ul<,, Ef(I) Ul<,, #(a, 5?(p"))). (C) There is an (infinite) indiemible set J E #(a, a) baaed on A m h

that tp#, A ) = tp*(I, A ) implies I', J are not orthogonal. (D) If J , c #(a,@ b an infinite indiscernible set, then for some 1, and

I , c ~ ( 6 , 8,), J, , I , are not orthqtmul. (E) If Av(I , U I ) is regular, then Av( J , lJ J ) is semi-regular. ( F ) If Q(Z; a) is @-regular, #(z; a) is @-simple.

LEMMA 4.22: 8 ~ p p ~ e K = [A]' + K,(T), bl is F,O-sdurded, p = stp(a, A ) is semi-rephr, then there ie an indiscernible set I E ldll based on p , Av(I , A U U I ) I- Av(I , M).

Proof. First choose a maximal set I of sequences from M realizing p, which is independent over A . Let B be a maximal subset of i l l satisfying

(i) A u U I E B, (ii) B is q-atomic over A u U I

(exists aa A U U I satisfies those conditions, and they are preserved by unions of increasing chains). By IV, 3.2, for no b ~ w 1 - B is

OH. V, $41 SEIUI-REGULAR AND SEMI-BEINRUL TYPES 283

tp(b, B) E-isolated, hence B = INI, N F,O-saturated. We can find a complete regular type q over !.Al not orthogonal to p, and let J, = (4:i < a} c be a maximal set of sequences realizing q independent over m. Let p , be the stationarization of p over m, clearly M omits p,, hence a < KO, and even a < w(p) .

Choose B c N, IBI < K ( T ) such that q does not fork over B, q 1 B is stationary, A c B, tp(B,A U UI) does not fork over A U (B n I), and p, rB,qrB are not weakly orthogonal. Let J c !.Al be a maximal set of sequences realizing q rB and independent over B. Now J U J, is a maximal set of sequences from PI realizing q 1 B and independent over B (1.16(3)). Next choose I, c I, J , c J , lIol = lJol = K such that I - I, is independent over (A u I, u B u J,, A) and J - J , is independent over (A u I, u B u J,, B). Let N* G M be Q-prime over A U I, U J , u J,. By IV, 4.11 there is a set I, G IN* I of sequences realizingp, independent over A, AV(I2, A u U l a ) l-Av(l,, p*l) (the hypothesis of IV, 4.11 holds as iF* is F:-prime over 0, by IV, 4.9(3)).

Now 1-1, is independent also over (p*I ,A) , and by IV, 4.15N* is Ff-atomic over A U UI,, hence N* U UI is F:-atomic over A U UI,, where I, = ( I - I o ) U I,. Clearly for every FE J, tp(4 B U UI) fork over B hence tp@, p*I U I,) fork over B, hence is orthogonal to p so also IN*] u u I u 5 is E-atomic over A u U I,. Repeating (and noting Jo c lN*l) we get that 1N*I u u I , u u ( J u J,) is E-atomio over A u 13. So we can find a maximal set B, s liKl E-atomic over A u I,,

and let p’, q’ be the stationarizations of p, q, resp., over lN1l. But J u Jo c lNll was a maximal set of sequences in M realizing q B independent over B, hence iK omits q’, hence p’ has a unique extension over WI. As by 4.10(2) Av(I,,A U I , ) t- Av(I,, WJ) we finish.

IN*] U u (13 U J U J O ) c B, and @n B1 = 1N11, N, E-&watd,

EXERCI8E 4.12: Show that in 4.18 and 4.19(1) the demand P = cl i (9) is necessary. [Hint: Let T be a theory with only one equivalence relation, with infinitely many equivalence c l m s , each of them infinite, and choose distinct a, E B, B C alEaa A -,alEa3.

For 4.18 let 9 = c13{tp(a,, a,), tp(aa, a,)}, A = 0, a = 2, b, = (a,), b, = (a2): and for 4.19(1) B = c13(tp(aa, a,)}, A = 0, B = {a,}, 8 =

(aa>.l

- -

EXERCIBE 4.13: Show the if in Definition 4.3( 1) (i) we demand only “stp(7i, B) is Orthogonal to every p E B which does not fork over A,” then 4.18 may fail. (Hint: Use the example from Exercise 4.12.)

284 MORE ON TYPES AND SATURATED YODELS [a. V, f 6

EXERCI8E 4.14: Look through Sections 1-4 and generalize the relevant theorems to the notions from Definition 4.5 and 4.4 (in addition to Theorems 4.1M.22) .

EXERCI8E 4.15: Suppose q, A are as in Definition 4.4(1) A , Dom q E B G C, 7i realizes q, Then low&& C) s lowg(& B).

V.5. Mdti-dimensional theories

DEFINITION 6.1: Let {J t : i E 8) sw I if for every %-saturated model (where K = K J T ) + 181 +), such that I , Ji E IMI.

min dim(J,, M ) s dim(I, M ) t€S

LEMMA 6.1: The following conditions on I , J , (i €8) are equivalent (letting K = K,(T) + I8l+):

(1) {Jt: i €8) s, I . (2) If dl is F:-8aturated, J* = {a;: i €8, j < K } d8 independent over 1M1,

I , J , E M and izf realizes Av(Ji , ] M I ) and N $8 Fg-prime over 1M1 U U J*, then Av(I , IMI) is realized in N .

(3) There are a FE(m-8aturated nzodel My I' c M equduab4 to I , and 8' c 8, 18'1 < K(T) and J; E M(i €8') equivalent to Ji, and J* = (8;: i EB', j < K(T)} idpendent over 1M1, a: realizes Av(J;, 1611) such tirat in the F&,-ppime model over 1M1 u U J*, Av(I ' , lMl) is realid.

(4) There are JI equivalent to Ji (i E B) and I' equivalent to I such that

(ii) I' is indi8cernible over Utes Ji and J; is indiscernible over I' u

(iii) Av(I', 1') has a F:(,-isolated complete extmion over I' u

(6) For 8 m e h 2 p 2 K , there i8 ru) %-saturated My I , Jt E M such

(i) p ' l Y I J I I = 4 T ) ,

U { J i : i # j ~ 8 ) ,

u { e l t : i€S}.

that dim(I, M ) = p, dim(J,, M ) = h (i €8).

Prmf. As the ideas essentially appear in Section 2, we leave it to the reader aa an exercise.

CONCLUSION6.2:(1){Ji:i~rS) 1,Ii,fJfor8melsll E 8,lS'I < K(T), {Ji: i €8') I , I .

(2) The holding of {J i : i €8) I, I depend on I , J,only up to equivalence.

Proof. Immediate by 6.1.

OH. v, 8 61 MULTI-DIMENSIONAL THEOFLIES 285

THEOREM 6.3:LYqpose I , Jf are indiscernible over A (i < K( T), a < K( T)) a d I I l , I J f I 2 Kr(T),IAl c ~ r ( T ) a n d t p , ( J ? , A u U ( I u U { J ~ : B < a or = a, j c i}) does not fork over A , and stp,(Jf, A ) = stp,(Jf, A). Let J * = Ur,cr Jr, B = A U U (I U J*). Assume {Jf: a, i c K(T)} &,I.

(1) If J = {ti?: a, i < K ( T ) ) is i-endent Over By 3: realizes Av( Jr, B), then Av(I, A U I) hae an l&&solated mplete extenaim q Over B u U J.

(2) If id additiort LIZ F&)-sdurated, B c (MI, {a?: a, i < K(T)) i8

independent over (1611, B) , then q I- Av(I, 1M1).

Proof. (1) W.1.o.g. we assume 111 = lJfl = A = (2ITI)+, and let p = Av(I, B). Clearly there are J' E J , I J'I < K(T) and ql, p G q1 E

Sm(B u u J'), such that ql has a unique extension Pa E P ( B u U J) and let 6 realize ql. By notational changes we can assume that J' = {a?: a, i < a, < K(T)}, and that we can choose C c B, ICl < K(T) C c B, = A u U (J' U I U U { J f : a, i < a,}) such that q1 does not fork over C. Clearly also does not fork over C. Choose JT = {k,: a, s a < K(T) , i < K(T)} such that J+ = U {JT: j c A + } is independent over B u lJ J and ?& realizes Av( Jf , B u lJ J). We shall show that q1 k tp(6, B u U (J u J + ) ) , for this it suffices to prove that for every n < o, q1 I- qn = tp(6, B U (J u U,<,,J?)), and qn does not fork over C and this we shall prove by induction on n. For n = 0 there is nothing to prove, For n + 1 choose an elementary mapping f such that f is the identity over B,, and for i , a, a, 5 a c K(T), f maps Jf onto Jf u {at,: j < n) and f (a?) = a&. As q1 k qa clearly f (ql) kf(qa), so iff (ql) = q",f(q,) = qn+ we finish. But this holde as q1 does not fork over C, C c Bo, f 1 B, is the identity, clearly f (ql) 2 q1 does not fork over C, hence doe8 not split strongly over C, but also qn satisfies this and q1 E q" ; hence qn = f ( q J ; similarly f (q , ) = qn+'.

We c m conclude that ql I- tp(& B u (J u J + ) ) . Let M be R-prime over B and Flf-constructible over B u J+ ; and N be Flf-prime over IMI u J+. As for a 2 a,, dim(Jf, N) 2 A + , dim(I, N) 2 A+ [clearly there is an elementary mapping f, f 1 A u (J I is the identity, and f maps Jf onto (a, i < K(T) ) hence {Jf: a, i < K(T)} sW I implies (JP: a. s a < K(T), i < K(!P)} sW I]. So there is I+ c IN1 indiscernible over lMl u 6, 11'1 = A + , I u I+ is indiscernible over A.

By IV, 4.10(2) tp(6, B u U J+) I- tp(5, INI); hence

ql I- tp(5, u (I u I + ) u A).

We can aasume that if 5 E I is not disjoint to C - A then 5 E C. By

286 MORE ON TYPES AND SATURATED MODELS [UE. v, 8 6

the proof of IVY 4.3 we o m deduoe that stp(6,A u C) k tp(6, A u U ( I , u I+)) , ( I , = { E E I : E E c)) hence stp(6, A u C) k tp(6, A u U I ) .

(2) The proof follows eesily now.

DEFZNITION 6.2: The (infinite) indisoernible set I is oalled multidimensional over A if whenever I j (j s K( T)) are suoh that tp,(Ij, A u Ut<j I j ) does not fork over A and stp,(Ij, A) = stp,(I, A) then { 4 : j < 4T)) $w4r4n.

DEPTNITION 6.3: The theory T is multi-dimensional if Borne I is multidimensional over some A.

LEMMA 6.4: I f I is orthogonal to A then I is multidimen&onal over A .

Proof. Immediate by 1.6 and 2.7.

THEOREM 6.6: 8wppo8e T is not multidimnakmal , and M is F&n- saturated, and A,, s 2ITI i8 the 3rd cardinal in which T i a stable. Then t h e i s N -c M , [IN11 s A, and a set J E M in&p- over M such that M is a e(n-prime over IN1 u u J .

We first prove a olaim:

CLAIM 6.6: 8uppe A,, is aa in 5.6, No E bl, c My and they are q,,-saturated and llNoII s A,,, and for every 2 E My tp(8, Jbl,l)forh over No or E E INol. There are N, , N o c N1 c M,, llNLII s A, and J c M idpendent over (IMol, lNll) m h that if E E (MI , 8 $ lblol, i? ~ e a l i z ~ i Av(I*, ~ M , ~ ) , I* any countable set indiacernible over No, I* not multi- di&alover INo! and Cb(Av(I*, (J I*)) c lMol , then tp(Z, M , u u J ) fork Over lMol or tp*(I*, IN,l)forh over IN,]. Also N , is F&-saturaterl.

Proof of 5.6. Let {(I,, J,): u < a,} be a maximal family such that: (i) I, is countable, and it is indiscernible over INol, and I, is not

multi-dimensional over lNol and Cb(Av(I,, (J I,)) c IMol. (ii) J, E bl is a maximal set independent over (lMol u UB<, JB,

lMol) of sequenoes realizing Av(I,, JM,I), and J , is not empty. (iii) tp,(l,,Up,Jp U pol) does not fork over POI. The number of

possible tp,(Ia, IN,!) is sA,,. [If A20 = A, by the A,-stability of T. If A$ > A,, then necessarily K(T) = No, so ohoose I: s I,, I: finite so that Av(l,, u I,) does not fork over I:, and let I: = {a;, . . . , z&-,}, a& E I, - I;; now tp(a,- - - - -- an(,), I N O l ) determine tP*(Ia, INol)*I

Similarly, for each i there is C, E INol, IC,l a KJT) and I: 5 lNol such that tp*(I,, INoI) does not fork over C, and stp,(I:,C,) =

So if a. 2 &+ then w.1.o.g. for every a < &+, tp*(I,, INo!) =

tp,(I,,, ",I). As I , is not multi-dimensional over INol, we get a contradiction by 5.3. (As U,:j < K(T)} I wI,*, Av(l,*, POI) is realized in M . ) So uo < &+ and we choose N,, N , E M , llNIII 5 Ao,

stp*(l,, Cf).

IN01 u u Cb(Av(Ia, u I , ) ) E {<a0

and our conclusion follows.

Proof of 5.5. We define by induction on i s K,(T), N,, M, , J , 80 that: (i) Ni _C dl, llNfll 5 A. and N , is F:r,,-saturated. (ii) J, E M, J , is a maximal set independent over (IN, I u Uf <, lMf I ,

(iii) M, is E(,-prime over (J J , u U,<, INf I , and M , E M . (iv) N , + , c M,, Nd 2 Uf<aNf for limit 8. (v) For every E E ]MI, E g 1M,I, and countable indiscernible set I,

Cb(Av(1, U I)) _C lMfl, Av(I, lMfl) = tp(E, IM,l), at least one of the following holds :

INfI).

(A) tp(E, lMfl u U J , ) forks over IM,]. (B) tp*(I, INf+ll> fork6 over

We choose N o to satisfy (i), and M , by (iii). J , and N , + are defined by Claim 6.6, a d Nd by (iv).

and a countable indiscernible Set I over NXr(T), Cb(Av(1, U I)) E M,,,(,, such that Av(I, MXdn) = tp(Z, ME,(,). By (v) we get a contradiction (when ~~(2') = No, we have to work a little more).

Remark. In 6.6 we can msume only M is E(,,-satur&ted.

THEOREM 6.8: For every nun-multidimensional T, there is a cardinal h 5 (2IT')+ m h that:

(1) A the $r8t c u r d i d for which there are no I,(a < A) m?& that { IE: a < p < A} $ I, .

(2) Let K,( T ) 5 Ha < HE, T stable in HE, and p i8 the number of nun- ~8onwrphk IQm-8aturated d l a of T of cardinality &. Then

If M,(, # M choose E E M, E $

288 MOBE ON TYPES AND SATURATED MODEL8 [OH. v, 5 6

Proof. Left to the reader as an exercise.

PROBLEM 5.1: Make 6.8(2) more precise. In particular try to prove that A is always a successor. (See IX, Section 2.)

THEOREM 6.9: If T i s multi-dimeneional, T stable in 8 8 , KJT) ,< Nay a < t h T has 1 216-al n o n - i s w p h i c Ga-8aturated d e l s of cardinality N, provided that p < N,.

Proof. Let I be (an infinite indiscernible set over A which is) multidimensional over A; and we can assume IAI < KJT), I countable. Choose I $ ( N a 5 p 5 N,, f < NB)y /I$ = p , such that Stp*(I$, A) E stp,(l,A), and tp,(lf,A U uA<pI: U 1 r<Elt) does not fork over A. For any set 8 E 8, = { A : N, 5 A < & y h regular or A = Ha}, let p = min(S U {K,}) and M , be a Fz-prime model over

A U U ( I : : r \ E S U { N B } y [ < NB}.

Clearly M , is a F(fa-saturated model (of T) of cardinality 8 8 , and it is not hard to prove that B ( Z , ) n 8, = S where 8 ( M ) = {dim(J, M): J an infinite indiscernible set, J s M } .

So we finish.

PROBLEM 5.2: In Definition 6.2, can we replace “over some A” by “over 0”.

PROBLEM 5.3: Suppose A , I, J? are aa in Theorem 6.3. Is it necessarily true that for some a, {J; : i < K( 2’)) < EIl.

EXERCIBE 5.4: construct a two-dimensional (b., A = 3 in Theorem 6.8) oounstable, stable but not superstable theory.

EXERCIBE 5.5: In 6.3 replace K J T ) by any K 2 K,(T) (so lAl < K

now).

EXERCI8E 5.6: Suppose A -c By tp(si, A) has no extension over B which fork over A, and tp*(B, A) is stationary. Prove tp(si, A) k tP@, B).

QUEBTION 5.7: Can we in 6.3 get F*,,,-ieOlation?

UE. v, 8 61 QUU"IEE8 AND TWO-0ARDINA.L THEOREMS 289

EXERCIHE 5.8: Suppose I is indiscernible over A. Prove that for some B c A, IBI < K,(T), tp*(UI,A) does not fork over B: provided that I is based on A.

V.6. Csrdinalify-quantifiers and two-csrdinal theorems

We first define a general context, and then get conclusions on the logics with cardinality quantifiers and two-cardinal theorems.

satisfying the following conditions: Let K = (F, W, A, p) be 8 quadruple (we write A = hK, p = FKs etc*)

Condition 1. F is aa in IV, Section 1, satisfying Axioms (I), (III.l), (IV), (VI), (VIII) and (Xl).

Condition 2. W is a set of triples (cp(z, g), $(g), x) where x is a regular cardinal 2 Ax and IAI < x implies that for any F-maximal F-comtruct- ible modelM over A, llMll < x, and ifM is F-maximal, A E M , IAl < x, then there is an F-maximal N , A C N C M , llNll < x. Remark. A model M is called F-maximal, if for no a $ bl, is tp(E, l M ) ) F-isolated; similarly for a set.

Notation. Let CarK be the set of cardinale appearing in W and 00. We say C"(3'xz)cp(z, a) " (or C,, or C, instead C) if for some triple ( ( ~ ( 5 , $), #(Z), x ) E W, C$p]. Clemly this is an elementary property of a. Define induotively C,,(cp(Z; a)) = min{~: Icp(Q, B)I = K < No or there me

$(y; 6), O(E; y; c') such that: O(Z; y; 5) k cp(Z; a) A $(y; 6), cp(3, a) k (3y)O(Z; y; C'), C"(3'ly)#(y, 6)" for some x s K and for each b,: n > 0, C,,-,(O(Z; bl, a)) s K or O(Z; bly a) is algebraic}. Let C(C~(Z; a)) = min,C,,(~(Z:a)),C(p) = min{C(cp(~,a)):p kcp(~; a)},C(a,A) = C(tp(zi,A)) and let for a4 (infinite) indiscernible set (of sequences) I, H ( I ) = SUP(K: if J is an infinite indiscernible set and cp(z, a) E Av( J , a), I=" (3 <xz)cp(z, a) ", x < K then I, J are orthogonal}, For a complete or stationary type p , let H(p) be H ( I ) , for any indiscernible set based on some stationarization of p , let H(a, A) = H(tp(ii, A)). Let C*(p)(z, a)) = min{K: there is a F-maximal model M , E bl, such that for every p , cp(z,a) ~p E 8(1611) implies H ( p ) s K} (clectrly C*(cp(z, a)) s C(cp(z, a)). Let B 4 A (or B 4 A) means that for any a E A, H(a, B) 2 h (B, A a n be replaced by models.) The sup in the definition of H ( I ) is taken on K E C ~ ~ K; similarly elsewhere.

290 MOBE ON W E 8 AND SATURATED MODELS [OH. V, 8 6

Condition 3. For any formula p(x, 7i) C*(p(x, a) ) = C((p(x, a)), i.e., for any F-maximal model M , 7i E 1 M 1 , there is a type r such that p(x, 7i) E

r E 8(l M 1 ) and H(r) = C(p(x, a)).

Condition 4. h 1 p+ + IT1 + + h(F).

Condition 5. I f M is F-maximal IMI c B , N is a F-constructible over B, c E IN1 - IM1 then tp(c, B ) forks over ]MI.

Condition 6. I f M, (i < p) is increaeing, and each M, is F-maximal, then Ut ,, M, is maximal.

In each theorem (Cn,, n,, . . . ) will denote the conditions on which the proof o f the theorem depends:

CLAIM 6.1: (1) For formulas #(x, a), p(x, 8) the following conditione are equivalent:

(i) If 3, 8 E 1 MI then 1#(M, a)l < Ip(M, 8)l + + KO (this we denote by $ J ( x ¶ ~ s p(x, 8)).

(ii) There are a model N, and k , 1 < o, and finite A such thcct in # ( M y a) thre is no ( A , 1)-indiscermible set over Q(M, Z ) of cardinality k. (a, 8 E 1 M 1 , of couree.)

(iii) Ira #(a, 7i) there is no infinite set indiscernible over p(Q, 8). (iv) There are no lnodele M , N , 7iu8 s 1x1 c INI, such that

Q(M, a) = p(N, 81, $(MY a) # $J(N, a). (v) There are M , u 8 E M and finite k , A such that there are no

6, E #(M; 7i) (i < k ) for which tp,(6t, U j, , Ej u v ( M , E ) ) increaee with i and 6, # 6i+1 and ( E = g ) E A .

(2) For every c a r d i d K 1 KO and formula p = p(Z,g) thre ie a set p; such that for every 9, C(p(1; 6 ) ) 5 K iff 6 eatisfie8 sorne #(g) ~ p ; .

(3) I n (2) P: = Unca P:", where (i) Cn(p(l; 6 ) ) 5 K iff 6 satisfies some #(g) E p:",

(ii) p;en is the set of the following formulas: for each #(1; g,), O ( 1 ; y; I ) ad f in i t e q = p;(:;,',,, tznd k < o and <#(x; g l ) , $ l (gl ) , x ) E W, X 5 K the formula saying: there are g,, I such that (VZg)(O(Z, y; 2) -t p(8; g ) A

+(Y; a,)) and (=) (~(z ; -+ ( 3 ~ ) 8 ( ~ ; Y ; 2)) and and (VY)(V q ( ~ , 2) v (3*k1)O(8, y, 2 ) ) (for n = 0, q = 0).

(4 ) p; is closed under disjunctionrr ( m e @Iy: finite disjunction is equivalent to another formula) and contains all formulas ( 3 < k ~ ) c p ( ~ , y). Clearly C(V:, , pi(% 7i,)) = maxi C(pf(3; %)). C ( A pi) 5 mini C(pf) .

OH. v, 8 61 QUANTIFIER8 AND TWO-UARDINAL THEOREMS 291

( 5 ) If e(a; g, a) F V(Z; a) A #(g; 6), V ( E ; a) F (3g)e(z, g; a), C(+(g; 6)) I K and for every d C(e(Z, d , a)) I K , then C(tp(Z; li)) 5 K .

( 6 ) If p E P ( B ) , A E B, p does not fork Over A then C ( p ) = C ( p 1 A )

(7) For every p , for somejinite q E p , C ( p ) = C(A q). and H ( p ) = Wz, t A) .

Proof. (1) Left as an exercise to the reader. (Hint: prove +ii) + ,(iii) * ,(i) e= -,(iv) =j +ii) and, using 11, 2.17 (ii) 9 (v).)

(2) Follows by (3). (3) Easy, by induction and compactness.

(5) By induction on the first n for which Ca((+(y, 6)) 5 K and for

(6) Left to the reader. (7) Left to the reader.

(4) Trivial.

every d C,(O(z, d, c)) I K (this suffices by (2) and compactness).

LEMMA 6.2. For any stationary type p E Sm(A) and formula #(it; a) the following condetiortf, are equivalent:

(i) For every model M , i f A u 8 c ]MI then dim(p, M ) < 1#(M; li)I +

(ii) There is an equivalence relation E = E(Z, g, d ) , d E A u 3 such

(a) i f A U a c B, tp(6, B) i8 any etationarization of p , then 6/E q! acl B

@) for any 6 , 6/E E dcl [#(a; a) u 23 (and by a $zed formula). (iii) There ia a model bl, A u ii E 1611, 61 i s F3mturded, K 2 K J T ) ,

and tp(6,IMl) is a e t a t h d & i o n of p , and N is c-pdme Over 1611 u 6, and $(My a) # +(N, a).

(iv) If I is an indis~rnible set b a e d on p , and is idhernib le Over a, u +(C, a).

(v) not every J c #(a, a) is orth0gona;l top . (vi) There 6 an equivalence relation E = E(Z, @, a), d E A, such that: (a ) FIE 4 acl A, where b realizes p ,

+ x,.

that:

(in Qeq, of course),

then I is not indiscernible over

@) C((3Q)S = @/m 5 Q(+(%m, ( y ) 8Ome 8 t d h r y type q, +(Z, a) E q, i8 not orthogorutl to tp(6/E, A).

Proof. (ii) s (i). Let I be an indiscernible set baaed on p , I E (611, IIl = dim(p, M ) . By (ii), if I = {6:: i < III} then the &/E’s (i < IIl) are pairwise distinct, hence 1{6/E: 6 E 1611}1 2 IIl. By (ii)@) 1{6/E: 6 E 1611}1 < 8, + l+(M, ti)[ + ; so (i) follows.

-(iv)*-(i) If (iv) fails, let M be F:-saturated (choose K =

((Al+ +lTI)+), I U {a} E WI. So in the F,"-primary model over U I U @ U @(M, 81, no non-algebraic q, +(z, a) E q ~ S ( a U +(M, a)), is realized. Choose a set J, indiscernible, over a U @(M, a), 1 E J, IJI > I@(M, @ ) I , and then the F,"-prime model over U J U @ U @(M, a) contradicts (i) (mapping A to M).

~ ( i i i ) =, ~ ( i v ) Let M, 6, N be as in (iii), except that $(M, 8) = $(N, 8). Clearly tp(6, A u 8 u $(N, 8)) is (A u 8)-definable, hence we choose &,,EM, such that 6, 6, realizes the same strong type over A u 8 u U,,, 6{, then they realize the same type over A u 8 u Ute, 6t u #(N,B). So (6,: n < 0) proves +iv) (we wsume w.1.0.g. lAl < K(T)).

(iii) + (ii) Choose z E $(N, a) - $(M, 8) SO tp(z, !MI u 6) forks over lMl hence tp(& A u ii u #(My 8) u 6) forks over A u 8 u $(My a), and let 9(1,6, a,, aa) forks over 1611, where aa E A u li, dl E +(M, a) and l=p[~, 6, dl, d,]. Let dl = d! - qndf, l=@[q, a],

J

We leave the checking to the reader. (v) o (iii) Immediate. (vi) - (v) Immediate (we do not use (vi) (p)) . (ii) + (vi) We work in Cq. Let a = ~l^l i , a, E A, I = {a,: n < U)

an indiscernible set over A, based on A, Go = li. Clearly C($(x, a,)) = C($(x, 8)) so for every n C(V,,, $(x; q)) = C($(x; 8)); and for every n, if En = E(z, g, a,, a,,), then for every 6, 6/E, E dcl[a,, u $(a, a,,)]. For each n let IP = V,wI,n,w,,n-, /\,,Z(~,ji, a,, at); so for every 6, 6/En E d~l[Ul< an - 1 (7iI U #(a, %))I heno0 C((3g)(x = 81E")) C($(X, a)) (by 6.2). By 111, 2.6 for some la, En is dmost over A. By a cosmetic treatment we get an E* which is over A and C((3ji)(x = g/E)) s C($(X, a)).

Remark 6.2A. In (vi) ( y ) we can have q regular and if tp(a,A) is stationary we can have tp(b/E,A) semi-regular (see 4.1).

LEMMA 6.3: (C2) (1) Tire fouowing cimditione on M, A, where I MI c A, are equivalent:

(9 M<"A, (ii) for no E A, 6 E I M l , ~ , $ dtm C(cp(x, 6)) c A, t(3x)#(x, 8, 6) and

$(x, li, 6) I- { ~ ( x , 6) A x # a: a E IMI),

OH. v, 3 61 QU- AND TWO-CARDINAL THEOREMS 293

(iii) for no a E A, 6 E IMI, 'p, $ and x < A doee 1="(3<x)'p(z, 6)", C(~Z)$(Z, a, 6) and $(z, a, 6) I- {'p(z,6) A z # d : d E 1611). (2) Working in Co, if H @ , A ) < A, then for a m formulas 1 =

E ( f , g; 6), cp(x; 6), E is an equitnzlenm re2ation, 6 E A, tp(a', A) I- 'p(f/E; 6), C('p(z; 6) ) < A and

( 3 ) H ( I ) = SUP{K: ijJisindiscemzibZe,q(z,&)EAv(J,a), C ( q ( ~ ; a ) ) + < K t h I , J are orthogonal).

Proof. ( 3 ) Let K be the supremum mentioned in (3). Trivially K 5 B(I ) , so suppose 'p(1; a) E Av(J, a), C('p(S; a) < H(1) ) . By 6.2(vi) and the definition of C, the rest is easy.

4 a01 A.

(2) !hivial by 6.2(vi). (1) By (3), (i) +i) * -@) So let 6 E A, H(6, l M ( ) < A, so by the definition there

is $(z, a), such that a($(%, a)) < A, and some J s $(a, a) is not orthog- gonal to any I b d on tp(6,1M1). Using 6.2(ii) to see what is the requirement on a, we see that we oan wume a E 1611.

By the proof of 6.2(ii), for some and FE WI, $(s, a, c) l -q(x, a) is consistent, but not realized in wl.

-,(ii) * +i) So let 'p(z; 61, $(z, a, 6) oontrctdiots (ii). As C('p(z; a)) c A, for some n C,,('p(z, a)) < A, and we can eaeily prove by induotion on n. that the existenoe of suoh cp, $ oontradiots (iiil.

COROLLARY 6.4: (C2) If M G N , then M <A N ifl 3 EM, C('p(z, a)) < A iwlies 'p(N, a) c 1M1, if E M, I = " ( ~ < ~ X ) V ( X , a))' , x < A i w l i ~

(ii), and trivially (ii) * (iii).

'p(N,@ c 1MI.

LEMMA 6.5: (1) [Cl, 2, 61. If bl 4" A, bl W F - m i ~ ~ t ' d , N k F- oorcetrzcctible over A, thm ikf 4" N. (2) [C2] If A, (i < a) W increming, B <A A, then B <* U,<# A,. (3) [ C l , 2 , 6 ] I f ~ l ~ A ~ a < A ~ d d , t h e r c M l < A M ~ ; a n d i f i < j < 8 3

(4) [CZ] The property H ( p ) 2 A ia preeervd by autcnnurphha of Q

(6) [C2] If for each i < a, H ( 4 , B U Uj<, aj) 2 A, t h B <" B U

(6) [C2] If A <A B <AC t h A <"C, a d i f i < j < 6 A, <"A,

(7) [Cl, 2, 61 If tpm(N1, N,) not fork Over IMI = lNal and M, N, are F d m l , M <" N1, a d M' i 8 F-c.omtructible over IN11 u INS1 tlien Na +M'.

<" Mj, 61 i8 F - d P U C d f i k Over u,<d Mi t h i < 8 * M, <A M.

and by repking p by a p r a l k l type.

U t < a 4.

then A, <" Ujcd A,.

294 YORE ON TYPES AND SATURATED MODEU [m. V, 4 6

( 8 ) Condition (6) lrolcEs for p = KJT) if condition 5 hO&, so we can wave condition 6, and replace condition 4 by h 2 !TI+ + K,(T)+ + W).

(9) [ c l , 41 Any regular p’ 1 h, sat~s$e% cond&%m 6. (10) The condition on x in C2 is equivalent to : if B is F-constructible

(11) [Cl] Any F - d d set ie (the universe of) a d e l . over A and IAl < x, then IBI < x.

Proof. (1) If the conclusion fails, then there are a E 1M1, Q and c E

definition of <”, tp,(A, IMI), tp(c, 1M1) me orthogonal, hence weakly orthogonal contradicting (C5).

IN1 - lMl such that ~.“(~‘*z)Q(z, a)”, krp[c, a], x < h (by 6.4). By the

(2) Trivial, by the definition. (3) Immediate: first part by 6.4, second part by 6.5(1) and 6.5(2). (4) Trivial, by the definition. (5 ) Define inductively BIf such that: M , is I TI + -compact, B E 1M,1 ,

a, E ~iK,+l~, tp*(M,, B u Uj<a Z j ) does not fork over B u U,<, i Z j and BI, + is FpT, + -constructable over IM, I u 4, Ma is PpT, + -constructible over Uf Mf . By 6.5( 4) a@,, I BIf I ) 2 A, hence M , <” Mi u a,, so as 6.5( 1)

for every n c w, i(O), . . ., i(n) c a. Hence we finish. ill, <” Mi+1, 80 by 6.5(3) ill, <” Ma SO H(Z,o>^. * *-8f(,,), 1Mo1) 2 A

(6) Left to the reader. (7) Left to the reader. (8)-( 11) Immediate.

DEBTNITION 6.1: A model M will be called good (or K-good) if a E 1N1, l = “ ( 3 ~ h ~ ) ~ ( ~ , ~ ) ” impliea Iy(iK,Zi)l < A; and it is called h-good if in addition for 8 E lMl ~[C*(Q(Z, a))] s ~Q(BI, a)l. (If condition 3 holds this is equivalent to h [ C ( ~ ( z , a))] s Icp(M, a)l.)

LEMMA 6.6: [C2] M &s god ifa E M implie% IQ(N, 8)1 < C(Q(Z, a)). Prmf. See 6.4.

THEOREM 6.7: [Cl, 2,4,5,6] If M is F-rrwxid and good, 7G a function from Car K to cardinals, h(h) < A, t k n thre i8 a nzodel N , BI < N , N i.9

h-good, F-maximal.

We first prove:

LEMMA 6.8: (1) [Cl, 2, 4, 5, 61 8wppme M is g d , IAI < hK, N i s F- constructible mer lMl v A. Then N is good.

OH. v, 8 61 QUANTIR'IEaS AND TWO-OARDINAL THEOREM8 295

(2) [Cl, 2, 4, 6, 61 8wppoee M , (i < 6) ie good and F - d m l , i < j < 6 * Mi < M,; cf 6 < AK and M* ia F-constructible over uf<d lMfl. Then M* is good, and if E IMfl, cp(M,, a) = rp(M,, 8) for every i 5 j < 6 then cp(M*, a) = cp(M,, a).

(3) [Cl, 2, 4, 6, 61 In (1) swppose in addition that #(z, a) E tp(a, [MI) , A = {a}, and 6 E IMI, cp(N, 6) # cp(M, 6), and let co 5 y ( N , 6) - cp(M, 6 ) . Then there is an equivalence relation E = E(x , y, b,) (blEq$V,6)) U 6 8Wh that I{C/B: C E cp(M, 6)}1 5 l$(bl, a)l and CO/B $ Meq.

Proof. (1) Suppose N is not good, so that is cpo(s, a)@ E INI) such that k"(3%)pO(z, a)" but jcpo(N, a)[ 2 x. Choose N* c N such that IIN*II < x (see Cl), A u a E IN*I, tp(IN*I, IMI) does not fork over IN*[ where IM*I = n IN*I, and M* 41' bl (see 6.4) and N* is F-constructible over PI U A, p*l u A, and N is F-constructible over (MI U (see IV, 3.3) and N* is F-maximal (use C6). By 6.5(7) N* <x+N, but Icpo(N*, ao),)l < x < (cp,(N, a)l, contradiction.

(2) A similar proof. (3) By (C6) tp(co, IMI u a) forks over IN], so let CB[co, a, &],

e(z, a, 6,) forks over [MI, and B(z, y, 6,) k cp(z, 6) where 6, E M , a 5 6,. So let E(z, y, 6,) = (Vz)[#(z, a) -P e(z, z, 6,) = B(y, z, 6,)], Then clearly co/E$Meq, and I{c/E: c ~ p ( M , a ) } l < l$(M,d)l. Using the definability Lemma 11, 2.12, we can get 6 1 ~ ~ ( M , 6) U 6.

Proof of 6.7. We define inductively models a6, (i 5 a,,) and elements a, such that Mo = M , M , is F-maximal, M i + , is F-constructible over Nf U {a,} and Md (6 limit 5 ao) is F-constructible over u { < d b1, and for each i there is $,(z, 4) E tp(a,, pq), pi(M,, a,)] < h[C*(#i(zs m, and C*($,(z, 4)) 5 Waf, IMfI).

Let us prove by induction that Mi is good. For i = 0 it is known, for i + 1 i t follows by 6.8(1). For i = S limit, let = U i < d M f . If cf 6 < hK, aa M , is F-constructible over I Mj I, the msertion follows by 6.8(2). SO suppose cf 6 2 hK, hence by 6.6(9) Mj is F-maximal hence

So if k"(3'xz)cp(s, 6)", 6 E 1Mi1, then Icp(M,, 6)1 < x for each i < 6, by the induction hypothesis hence cf 6 = x . As by (C2) x is regular, and x 2 AK > IT1 [(C4)] there is a < x such that for each equivalence relation 4 = E(z , y, a) a E ?(Ma, 6) u 6 either I{c/B: c E p(dA,, 6))( < x and for each c E (p(Md, 6) there is c' E tp(Ma, 6), kE[c, c', a] or c E

constant for a < i < 8. In both cases we get that is good. Let a.

Md = M i .

cp(M,, 6))1 = x, I{c/E: c E 9 o f a , 6)}1 2 N x ) . so by 6.8(2), (3) c p ( J f , , 6) is

296 MOBE ON TYPE8 AND SATURATED MODELS [UE. v, 4 6

be the first a for which we cannot define Ma + (clearly if A 2 11M11 , and h 2 x for every x E Car K, x < a, T stable in x then a. < A+) . Clearly Ma, is aa desired.

D~~lwrrrow 6.2: If h is a function into h: x regular, and IAI < x , N F-constructible over A implies llNll < x} ; let Kh = (F, Wh, h(h), p),

where Wh = ( ( 9 9 $ 9 h(x)): (cp, #Y x> E w).

LEMMA 6.9: If h is imrm'ng h(hK) 2 I TI + + p+ + h(F) and ae above;

Cg+p(z, a)) and eimilarly for C*(tp(x, a)).

Proof. Trivial.

CONCLUSION 6.10: [Cl-61 If h i8 cc8 in Dejnition 6.2, A(A) = hl(h)+, h ( a ) < 00, then there i 8 a F - d m u l model M such that for every a E ]MI,

Proof. Apply 6.7 to Kh.

then Kh 8&i8@ cn if K 8&&@ c n for n = 1, 6 and h[CK(v(x, ti))] =

h"d~9 a111 = IdW all.

Unfortunately not all the conditions necessary for 6.10 are satisfied in all interesting cases. E.g., F; satisfies C1-6 for K 2 K,(T), but for E", we have difficulties in C1, 3, 5, and for FfTl we have difficulties in C1, 3. Hence we suggest here an alternative set of conditions, denoted by C*i (if it is the same as the previous condition, we do not redefine it). Then we prove a parallel to 6.7 and 6.8 this time ensuring only the existence of a suitable model. We work in P a .

Condition* 1. F is as in IV, Section 1, satisfying Ax(III.l) and Ax(X.l).

Condition* 2 . (a) W is a set of triples (p(z,tj), @(tj),x).

IBI < x; and x > IT1 +A, x 2 A, and x is regular also.

We define C(p(x,a)),H(I), C@), < A as before; and

(b) If x appears in W, IAI < x, B is F-constructible over A then

Condition* 6. At lest one of the following holds: (a) If A, E B (i < p) is increasing and eaoh A, is F-maximal in B,

then &,, A, is F-maximal in B. (A is F-maximal in B if i! E B, 5 $ A implies tp(5, A) is not F-isolahd.)

(b) If A, (i ?j x E C d K) is increasing and contimoue then for some i < x , A, is F-maximal in A,, provided that lA,l < x . Note that we weaken C1, change C6, and add:

m. v, 8 61 QU- AND TWO-OABDINAL THEOREMS 297

Condition* 7. If p eBrn(A), a E A, A = dcl A, tp(z, a) ~ p , C = A n dcl@ u p((Eeq, a)) and p is F-isolated then p C is F-isolated.

Condition* 8. If c E dcl(A u a) - dcl A, tp(8, A) is F-isolated then tp(c, A) is F-isolated.

DEFINITION 6.3: A = dcl A is called good if for every tp and Itp(A, 7i)l < C(tp(Z, a) (remember we are working in F).

LEMMA 6.11: (1) (C*2, 68) If A = {af: i < x}, x appears in R, then there is a closed unbounded 8 c x such that 6 €8, cf 6 = p, impliee {aj: j < 6) is F-?ruz&nd in A; 80 C*6(b) holds.

( 2 ) We can replace in 6.3, 6.4, 6.5 and 6.9 (C2) by (C*2a), (C,) by (C*n) (for n # 2 ) (and get true lemmas).

(3) (C*2) If A is good, IBI < A=, then dcl(A u B) is good. (4) (c*2) I f A, (i < 6) ie increasing, each A, is good and of 6 < hK

(6) (C*l, 2, 6, 7 , 8 ) If A is good, B is F-cawtructible Over A then B ia

E A,

then UfCd A, is good.

good.

Proof. (1) For every 4 < x we define inductively on a s p i, < x, Aa c A, [A"] < x. We let i, = 6 and if A" is define, i,,, is minimal i > i, such that A" E {a j : j < i,+,}, and for limit a i, = Uflcplifl.

Now Aa is a maximal set E A which is F-constructible over {uj: j < ia}, so by C*2 ]Aa] < x. Clearly i, satisfies the demand on 6; so i, ~ 8 , = (6 < x : cf 6 = p, {a,: i < 6) is F-maximal in A}. Hence 8, is an unbounded subset of x. Clearly the closure of 8, prove the lemma.

(2) Easy. (3) Suppose not, then for some E C = dcl(A u B), h and tp, C(tp(z, 8))

s h s Itp(C, a)], so w.1.o.g. we can wume [ A ] = A, and we find A, c A, lAll < h such that 8 E A,, C(&c, a)) s himplies #(A, a) E A,. We can wume that tp,(B, A) does not fork over A,. By Exercise 6.7 we get A , <A A, hence cleazly C, 4 C where C, = dcl(A, u B). So v(C, a) E @,, 8) but Iq(Cl, Z)l s IC,l + 12'1 < A, contradiction.

(4) Trivial. (6) Let JZ? = (A, {ui: i < a}> be a F-construction of B over A,

s.t. A U { a j : j < i} # dcl(A U {a j : j < i}) implies a,Edcl(A U { a j : j < i}), and dcl(B) is not good, and a is minimal under those conditions, so by 6.11(3) a is limit. Let Bo = dcl(B).

be such that C(q-~(z, 6)) = x < Iq(B0, b)l, so by the minimality of a, Iq-~(B',,b)l = x, and for every a < a, lq-~(B~,g))( < x,

Let ~ E B , x and

298 MORE ON TYPES AND SATURATED MODELS [OK. v, 8 6

where B, = dcl d,. Let C = dcl(6 u rp(CQ, 6)), C,, = B, n C; again by the minimality of a fl < a implies IC,l < x , and clearly for limit 6 C, = Us<(lC,. Hence by 6.11(1) (or C*6(b)), as cfa = x , for some 6(0) < a, Cd(0) is F-maximal relative to C,. Let y < a be the la& ordinal such that C, = C,,) (there is one, as for limit 6, c d = U, < d C, and lCol > IC,o)l) and let c EC,+~ - C,. By C*8 tp(c, By) is F-isolated, . hence by C*7 (after a cosmetic treatment of rp) tp(c, C,) is F-isolated; so tp(c, CN0)) is F-isolated, c E C, - cd(o), contradiction to the choice of w. THEOREM 6.12: The parallels of 6.8(2), 6.8(3) and 6.7 (even if we repkcce M by a good 8d) b&, hence abo of 6.10.

Proqf. Ewy, by checking the proofs.

THEOREM 6.13: Let K = (F, W, A, p), K 8dbfi C2, and h > IT]. (1) If F = E, K ~ ( T ) S K < h, p = K,(T) < h t h (a) C1-6 are satie$ed, and ale0 C*1-8, C*6(a), (b) in C2, the Conddth on x is equivalent to: x ie regular, and T ia

&able in arbitrarily large x1 < x . (0) F - d m Z &I equivalent to F-8atUTated. (2) If F = q, &(T) + Kr(T) s K < A, p = K J T ) < A, then (a), (b),

(3) If F = pK, ha(T) + K ~ ( T ) S K < h, p = ~~('ll) < h, t h (a) C1-6 b&, and C*1-8 b&, (b) in C2 the condition on x i s equkxdent to: x > IT1 iS regular and for

(0) F - d d is equivalent to F-eaturded. (4) If F = F b l , h > IT/, p = ITI, and c 2 kg eatiecfied, then (a) C*l,2,4,5,7,8aresatisJied (andC2,4,5)and whenJTIisreguZar,

(or h > I TI +, and we cirange p to I TI +) C6, C*6 too is 8dis@?d, (b) in C2(b) the condition on x is equivalent to: x 2 h is rqu.lar,

min Car K satisfi the condition ae etated and for euey xI < x, x;lT1 < x orx;=(m < X ,

(c) of (1) holds, except possibly (C*7).

every x1 < x, x : " ( ~ ) < x, and AA' < x when A' c A,(T),

(0 ) F--'ml i s equivalent to ITl-compccct.

Proqf. (1) Parts (0) and (b) are quite immediate, so let us prove (a) remembering C1* C*l, C2 =$ C*2, C3 = C*3, C4 = C*4, C5 = C*5, and C*6(a) * C6, C*6(a) A C*2 C*6(b).

Condition 1. Immediate (see Table 1 in IV, Section 2).

OH. v, 8 61 QUANTIFIERS AND TWO-CARDINAL THEOREMS 299

Condition 2. Easy by (b).

Condition 3. Let Af be F-maximal, a E IMI, t(3z)tp(z, a). We can clearly find A E [MI, a E A , IAI < K(T), and r € # ( A ) such that (a) C(r) = C(tp(z, a)), i.e., r k #(z, 6) implies C(tp(z, 8)) 5 C(#(z, 611, (p) no extension of r which is a type over IMI, satisfies (a) and forks

over A. Choose any T,, r E rl E .S1( IMI), which satisfies (a). Suppose H(r,) <

C(tp(z, a)) and let c realize r,, so for some E = E(z, y, 6) (6 E 1Af1) c/E $ lAfeql and for some #(z, 61) (in iKeq), C(#(z, 6')) < C(tp(z, a)), C#[c/E, 611. Choose c' E !MI such that tp(c, A u 6 u 6') = tp(c', A u 6 u 6') and then r u {E(z , c', 6)) will contradict (a) or (8).

Condition 4. Immediate.

Condition 5. See IV, 4.10(2).

Condition* 6(a). Easy, by IV, 4.3(2).

Condition* 7. Let ( p , B) E F, IBI < K , and choose B, E C, lBll < K

such that tp,(B, C) does not fork over B,. By the definability Lemma 11, 2.12 and IV, 4.3(2), it is clear that (P~C,B , )EF.

Condition* 8. If ~(z, a, 6) (6 E A) define c, and r I- tp(si, A), r closed under conjunctions then r' I- tp(c, A) where r' = {(3g)(#(g, 6') A ~(z, a, 6)): #(ij; 6') E r} so the result is clear.

(2) The proof is like the proof of (1) only in the proof of C1, Ax(X,l) we need ha( T) 4 K ; and in the proof of C3 we need ha( T) 5 K (remember that by IV.2,16(1) A2(T) I Aa(T)). Those are the only places we use

(3) The proof is like the proof of (l), only in the proof of C1, Ax(X.l) we need A3(5!') 5 K (remember P ( T ) 5 A3(T) by IV, 2.16( 1) in the proof of C3 we need A3(T) 5 K .

&(T) 5 K .

We use h3(T) 5 K in no other place. (4) Parts (b), (c) are easy.

Condition* 1. Immediate (me Table 1 in IV, Section 2; for C1 Ax(VII1) is missing and Ax(V1) for singular K ) .

Condition 2. Holds by an assumption.

300 MORE ON TYPES AND SATURATED MODELB [OH. V, 3 6

Condition 4. Immediate.

Condition* 5 . By IV, 3.2, N is F-atomic over B, hence tp(c, B) is F- isolated. By IV, 4.3(5), tp(c, M) is F-isolated, contradicting M being F-maximal.

Note that IV, 4.3(5), implies C*6a holds when p > 12'1.

Condition* 6. Part (b) holds, 80 suppose A, (i I x ) is aa mentioned there, and for each i there is Z, E A,, 5, q! A,; and tp(Z,, A,) is F-isolated. Then for each i there is p , c tp(Z,, A,), 1p,1 < I TI such that p , k {Z # E: 5 E m(oA,} where m(i) = @,). If cf i = IT1 there is h(i) < i such that p , is over Ah(,), hence by 1.3 of the appendix there are my j, so that So =

{i < x : cf i = 12'1, h(i) = j,,, m ( i ) = m} is stationary. By C2 there is a F-maximal bl, A,, c ]MI, lldlll < x.

As x 2 A, > 12'1, x and IT( are regular, there is L' E L(T) , IL'I < 12'1, and a stationary 8, G So such that p , is a L'-type for each i €S1. By part b of 0.13(4) it is eaay to check that IAI < x implies 18; (A) [ < x , hence by Exercise I, 2.6 there is an unbounded 82 c 8, such that {Z,: i ~ 8 ~ ) is E-indiscernible over Are, an easy contradiction.

Condition* 7. Eaay by the definability Lemma 11, 2.12.

Condition* 8. As in 0.13(1).

THEOREM 6.13(5). If A > K 2 K,(T), F = F:,p = K,(T), then (a) C*l, 2a, 4, 6, 7, 8 hold and, in fact, C*6a holds. (b) In C2 for regular x > 2ITI, the condition on x is equivalent to

(VX1 < x) (x:""' < x), and for regular x < 2ITI the condition on x is satisjied if

(c) F maximal is equivalent to K-compact. (d) Though C5 is missing, the parallel of 6.7 holds.

Proof. ( 5 ) (a), (b), (c) No new point (we use IV, 4.3(4)). (d) As in the proof of 6.7 we define by induction on i < a. M,,Cr;

such that N , is increasing, M , = M , M t + l is F-constructible over M , U a,, for i limit M, is F-constructible over u,,,M,. By the correct choice of the at's we can ensure M., is h-good provided we have proved each Mi is good. We prove this by induction on i , so by

OH. V, 8 61 Q U ~ T ~ ~ I E B S AND TWO-OILBDW~ZI THEOREME 30 1

Lemma 6.11 the only problematic case is i = S, S limit, cf S 2 A,. Clearly, U,,,M,, is K-compact (by 6.13(5)(c)), henceMd = uj<dMj. SO suppose EM, I= " (3'Xx)q(x, a) " and b(Md, a)l = x and we shall get a contradiction. Clearly, for some jo, EM,^, and for any B,j0 < /3 < S let BB = MB n dcl(@ U ~(6, a)). So pB1 < x. By C*2 over any BB there are < XFf-isolated types. By 6.13(5) (b) there is /3 < 6, cf /3 = K (or cf /3 = y if K is singular) such that : for any FEBd and r s tp(c,B,,), Irl < K , Y < /3, finite d and C E Ba, IcI < A+K,(T), there is realizing r such that

R"(stp(C,Bp),d, 2) = R"[stp(C',C*),d,2]

for some C* E A, C c C*.

singular use IV, 4.3(4)). Continue as in the proof of 6.11(5) using IX, 1.1, IX, 1.lA (if K is

CONCLUSION 6.14: Suppose M E N , M # N , P(M) = P ( N ) (Y-a one place ~ e d i d e ) and P(x) is not algebraic.

(1) If T ie countable, for every h 2 IlNll there is a mode2 N, , N c N, ,

( 2 ) Poreueryh 2 p 2 ITI,tlrereieanzodelN,, llNIII = A, IP(Nl)I = p. (3) If M, or N, is ITI-mwpct, then for every h 2 IlNll there i s N, ,

(4) If M or N is F - d d , and F satis$m C6, l then for every A 2 IlNll

llNlII = A PW,) = P ( W .

N c N, , P(N,) = P(M) and llNIII 2 A.

there is N, , N s N,, P(N,) = P ( M ) and llNIII 2 A.

Remark. Note that C5 deals with F only so it is meaningful to say "F satisfies (26".

Prmf. (1) Let K = (Fko, m, K,, KO>, wy, = { ( P ( Z ) , (Vy)(y = y), x)). C h m c E IN1 - 1611, A c IMl, IAI < K ( T ) such that tp(c, 1M1) does not fork over A ; now add the elements of A aa individual oonstmts of Q, and get a,. In Q, for every model X,, A E 1M,1 henoe there is Y E

S(lM,l) p d e l to tp(c, 1M1) so H(r) = 00 henoe C*(z = x) = 00. So apply 6.13(4), and 6.12. (2) For oountable T it follows d y by (l), for unoounthble T apply

6.7 for R = (F&(m, Wp, I TI + + K ~ ( T ) + , KJ T ) ) and get the result when T is stable in h and y . For the general case apply Exercise VII, 5.1 to the ( ~ ~ 8 8 just mentioned.

holds). (3) Apply 6.14(4) to F = FfrI (by 6.13 the hpthee i s of 6.14(4)

302 MORn ON TYPE8 AND SA!CURATED MODELS [OH. v, 8 6

(4) Define WX, as in (1) and let K = (Fly W$, x, x ) where x = (21Tl)+. Choose c E 1M1 - IN1 and define c,, Ni (i s A) inductively: No = N, Nd is F-constructible and F-maximal model over UjCdNj; tp(ci, Ni) is a stationmization of tp(c, IMI), and N,,, is F-constructible and F-maximal over Ni u {c}. We should prove by induction on i that N + N, and N ( x IN,I u {c,} which is easy by 6.6.

CONCLUSION 6.16: Suppme h ( i ) (i < a) i s an increasing sequenoe of rqular cccrdinala, M a rn.o&l and for every cp(x, a) and i < a there i8 I&, m h that for every B E IMI M C $!!.1,[7i] if Itp(M, B)l z h(i) (and of couree T = Th(M) i s etable). Then for any non-decreaSing sequence p( i ) (i < a) ofcardinale ZITI, Tlrasanzode1N;mhthatforanya~ INIandtp(x, y ) Itp(N, ~ ) l = min(tc(i): N C It",[i%]}.

Proof Left to the reader (use VII, Section 5) . Note that w.1.o.g. I4 G ITI.

THEOREM 6.16: suppo8e T i8 8't+Y8tabkS 11M(1 = A, IP(H)I = p, h > p. suppose ah0 that

(i) A* 2 p* 2 2ITI, (ii) h 2 A*, p 2 p*, (iii) A > h * o r h = h * , c f h > p o r h = h * , c f h s p * .

Then H hae an (elementary) 8ubmOcEel N , IlNll = A*, IP(N)I = p*.

Remark. See 6.17.

Proof. Suppose first that h > A* or h = A* = of A, so by Exercise I, 2.6 there is I c 1M1, 111 = A*, I indiscernible over P(M). Hence I is orthogonal to any J c P(a), so if N1 is the &,-prime model over I then IP(N,)I 1; 2ITI. Now over I there is a Fg,-constructible model N, E M , so there is a F;to-prime model N , over I, N, c N,. Hence IP(N,)I s IP(N,)I < 2ITt, and llN,II 2 111 = A*. If we choose A c P(M), IAl = p*, and N &B Fh-constructible over A u I, N c M, then llNll = A*, IP")I = P*.

NOW suppose h = A*, then let h = Z:(<dh 4, (4 < A), and choose

Uj<i J p Choose I f s J I , IIiI = KO, and choose J; c Ji, IJiI = A,+, J; induotively Ji E lMl, lJil = A,+, J, indiscernible over P(M) u

indiscernible over P ( M ) u U, < J , u U, < It . It is w y to check that J; is indiscernible over P ( M ) u U,+{ J;; and if of A s p*, we can choose N &B any submodel of My which is P~-conatructible over A u UlCcl J;,

CH. V, 5 61 QUANTIFIERB AND TWO-CARDINAL THEOREMS 303

where A is m y subset of P ( M ) of cardinality p* (we leave the cheoking that N eatiefies our conclusion to the reader).

We are left with the case cfh > IP(M)I; and then we can choose JI G J,, 1511 < No and J, is based on u JI and Av(J,, u JI) is stationary we can assume that I J;I does not depend on i, and that for some finite A c M , tp,(JI,P(M) U A U Ul<* Jy) does not fork over A and stp(J;,A) is constant. Then we proceed as in the previous case.

THEOREM 6.17: I n 6.16 we can replace (i) A* 2 p* 2 2ITI by (i’) A* 2 p* 2 ITI.

Pro($. Let F = F&Y Tv = {<P(4, (VY)(Y = y), (p*)+)},

K = (F, w, I TI +, HO),

so the conclusion of 6.13(6) holds and let us work in CQ. Now we can find I E M y 111 = A* and Z E M such that I is independent over (P(M) U Z, Z) (for A > A* or A regular use a variant of 111, 4.23, in the other cases work analogously to the proof of 6.16 and replace Z by a set).

Choose M, < M y IIMoII = p, Z u P ( M ) E IMol, and w.1.o.g. I is independent over (lMol, a), so it is easy to show a E I 3 H(a, Z) = co.

Choose Ml < M,, C E M,, IlM,II = p* = ~ ~ P ( M l ) ~ ~ y and I is independent over (lM1l u P ( M ) u Z, a). Let A = dcl(lMll u U I), so A c H, A = dcl(A), we shall prove that A is good, and this suffice (take N s M F-constructible over A, so by 6.11(6) N is good, so

IP(NI hence IP(N)I = p*,andA E N * 2 IAI = A*,andclearly IlNll s IAI + IT1 = A*).

Suppose A is not good, so for some si E lMll u U I , C(&, a)) s p* but Itp(A,si)l > p* (see 111, 6.3(1)), and choose a finite I, E I, s i ~

1M11 u U I,. As a)l > p* 2 l(lJfll u U I , ) ] + IT!, there is c ~ q ( A , a ) , cEdcl((M1l U UI)-dcl(lM,I U UI,). By changing if necessary I,, we can assume cEdcl(lM1l U I, U {6})-dcl(lMll U UI,) for some ~ E I . SO clearly tp(6, I M , I u U I ~ ) , tp(c, wlI u UI,) are not orthogonal (as they are not weakly orthogonal and the first, not forking over M I , is stationary). But the first, being a stationarization of stp(6, IMll) E stp(6,M0) satisfies H(6,1MlI U UI,) = 00, whereas by the choice of c,

p*, by P ( N ) = P(A) 2 wfl), 80 IP(N)I 2 IP(NlI 2 p*,

H(c, IMll u u 1 0 ) C(P)(2, a) 5 (p*)+

contradiction. Note that instead of 6.13(6) we can use IX, 1.2, 3.

304 MORE ON TYPES AND SATURATED MODELS [OH. v, $ 6

Remark 6.17.4. The superstability assumption in 6.16 is necessary, counterexamples can be built as in IX, Exercise 1.2.

QUESTION 6.1: Try to eliminate the stability assumption from the general presentation.

EXERCISE 6.2: Generalize the theorems in this section to the c&88 which deals with dimensions of indiscernible sets instead of cardinalities of sets v ( M , Si). That is suppose not W is given, but a function C, whose domain is a family of indiscernible sets, and C ( I ) is a infinite cardinality aa in Condition 2. Assume that if C(I ) , C(J) are defined and distinct then I , J are orthogonal. Let H ( I ) be the supremum of A such that H ( J ) < A implies I , J, am orthogonal; and define H(p) , H(a, A) and B <A A aa before; and C*(I) parallelly. M is good if for every I E IMl, Av(I , / M I ) is F&,,-isolated when C ( I ) is defined, and h-good if dim(I, M ) 2 h(C*(I)) in such caw.

Generalize also Exercises 6.10-6.20.

QUESTION 6.3: Try to weaken the demands we mention or get naturally in Exercise 6.2. Alternatively, weaken the orthogonality demand to { J : C ( I ) < C ( J ) } g W I (see $5) .

EXERCISE 6.4 :Prove that in 6.16 assumption (iii) cannot be deleted.

EXERCISE 6.5: Adapt 6.7 to deal also with I{a E M : a mlizesp}I for “small” types p over lMl (.e.g., F = pIch, A regular TK(!P), IpI < A).

EXERCISE 6.6: There is a countable No-stable theory T such that (i) T haeamodelM, IlMll > IPMI 2 KO. (ii) T has a model M , such that for no N M c N , M # N , P ( M ) =

P(hT). [Hint : InL( T) there are only an equivalence relation E and a one place prediccLte P.]

EXERCISE 6.7: Prove 6.2 for not necteeearily stationary p, when SiEA.

EXERCISE 6.8: Prove that in 6.14(1) the countability of T is necessary even when it is superstable.

OH. v, o 71 RANKS REVISITED 305

R" = w + l ,

= P" U Q" U R", Pr = {w:v Q 7 or 7 4 U E " ' ~ } ,

E is a three-place relation E RN x RN x Q",

EN = { ( n , B , ( p , { n , P } ) ) : n : n < P G w , 7 Q p E "21,

Ff is a partial one-place function,

F,N((p,{n,B})) = p r k when k is maximal such that p r k 4 v.

Using the automorphism of saturated models we can see that T is superstable. Let M be the submodel of N with universe

PN u w u U "2x{{n , /3) } . n < p < w

Then M, N is a counterexample.]

Remark. On categoricity for logics with generalized quantifiers, see the end of VI, $6,

We use the convention of Section 6, and A = A;, be a function aa in Definition 6.2, K = K". For simplicity we deal with p(z; jj) rather than 94% !i)*

DEFINITION 7.1 : We call q(z; 8) K-algebraic if C(p(z; Ti)) < 00; we call p(x; a) K-minimal if it ie not K-algebraic but it hae no two contradictory extensions which are not K-algebraic; we call p(z; 8) weakly K-minimal if it is not K-algebraic but whenever #(z, 6) forks over 8, p(z; a) A #(z; 6) is K-algebraic; and we say p(z; a) sK #(z; 6) if for some x for every K-good model bl 3 8 u 6, Ip(bl; a)( 5 ($(bl; 611 + x.

306 MOEE ON TYPES AND SATUBATED MODELS [a. v, 5 7

CLAIM 7.1: (1) The only information f r m K needed to Dejinition 7.1 is

wg = {(dx; 9)s $ @ ) ) : f o r some x (dx; 91, $(@, x ) wK}

(provided that K satisfies C1, 2, 4, 5, 6) for 5, being aa we want. Alterna- tively we can define ~ ( x ; a) I , $(x; 6) by: in (Q, 6 ) i f we add to W, ($(x; g), 9 = 6, Xo), a d get K 1 , ~ ( x , a) is K,+ebraic.)

(2) For every set Wa = {(tp,(x; &), $,(&)): i < a} we can jind a K satisfying conditions 1-6, such that Wg = Wa. So henceforth we aamme

(3) I n particular, for the empty W", in (2) we get the trivial K, which we denote by KB. So K,-algebraic, KB-minimal, weakly KB-minimal, ~ ( x ; a)

$(x; 6), means algebraic, minimal, weakly minimal, p(x; 3) I $(x; 6) respectively; and for KB, pgcs,o, = {(Fkx)p(x; 9): k < KO} (see 6.1 (2)).

(4) If p(x; a) is weakly K-minimal, the dependence relation on V(Q, a) " b depends on {bl, b,, . . . , b,} iff H(b, {b,, . . . , b,) u 3) < 00 (or equivalently, C(b, a U { b l , . . ., b,}) < 00) satirrfies the axbm.9 of dependence relation mentimed in 1.14 with transitivity ale0 for non-inde(pendentsets.

(6) If p(x; a) ie weakly K-minima2 and hp[c; a] and H(c, 8) = 00, then stp(c; iZ) M regular. Tire family B = {p: p a K-minimal type} is regular and ~ ( x , a) is @-regular (see DeJinition 4.5).

(6) For every Q = Q(x;%): p(x;a) is not K-algebraic iff ii realizes { - y ~ ; p ~ % } where p t = pv =del{rp: ( 3 ~ ) q ~ p ; } see 6.1(2)).

(7) The formula p(x; ii) is not weakly K-minimal nor K-algebraic iff there are f m u h #(x; 6,) (i < lT l+) such that:

K Satisfies C1-6, h k > (2'IT')+.

(i) $<x; 6,) Q(X; a), (ii) #(x; 6,) is not K-algebraic,

(%) for i < j < l!Z'l+, $(x; 6,) A #(x; 6,) is K-algebraic. ( 8 ) ~ ( z ; a ) G K $ ( z ; 6 ) iff for any F:r,,,-saturated modeZM<,"N, if

ii u b E 1611, $(N, b) c 1M1 t k n p(N, a) c 1611.

Proof. Easy, hence we leave it to the reader. (For (7) use 11, 3.12.)

DEFXNITION 7.2: (1) We define Dm(p,d, K ) as Definition 11, 3.2, but Dm(p, A , K ) 2 a + 1 iff for every A and finite q E p , there me m- contradictory #,@; a,) for i < A such that:

(9 D"[q u {$@; a,)}, L, KI 2 a, (i) I = {a,: i < A} is an indiscernible set,

(iii) H ( I ) = 00,

(iv) #,@; 9) E d.

Equivalently, there are a K-good M and pairwise explicitly contradictory r, (i < A) such that q, r, are over IMI and Dm(q u r,, L, K ) 2 a. W e define Mltl(p, L , K ) similarly.

( 2 ) For a stationary type p, let wK(p) = 2 flow&): q a regular stationasy complete type, H(q) = GO) and wK(a, A ) = wK(stp(a, A) ) .

(3) W e call cp(x; 8 ) K-weakly-minimal i f D[cp(x; a), L, K ] = 1. The following claim sum up the trivial facts concerning Definition 7.2.

CLAIM 7.2 : ( 1 ) The two ver&olze of DejEnition 7.2(1) are really equivalent and the parallels of 11, 3.4 and 11, 3.5 holds. If M is K-eaturated, K ( T ) 5 K , A c 1M1, IAl c K , p is over A not K-abebraic, then for some q E Sm(M) extending p, H(q) = a, D(q, L, K ) = D(p, L, K) .

(2 ) If T is &uperetable, wK(p) b dway8 defined, Jinite, and S w ( p ) ; and it eatieJies the parallels of 3.10, 3.1 1 , 3.13 and 3.16 when " I independent over A" is replaced by " I K-independent over A" which ~ n z e a n s I ie independent over A , and t p ( 4 A ) ie not K-a€gebraic for any 5 E I .

(3) If K ie trivial, wK(p) = w(p) , and D(p, L , K ) = D(p, L , a). (4) If ~ ( x ; 6 ) ie K-weakly-minimal, b [ b ; a, then w,(b, 6 ) = 1. Ah.xzye

(6) Any weakly Kg-minimal fonnula ie K+ueakly-minid, but nd wnverady. However for trivial K , a fornzula ie K-weakly minimal iff it ie weakly K - m i n i d iff it w weakly m i n d .

(6) f l w e ( i ) O ( X , 9; 2) t- cp(z; 6 ) A $(S; 61,

(ii) c p b ; 6 ) (3%)8(x, g; a), (iii) for euey a, O(x, a, a ) ie Kdqebraic.

Then D[cp(x; a), L , K'j 5 D[$(Z; 6 ) , L , K].

Proof. Easy, so we prove only (6).

(6) Now we prove b y induction on a that (for every suitable cp, $, 8, Z?)

(*I Dm[+; a), L, K ] r a implies D"'[$@; b), L , K'j r a.

For a = 0 or a limit there is no problem, so suppose a = p + 1, and (*) holds for 8.

Let p be any cardinality, and we can assume it is regular and bigger than 2", x = X, (the supremum o f all cardinalities x appearing in W,). So there are explicitly contradictory types p, (i s p) over a K-good

308 MORE ON TYPES AND SATURATED MODELS [OH. V, 5 7

model M, euoh that cp(x; a) E pi, and D(pi, L, K) r 8. We osn assume pi is closed under conjunotions, and let

(eo olearly qi(@ k #(g; 6)). It sufiioes to prove: (a) Drn(qt9 LY 2 p. (b) The union of any p qiYe is oontradiotory,

be&uee then by 7.2(1) we om find A E 1 M 1 and q:, q, c q: E @(A), D(q:, L, K) = D(qi, L, K), hence I{q:: i s Cc)l = p. A8 this holds for every suitable p, this implies Dm(+(g, b), L, K) 2 p+ 1 = a by the parallel to Th. 11, 3.11, Ex. 11, 3.27.

Proof of (a). It euffioee to prove that every finite subtype of q, haa degree r p , eo aa pi is o l d under conjunations, it eutlFioee to prove that for every TI(%; a,) E pi -[(35)[e(x; a; a) A y1(x; a,)], L, gl r p. we ~ u m e cpl(x; a,) cp(x; a). Now cpl(x; a,), (3~)[9(~; #; 5) A a,)], O(X, g ; a) A cpl(x; a,) satbii8~ the hypotheeia (for V(X; a), #(g; 6), e. (g ; a) map.). Henm by the induotion hypotheeis

Proof of (b). If not let a* recllizes p q,'e, no by notational ohange we osn aaeume it E M (6.8(1)) and satisfies all of them. So for wah i f i u {8(x, d*; E)) is finitely satiefhble in M; hence it definea a filter D, on 8(M, a*, a), generated by {a E B(M, a*; a): hpl[a, a,]) for cp,(x, a,) €pi. Let le(M, a*, E)( = X , eo 2'- p, 80 for some i # j D, = D,, but then p, u p, is finitely satidlabley ctontradiating their choice.

LEMMA 7.3: (1) Bwppoue p(x; a) s, #(x; 61, then there w a fopmula e(x, 9; 5) BW:A thad:

(i) e(x, Y; 5) k cp(x; a) A #(y; 61, (ii) v(x; a) k (330e(x, y; i),

(iii) for each a ~ c p ( 6 , a), except < IT1 of them, there is b euch that C8[a, b, 07 and 8(x, b, c') forb over a.

(2) If in addition cp(x; a) ie toeakiy K-minid , then for come el(y, Z) E

PO(,;,,^, t= ( V Y ) ~ , ( Y , Q . (see 7.1(6)). (3) If in (I), in addition #(y, 6) b weakly K-minimal, t7m for eonte

ea(Y, Z) E Pe,,cy;z,#p C(VY )ea(Y, a), ~ o ( Y ; X, 2) = e(x, y, f 1. ( 4 ) There ia a set p(Z; a) 8w:h thad cp(x; a), #(x; 6) eatbJies the con-

OH. v, S 71 RANKS BEVISITED 309

c l d o n of ( l ) ( i ) , (3) iff iin6 satiejkz at leuat one formula of p(Z, g). 8imihrly for (2) , and for (3), and for their combination.

Proof. There is no new point (see 7.1(8) and the proof o f 6.2).

DBIFXNITION 7.3: An n-witness (or just a witness) for ~ ( x ; ii) sK #(x; 6) is a sequence 8 = (O,(x; ji,, gl, a): 1 s n ) such that -

( i ) PI = (yo,. . ., Y , - ~ ) , 2, = (zO,. . ., z ~ - ~ ) , E = iihb, (ii) ke0(x, go, ZO, a ) m ~ ( x ; a),

(iii) e,+,(x, g,+,, z,,,, a) t- e,(x, g,, $, z) A #(Y,, 8) A el(zl, g,, 81, a), ( iv) 8,(x, g,, Z l , Z ) k ( ~ Y , ) ( V ~ I ) [ ~ I ( ~ I , #I,%, 5) + e,+l(x, jit,,, Z ,+ l , a )] , (v) for some e*(gn, 2,s 2 ) '=(V&, Zn)e*(gn, Zn, a).

LEMMA 7.4: ( 1 ) If 8 = (8,: 1 s n ) is an n-witness for ~ ( x ; ii) s, #(x; 6) , then for any a, b, (8,: 1 r n - 1) is a witneas for ~ ( x , ii) A

el(%, b, a, a) sK #(x; 6), where

(so yt take the role of yt + ,). (2) ~ ( x ; ii) sK #(x; 5) iff there is a witness for it.

Proof. ( 1 ) Trivial. (2) Eesy, by 6.l(v) and the dehability lemma.

LEMMA 7.5: (1) 8wppocre q(g) is an rn-type, ~ ( x ; ji) a f m u l a and for every ii realizing q, ~ ( x ; ii) 6 ,not K-algebraic; and for some &, for every h,-goo& M

{lv(M, Z ) l : a E A2 realizes q},

has cardinality c A, < Ag. Then there are # ( j i ) E q (msuming q i s closed u d r con junc th ) and k < KO 8 ~ & h that for every good M , {Ip(M, ii)l: M C #[ii]} hm cardidity s k, and C#[ii] implies ~ ( x ; ii) G not K-algebraic.

( 2 ) I f i n ( I ) , whenever ii realizes q, ~ ( x ; ii) is weakly K-minimal then we can assume:

( i ) E(Z, ji) ia an equivalence relation on #(Q), E ( f , y) k # ( f ), (ii) E has finitely many equivalence claeses, and i n each of them there

is a sequence a realizing q, (iii) i f k#(a) A $(6), then CE(2iY6) i$ in any hK-good M 2 ii u 6 ,

Iv(M9 811 = IdM, 6)1, ( iv) there are 8(x, y, Z ) , 81, 8, such that if E(8, 6) then for some 5,

7.3(1)(i), (ii), (21, (3) bid8 (with 1,4 = p), (v) C#[ii] implies y(x; ii) is K-weakly-minimal,

310 MORE OW TYPES AND SATURATED MODEL8 [OH. v, 8 7

(vi) if K is trivial, R[q(x; si), L, A] = 1 for every a realizing q, then c#[a'] impziea R[q(z, a'), L, A] = 1.

Proof. (1) Let for every a, r, be the union of the following sets of formulaa :

(a) { q m , 3,): n < w, i < a}, (b) the set of formulas saying: {z?: n < w } is a non-trivial indiscern-

(c) the set of formulas saying H({z;: n < o}) = m, (i.e., for each formula # = #(q g ) and 8(@) cplr and i < a let n(0) be as n in 11, 2.20 and we have the formula

ible set over u,<, dQ, @,) u u,<, g,,

(VP)[ e(g) -+ V A +(ZT, g)], w E anco, + 1

Iwi > n(0) nEw

(dl q(!i,f)- Clearly if bl is an h,-good model, a, (i < n) realizes p, and Iq(M, ao)l

< Iq(M, sil)l < - . . < Iq(M, Z,,-l)l, then r,, is consistent. It is also clear that if each F,, is consistent then each Fa is consistent. Lastly, it is clear that if r, is consistent, A < A,, we can find a model M and a,, I, (i < A) such that si, realizes q, I, E q ( M , si,) is indiscernible over U,<, q ( M , a,) U U,<, a,, H(I , ) = 00. Then we can contradict the hypothesis by A = A,,, and we find # easily.

(2) Let us defhe - 6 if for every Is,-good M 2 a u 6, Ip(M, a)l = lq(M, a)]. First we prove {a/-: iz realizes p} is bounded, in fact < Ao = (2a1Ti)+ < A,, otherwise there are a, (i < (219+) realizing q, pairwise non -equivalent. By I, 2.8 we can assume {a,: i < ( 2 2 i T 1 ) + } is an indiscernible set. As not sio - a,, and tp(sio-sil, 0) = tp(silnBo, 0) for no i # j does q(s, a,) S, q(s, a,), hence there is an indiscernible set I{ E q(Q, a,), H(I{ ) = 00, I{ indiscernible over a, u si, u q(Q, a,). By Erdos-Rado theorem (2.6 of the Appendix) (as the number of J E

p(Q, a,), H ( J ) = 00, up to equivalence, is 5 21Tl) we can aasume that for a < f l < y 0), and by this we get a contradiction to the hypothesis by 7.6(1). Let si, (i < a < A O )

be a maximal family of pairwise non-- -equivalent sequences realizing it. Let r = r(Zl, 4) be such that p(s, El), p(s; 6a) satisfies the conclusions of 7.3 (1) (i), (ii), (21, (3) iff satisfies at lemt one formula of r. Let

ri(%, {$I(%; 2,) A #a(%, 31): #i@i, %a), #dzi, %) E r};

OH. V, § 71 RANKS REVISI!CED 31 1

so if 6,, 6, realizes q, 6, - 6, then 6,6, satisfies at least one formula of rl. Hence q(Z) u {-,$(Z, af): i < a, $(Zl, Z,) E r,} is contradictory, so there is a finite, contradictory subset. Clearly it involves only finitely many at’s so a < o and let k = a, and we can replace q by some $(if) E q (or $ = A ql, q1 c q, q1 finite). Also only finitely many formulas #:(a,, it,) E rl(Zly Z,) me involved. Let

E(%, 32) = $@I) A A (%)[v $%I, @ A v $%%Y g ) ] . n n

Now if CE[6,, 6J, then for some n(l), n(2), 8, C #$&, 4 A $n(aJ6a, 81,

(i) cp(z, 6JY cp(z, a) are both K-algebraic, or for every A,-good M 2

(ii) similarly for cp(z, 6,), cp(z, 3). Hence a similar conclusion holds for ~ ( z , 6,), cp(z, 6,). Now if k#[6,] there me ai and n such that k#:[6,, 4 3 , hence cp(z, 6,) is

not K-algebraic and for every ir,-good iK, Iq(M, &)I = Icp(iK, S)l. Hence if#:[6,, at] A #@,, 43, i # j then k,E[6,, 6,].

So clearly 1 is an equivalent relation on #(a), and a,, . . . , a,,-, am representatives, and k8[6,, 621 iff C$[6J A $[6,] and for every hK-good ,iK 2 6, U 82, lcp(iK; 6,)l = Icp(iK; 6.JI. So 7.6(i), (ii), (iii) holds and (iv) is proved similarly. For (v) use 7.2(6) and for (vi) Exercise 11, 3.32.

henpe :

6, u a, Icp(M, 6,)l = lcp(Jf, q,

-

LEMMA 7.6: (1) 8 q g w e that for every realizing q cp(x; a) is weakly minimal, and R[cp(s, a), L, A] = 1, Mlt[cp(s, a), L, A] = p and l{ltp(iK, a)l: a realizes q, a E Af)1 s A, for 80me &,. Then there are #, E a8 in 7.6(2) (for K = K,, A, > A,,) and k WA that in addition:

C#@] i m p l i ~ R[cp(z, a), L, A] = 1, Mlt[cp(s, a), L, A] 5 pk.

( 2 ) Suppose in addition that (i) for a realizing q, ~ ( x ; a) is minimal,

(ii) (a) !=+[ti] and rpl(x, al) l-rp(x; a) implies {lrpl(M, &)I : & E M realizes tp(al,pl)} is bounded or

Then there are formulas @) T doee nd have t k f.c.p.

(a) !=t,hl[6] impZies vl(z, 6 ) i8 m i n i d , (b) the fornculcG

cpl(x, 9,) ( I < n) m h that:

( 3 ~ 0 9 * * .s R-1)[ A #I(R) A (v~ ) (cP(~ , g) V ‘~1(z, R))] I I

i.9 wnahtd with #(@.

312 MORE ON TYPES AND SATURATED MODELS [CE. v, 8 7

Proof. Left to the reader.

THEOREM 7.7: (1) Buppose for a m A, < Ax for every hx-good nzodel M {Irp(M, a)l: 6 E iK, rp(x, a) not K-algebraic} lras cardinality < A,. Then for a m formula #:,for every 6, cp(x, a) iS K-algebraic iff I=#@].

( 2 ) 8-e the hypothe& of 1 ?told% for every 9. 8 u m e rp(x, a*) iS K-minimal, T totally tranecerrdenta. Then in addition to the conclueion of 7.6(2) (for q = 0 ) ) there are rpl(x; g1), El, #ly &o eatiafving the wn- cluSione of 7.5( 2) and

(v') h,h1[8] imp&ee pl(x, 8) ia K-minimal, (vii) if 1=#~[6] then for 8 m 6, C#[G] and Q ( X ; a) sK rpl(z; 6) sK rp(z; a);

and conversely for every such a there iS euch a 6.

Proof. (1) Let q(z,g) be a formula, and q be a type such that q ( x ; a) is not algebraic iff 6 recllizes q (see 7.1(6)). Now apply 7.6(1), 80 we get +(g) = A ql, q1 c q finite, and + aatiafiea our demands.

( 2 ) Let $ be as in 7 .5 (2 ) . We now define by induction on n < w , formulas rp,,(z, g) for q E ">2, such that, letting Cp"(z; y") = rpv = rp(z; g) A At<iCn) ~ n l t @ ; ~ ~ l t ) r r r J for 7 E ns2:

(*)

which by (1) is equivalent fo:

(*')

~(3g)(3.. .g,,. - -),,sn>ap(gy a*) A ttcnza A $WV]

There are 6, a,, (q E ">2) such that CE(& a*) and for each q E "2 r p ( q 6) A AIet(,,) rpnll(z, iZ,,ll)*[ll is not K-algebraic.

If we can dehe them for every n, we get a contradiction to the total transcendency of T. So for some n Q,, is defined for q E ">2 but we cannot choose for 7 E "2. Choose a maximal set S E n2, such that

(**I) there a m 8, a,, (q E "'2 u B) such that E(6, a*) and for every 7 E "2 u {qn<i): q €8; i E (0, l}},

d z ; a) A A ~ ~ l i ( ~ ~ ~~ l i )wc i l 1 c Kn)

is not K-algebraic.

B exists aa "2 is f i f e by the choice of n, 8 # "2, so we can choose V E " ~ - S , and let e*(g",g*) be (3 . . . ~~ . . . ) , J [E(g ,~*) A &EJ7&g)],

NOW let #l(gv) = (3g*)[e*(r, g*) A #@*)I, rpl = pv; we leave the whereJ= "<2 u #-{vrZ;Z < n}.

rest to the reader.

m. v, B 71 RANKS REVISITED 313

THEOREM 7.8:8tqpose that (i) e(z, y, a) I- d z , a) A #(?I, 61, (ii) ~ ( 2 , a) I- (3y)8(~, y, a), (iii) B = R[#(z, 61, L, KO], (iv) for every b, R[8(z, b, a), L, No] I; a.

Then R[cp(z, a), L, No] I; a(@ + 1) or a = 0, R[cp(z, a), L, No] I; 8.

Proof. We can aasume a > 0 as for a = 0 the conolusion follows by Exercise 11,3.32 (or 7.2(6) forKg). We prove the assertion by induction on f l for all cp(z; a), #(y, 6), 8(z, y; a) and a.

For B = 0 #(y; 6) is algebraic (by Exercise 11, 1.3), so let #(a, 6) = {bo, . . . , bn}, hence cp(z; a) I- Vlsn 8(z, b,, a), hence by 11, 1.7

R[cp(z, a), L, KO] s max R[8(z, b,; a), L, KO] ?j a s a(0 + 1). ISn

So suppose p > 0, and we have proved for every 8' < 8. We can assume Mlt[#(y; 6), L, No] = 1 (otherwise we can find n c No and

= 8, Mlt[#,(y; &), L] = 1, we use the assertion for cp(z; a) A (3y)

[e(% Y; a) A #i(Y; FJI, #dY; 61), Y, 5) A #i(Y; 6,) instead V(z; a), #(y; 6), 8(z, y; c') resp., and use 11, 1.7:

R[cp,(z, a), L, No1 = max R[v(z; a) A (3y)(o(z, Y; a) A #&, h)), L, KO]

By Exercise 11, 3.21 it is easy to prove:

CLAIM: Rm[cp(z; a), L, No] 2 y iff there are formulas cp,,(z; a,,) for 7 E ds(oy) (see Def. 11, 3.3) such that

#dy; 6,) (I < n) such that M Y ; 6) = v , < n M y ; Q, R[#,(?/; 611, L, No1

I<n

(i) Q()(z; a()) = d z ; a), (ii) for v 4 7, cp,,(z; a") I- cpv("; a"),

(iv) for 7 E h(coy), VWcp,(z; a,,).

(iii) if 7, u ~ d s ( w y ) are incomparable by Q thenq,,(x; a,,), q,(Z; a+) are contradictory,

Suppose R[q(z, a), L, No] > a(p + 1) = a/3 + a, and we shall get a contradiction. Let y = ap + a + 1, and choose q,,(z, a,,) for 7 E &(coy) as in the claim, and let

Notice that R[cp,,(z, a,,), L, No] 2 y,, where Z(7) > 0, coy,, I; q(Z(7) - 1) < w ~ , , + w , or l(7) = 0 , y,, = y = ap+a+ 1. Hence y,, > ap implies R[#,,(z, &,), L, KO] = p (if it is < 8, by the induction hypothesis yn s R[cp,(z, a,,), L, No] 4 a@' + 1) I, ap contradiction). Now as Mlt[#(z, 6), L, No] = 1, there is a unique q €#(a u 6 u U {a,,: 7 coy)})

#,,(Y, 6,) = #(Y, 6) A (W(Q2, Y, a) A P"(", a,,)).

-


such that #(z, 6) E q, R(q, L, KO) = 8, and let b realize q. So if 7 E

&(coy), y, > a8 then #,(z, 6,) E q hence k#,[b, 6,], therefore v,(z, a,) A

e(z, b, 8) is consistent. If y, > a@ implies v,(z, a,) A O(z, by 2) is not algebraic, then by the "if" part of the claim,

R(v(>(z, a( >) A e(z, by a), L, KO) 2 a + 1

contradiction. Hence there are Y E &(wy), k < W ; yv > ap, k ( 3 s k z ) [vv(zy a,) A e(z, by a)]; hence +*(z, a*) = (3 ~ [ v ~ ( y , av) A e(y, z, a)] E q.

RCvv(z, a,) A (32)[8(z, 2, 8) A #*@, 2*)1, L, K O 1

By Exercise 11, 3.32 and monotonicity,

5 "*(% a*), L, KO1 5 R[@(x, 6), L, KO1 5 /9 s a@.

As p' =defR[@(x, 6) A i@*(x , e), L , KO] < /3, by the induction hypothesis

R[vv(x; 7iv) A (32)[8(2, 2, a) A +*(z, a*)], L, KO] s a@' + 1) s 4.

So by 11, 1.7, R[qV(z, aV), L, KO] a8 < yv, a contradiction.

LEMMA 7.9: Let cp(x;Z), @(x;6), O(x, y , q be as in 7.3(1) and assume

(1 ) If D[@(x, 6), L, K ] = B, and for every b D[O(x, b, c), d , K ] < n 5 = Y€d.

then D[v(z, a), d, K] s n(/3 + 1) except when n = 0, and t h

(2) If in (1) /3 = k c w, then D[q(z; a), A , R] I n + k. D[v(z, a), 4 RI 5 8.

Prmf. (1) Using 7.2(1), (6) we can w u m e n > 0, 8 > 0, and prove by induction on 8; suppose we have proved for every g < 8, but D[&; a), d, R] > n(/3 + 1) = n/3 + n, and we shall get a contradiction.

V 6 u 2 E 1M1, mi < W , p i ~ c l , d , Z , , E ] M ~ for ~ E I x , 0 < 2 s n, x = xg, such that :

So we can find an hK-good F&,-saturated model My

(i) pi+l(zy (ii) D[v*(z, a"), A , R] > np where r) E "x,

(i < x ) are m,-contradictory for 2 < n, 7 E ' x .

Q*(G w = ~ ( z , a) A

By 7.2(1) we can h d a model N , M <a N , and E N , (V7 E " X ) D[tp(a,, ]MI) , d, K] > nfl, Cv*[a,, an], and b, E INI, Ce[a,, b,, a]. If for some 7, D[tp(b,, (MI) , L, K ] < 8, we get a contradiction by the in-

A ~pi(z, a WIi), 0 < i s n

OH. v, f 71 RANKS REVISITED 315

duction hypothesis. Hence the number of possible tp(b,, IN]) is <x, and x is regular, hence by 2.6 of the Appendix we can assume that for every r ) E " x , q = tp(b,, IN[); hence we can msume b, = b. If for each r) E "x, v*(z, a") A 8(z, by a) is not K-algebraic, then D[q(z, a) A O(z, b, a), A, K] > n contradiction, hence for some r ) it is K-algebraic, so there is 0*, CB*[b,@, a] and CO*[b',a',Z] implies v*(z; a') A

B(z, b', E') is K-algebraic. Let

v+(z; a+) = v*(z, an) A (3y)[e(z, y, a) A e*(y, a", a)], #+@; 6') = I/@, 6) A 8*@, F, a),

e+(z,y,a*) = ~ ( z , y , a ) A p+(z,a+) A #+(Y, 6 + ) .

Clearly v+(z, a+) E tp(a,, 1611) hence D[v+(z , a + ) , A , K] > n/3, and clearly D[#+(z; 6+), L, K] 5 D[#(x, 6), L, K] = 8, and for every b' D[O+(z, b', a+), A, K] = 0; and thus our hypotheses me satisfied; contradiction.

(2) It is easy to prove that if D[v(z, a), A, K] 2 1 c w , then for some fhite A' E d, D[rp(z, a), A', K] 2 1. Hence we can assume A is finite; so for every formula vl(zs &) and 1 c w there is a type qZ(yl) such that D[vl(z, Bl), d, K] 2 Z if€ iZl realizes &(cpl). We now prove the assertion by induction on k; and the cams rn = 0 or k = 0 are easy.

So suppose D[v(z, a), A , K] > n + k, hence there are rn < w, c p l ~ o l l A and (i c x = x:) and an +good model 111, a i U u E , Uf c 1N1, suoh that: (a) D[&, a) A vi(2, at), A , KI 2 n + k, (b) the vl(z, 4) are rn-contradictory.

We oan now continue as in (1) (for n = 1). The only difference is that instead of m r t i n g "for some r), p*(z, a") A O(z, b, a) is K-algebraic" we aasert "for some i, B(po(z, b, 8, a), A, 9) < n," where

v ~ z , b, s, a) = v(z, a) A cpl(s, a{) A e(z, b, a), and we replace cck8*[b',iP, i5J implies cp*(z, a') A O(z, b', E'] is K- algebraic" by " CB*[b', a', a'] implies D[tpo(z, b', a', E'), A, rcJ < n" (SO -,e* E qvp)) .

CONCLUSION 7.10: (1) If v ( ~ , a) SK #(z, 6), #(Z, 6) is K-weakly- minimal or even D[#(x; 6), L, Kl = k < w, then D[p(z; a), L, K] is finite too. If there is an n-witness for v(z, a) sK #(z, 61, then D[v(z; a), L,K] < kn.

(2) 8-e

316 MORE ON TYPES AND SATURATED MODELS [CH. V , 5 7

(i) $(x, 6) ie K-weaUy-minimal, (ii) rp(xs 8) SK $(x, 6)s (iii) for m y B(x, 8) there h e*(g, 6 ) euch that $(x, 6 ) A 8(x, ~ ) * h

K-algebraic iff t=O*p, 61. Then tirere h a forrnUla rp*(i~, 6) euch tW +*p, 61 and +*[a1, 61 implies D[rp(x, a'), L , Kl = D[rp(x, a), L , K].

(3) If for mne $*, ~$*[6*] impliee $(x; 6*) s rp(x; 6*) I #(x; 6.) then we can omit 6 from rp*.


LEMMA 7.11: ( 1 ) 8-e ( i ) r p ( ~ 8 ) s $(x, 61, cp(x, a) not dgebraic,

(ii) R[#(x, 6), L, a] = R[$(x, a), L , A] = 1. Then R[&, a), L , a] = R[rp(x, a), L , A].

(2) If v(x , a) < $(x, b), then R(p(x, a), L, K O ) < [R($(x, 6), L, K,)lw.

Proof. ( 1 ) By induction on the length of the witness, refining the proof o f 7.10 to deale with multiplicity.

(2) By 7.8 and witnesses.

DEFINITION 7.5: (1) For every complete type p E rSm(A) we define a rank L ( p ) as an OW or a by:

2 0 i f p is a type

L@) 2 8 if L ( p ) r a for eeah a < 8,

L(p) 2 a + 1 if there is p,, p G p,, p, forks over A and L(p1) 2 a.

( 2 ) L(a, A ) is L(tp(8, A)) .

DIFINITION 7.6: W e define a Q f l (the natural sum) b y double induction on a and 8:

EXERCIBE 7.1: Show that Q is commutative and aseociative.

LEMMA 7.12: Here the t y p wiW be mpZete. ( 1 ) p E q impliee L ( p ) r L(q); and i f for nop 1 po, L ( p ) = a, then

p 2 po, L ( p ) > a impliee L ( p ) = m; L ( p ) = 0 iff p w algebraic.

(2) If p E #"(A) does not fork over B E A t h L(p) = L(p 1 B). (3) L(p) R(PY LY (4) T ie euperekzble i$ for every p, L(p) < oo. (5) If R[v(x, a), L, GO] 2 n, then for eome 6, b[6; a] and L(6; a) 2 n. (6) &(a-6, A) s L(a, A u 6) 0 ~ ( 6 , A).

Proof. (1) Trivial. (2) By (1) it s d c e s to prove by induction on a that L(p 1 B) r a

implies L(p) r a; and for a = 0, a = S this is trivial so let a = /3 + 1. As L(p B) athereareZ,C, B E C,c'realizesp 1 Band L(4C) 2 By tp(Z, C) forks over B. Let 7i realizes p ; so we can assume firet that stp(a, B) = stp(Z, B) and then also tp,(C u 5, A) does not fork over B (by 111, 2.6(3)). Then by 111, 4.14 tp(Z, A u C) does not fork over C = B u C henoe by the induation hypotheah, L(Z, A u C) r: 8. Now tp(a, A) E tp(Z, A u C) and clearly tp(B, A u C) forks over B; hence it forks over A (by 111, 4.4, aa tp(5, A) = tp(a, A) does not fork over B). So by definition L(a, A) > L(E, A u C) r 8, so L(Z, A) r a.

(3) See 111, 1.2(4). (4) The only if follows by (3) (and 11, 3.14). The if pcL1.t is left to the

reader. (6) We prove by induction on n: for n = 0 it is trivial, for n + 1 we

cutn find, n-contradictory formulaa v*(x, q) (i < (2ITI)+) such that R[tp(E. ii) A tp*(Z, a,), L, co] 2 n. So for some i q*(E, at) forks over 8. By the induction hypothesis for some 6 L(6, u ii;,) r n, kF[6, a] A tp*[6, 41. Clearly tp(6, a u ~ i , ) forks over a, hence L(6, a) r n + 1.

(6) It suffices to prove by induction on y that L(an6, A) = y implies L(a, A u 6) @ L(6, A) r y. For y = 0, y = 8 there is no problem; so let y = fi + 1. As L(B-6, A) > /3, there are By af, 6: such that A c B, tp(t~,^6~, B) extend tp(7in6, A) and forks over A, and L(aln6,, B) 2 y.ByIII,4.15tp(al,B U ~~)forksoverAortp(61,B)forksover~.As clearly L(B,, A u 61) = L(a, A u 6); this implies L(7il, B u b,) < L(ti;,fl U K 1 ) = L(a1,A U 6) or L(6,B) < L(b,,A) or L(a,A U 6) = co or L(b, A) = a. Hence in all caees

L(a,, B u 6,) 0 ~ ( 6 , ~ B) < L(B, A u 6)

But by the induction hypothesis

318 MORB ON TYPES AND SATURATED MODEL8 [a. v, 8 7

DEFINITION 7.7 : Let acl,(A) = {b : tp(b,A) is K-algebraic}.

THEOREM 7.14 (Qeq): (1) Suppose K satisfie0 C1-6, p = K,(T), Card K = {(2IT1)+}, T superstable non-multi-dimensional and ~ ( x ) = (x = x) is not K-algebraic. Then there is a formula O(x) (with no parameters), types q t ~ S ( a c l p ) ,

(i) If M + N , N, N are F&,-saturated, thew O(N) = O(N) iJ

(ii) r, is K-minimal, q, is o13{r,}-Bimple and q, (Pq) s acl,(r,(QeQ) u Z,). (2) Suppose in addition T is totally tranacendental, then we can aasume

there are K,-minimal p,(x, 8,) E r,, and #i(x, Zi) E q,, Z, E a01 8, w h that

c q@), and types r l , qi c rrES(ci) such that

q,W) = q, (N) for every i.

Qr(% a,) I- *&, a, ! b w q , a,) "cl,(q),(ceq, Gc) u a. Proof. We hereby prove Theorem 7.14(1) by a series of claims. Of course T is stable, but the use of stability is mentioned explicitly. We shall assume K satisfies C1-6, p = K , ( ~ ) , Card K = {(2ITl)+}, and works in

FACT A : B = ac1,A implies B = acl B = acl, B and 6~ B if tp(6, A ) is K-algebraic.

FACT B : If 6~ acl,(A U 6) - acl, A, then tp(a, A U 6) forks over A .

Proof. Suppose tp(a,A u 6) does not fork over A. Let C, = acl(A U a) n acl,A, C, = acl(A U 6) n ac1,A. As we are working in Ceq, tp(a,acl,A) does not fork over C,, and similarly tp(6,aclKA) does not fork over C,. As we assume tp(a,A u 6) does not fork over A, also tp,(acl(A u a), acl(A U 6)) does not fork over A , hence tp,(a U C, ,A U b U C,) does not fork over A, hence tp(a,A U 6 U C, U C,) does not fork over A U C,.

By 6.3(2) A U C, < A U C, U a, and H(a,A U C,) = 00. Similarly, H(6,A U C,) = 00. Let M be h,-good, {a$:i < x} be an indiscernible set over A U C, U C, based on stp(8,A U C,) and {6,;i < x} be an indiscernible set over A U C, U C, based on stp(6,A U C,), where x is large enough, w.1.o.g. A U a U 6 U C, U C, U {a6,6$:i < x} G M . Then M n acl,(A) has power < x, hence for some i < x tp(at, acl,(A)) does not fork over A U C,. Now M n acl,(A U at) has power < x, hence for some j, tp(b,,M n acl,(A U at)) does not fork over A U C,. Hence b,#acl,(A U at), but clearly stp(ainb,,A U C, U C,) = stp(a-6,A U C , U C,), hence tp(ai"b,,A) = tp(an6,A), contradicting an assumption of Fact B.

RANKS RBVISITED 319 m. v, Q 71

FACT C: If T is superstable there is a weakly K-minimal ~ ( x , a).

FACT E: Suppose 1{1tp(M, 6): 6 E M)1 < KO for any M, ~ ( x , 6 ) is weakly Kminimal, then there are n < w , a, realizing tp(a, 0) (1 < n) and non K-algebraic f m u l a B(z) (with no parameters) B(Qeq) E acl(rp*(Ceq, a*) u a*) where ‘p*(x, a*) = V,<, y(x, a,) is weakly K-minimal too.

Proof Like 4.11.

FACT F: If ~ ( x , 6 ) is weakly K-minimal, B(x) not K-algebraic, T superstable, B(Qeq) E acl,(cp(Qeq, a) u a), then there are complete types pi (i < io), e(x) €pi, pi K-minimal such that: if M <; N are Fio-saturated, u1 Dom pi u 6 s 1x1, then B(M) # B(N) iflsome c E IN1 - realizes some pi.

Proof of Fact F. Let {pi: i < io} be a maximal list of K-minimal types to which B(x) belongs and are pairwise orthogonal. The “if ” part is clear, so let us prove the “only if”. Let c E B(N) - B(H), be with minimal R(tp(c, IMI), L, ao), so tp(c, 1611) is regular. By a hypothesis there are b,, . . . , b, E ‘p(Qeq, a) such that c E acl,({b,, . . . , b,} u /MI) , and w.1.o.g. we choose n, c, bl , . . . , b, such that n is minimal. As c E N - M, M N necessarily n 2 1. I f n = 1, then clearly tp(c, ]MI) ie H-minimal, hence not orthogonal to some pi (i < io), so some

Now we suppose n > 1, and we shall get a contradiction; let 6 =

acl,(B u b u {b,,}), tp(6 (0 , b,), IMI) does not fork over A and tp(6 (c, b,), A) is stationary. As n is minimal c, b, 4 acl,(lMl u 6), so (as ~ ( z , 8) is weakly H-minimal) {b,, 6} is independent over (IMl, A). Choose 6’ E !MI, 6’ mlizing stp(6, A), then

C’ E IN1 - realizes pi, SO we finish.

- ., b?-1), and choose finite A E 1x1 such that zi E A, c E

tp(6, A u {b,,}) = tp(6’, A u {b,}),

and we can find c’ E N, tp(6’ c‘ E acl,(A u 6’ u {b,}) - acl,(A u b’), hence by Fact B,

(c’, b,?, A ) = tp(6 <c, b,), A). So now

tp(b,, A u 6’ u {c’})

320 MORE ON TYPES AND SATURATED MODELS [OH. v, $ 7

fork over A u 6', hence tp(b,, IMl U {c'}) fork over A u 6', but tp(b,, IN)) does not fork over A henoe over A U 6', so tp(b,, 1M1 u {c'}) fork over IMI, so c' $ IMI. So c', b, contradiot the minimality of n.

FACT G: 8wppose p = p(x , a) is K-minimal, and stp(8,0) = stp(a', 0) implies p ( x , a), p ( x , Z') are not orthogonal. Then there is q E 8(acl0), Z E acl(0) u q(CQ) , q -c r E ~ ( Z ) such that: r is K-minimal, q is clj{p}- eimple, and q ( 4 O q ) c acl,(r(CQ) u a); and there are n < w , Zl (I < n) realizing stp(a, 0) such that

So q is c13{r}-simple. (We wume T is superstable.)

The proof is like 4.1 1, except r . As q is not K-algebraic, there is a K- minimal r E S ( q , E G q(aeq) extending q (we can have F G q(EeQ) by the Definability lemma). Necessarily T , p ( z , a) are not orthogonal (as q(M) # q(N), M C N are FGo-saturated, al E M implies p(N, a) = p ( M , i ~ ) ) . As they are K-minimal necessarily for some finite B, (a U E E B)p(a"'J,a) E acl,(B U r(aeQ), henceq(aeq) G acl,(B U r(aeq). Let E' E q(aeq) be such that tp*(B,q(aeq)) does not fork over I?', E G C'. So tp,(B, acl(q(aeQ)) is definable over acl(F'), hence q(aeq) E acl,(c' U r(aeQ)) .

Remark. ( 1 ) We could, by choosing the p , looking at tp(z,al), have that t3E qr.

(2) The proof of 7.14(2) is similar, remembering 7.7(2).

THEOREM 7.14(3): We can add to 7.14(1) and 7.14(2). (iii) q,(aeq) c dcl O(a). (iv) For any model M , and c:, G E M realizing tp&, aclq) letting

Br = c13{rr}, we have : low,(q,(M), aclq) = low,(@, aclq) + low9(rr(M, G), $ U aclq).

(v) For i Zj, qj i s strongly orthogonal to B,, so qj is 9',-simple and for d € q j ( a ) , 10w,~(d) = 0.

CHAPTER VI

SATUaATION OF ULTRAPRODUCTS

VI. 0. Introduction

The ultraproduct construction is one of the most important methods in model theory, and unlike the primary model, the ultraproduct does not depend on the specific theories we deal with.

The ultraproduct n, , ,M, /D, for D an ultrafilter, M , L-models, is the product when we identify any two elements of n , , , M , (i.e., functions from I) which me equal for “almost ” every i. The important properties of the ultraproduct me:

(1) Log’s Theorem: a (first-order) sentence is satisfied by the ultraproduct 8 it is satisfied by almost every M,.

(2) The L,-reduct of the ultraproduct is the ultraproduct of the L,- reducts .

It is also important to know: (3) The ultraproduct is A-compact if the ultrafilter is A-good and

N,-incomplete (thus we can construct A-mturated models). (4) Two L-models are elementarily equivalent iff they have iso-

morphic ultrapowers. So (4) gives an “algebraic ” characterization of elementary equivalence. It is also important that by ultraproducts, we can get an ccalgebraic’y proof of the compactness theorem.

The ultraproduct is not central in this book (and outside this chapter, we shall use it only once or twice), and the first three sections of this chapter have nothing to do with stability. However we have developed here a classification of theories using stability and associated notions, different from the one Keisler [x67] gives (using his ordering a). We show that our classification is helpful in the analysis of Keisler’s order, thus giving more evidence of the naturalness of the classification. We also deal with the categoricity of pseudo-elementary classes, and saturation of ultraproducts.

In Section 1 we give the basic properties of ultraproducts and reduced products and regular filters.

In Section 2 we deal mainly with A-good filters. This property is 32 1

defined combinatorically, but its main property is: an ult rdter D is A-good iff for every family of A-saturated L-models M,, i E I, M,/D is A-compaot iff for every A-saturated atomic Boolean algebra M y Mr/D is A-saturated. If we require D to be N,-incomplete aa well, we .can omit the condition that the models am A-eaturated. The Boolean algebra comes quite naturally, but we can replace them by dense order. This is done aa follows: suppose M1/D is A+-compacf for every dense order M. For any model N, we expand it by encoding the set of finite sequences of formulae (with parameters) which are consistent, and some natural relations and functions and get Nl. Note they form naturally a tree and by it we can “ s p k ” on sequences of formula in N’,/D which are not n e d y “standard”. We look in N’,/D, and any 1-type p in W / D , 1p1 < A, and define by induction on i s lpl elements a, of the tree, increasing with i , and “consistent ” with p (in the inner sense we demand this for every finite subset of p separately and this is expressible): and letting p , = {v,: i < IpI}, we want that 9, wi l l “appear” in the “sequence” a,, the element “realizing” alpl is as required. The translation of the tree to the dense (linear) order is by intervals.

Now from this we can deduce for singular A that A-goodness implies A + -goodness.

The importance of constructing ultrafilters is clear, and this is the subjeot of Section 3. Some of those problems are connected to problems on large oardinale, and consistency results and are outside our scope (e.g., N,-complete ultrafilters, non-regular ultrafilters).

It is easy to construct regular ultrafilters as they sped on the existence of a family of sets, and this is done in 1.3(4). Good ultrafilters posed a more difficult problem ae their definition says: for every function into D there is a function into D . . . . Keisler [K 641, constructs a good ultrafilter on A when A+ = 2” in A+ steps; in each step we have a uniform filter generated by A subsets of A. When 2” > A+ , this seems to fail, but Kunen [Ku 721 suggests an alternative proof he takes a family of 2” independent functions from A onto A; and construct the ultrafilter in 2* steps, after the ith step, we have to delete s 8, + li I of them only. Now it is natural to ask where there is a (regular) p-good ult rdter over h whioh is not p+-good. But p s A + , and by 2.10 p is regular. For p a successor we use the product of ultrafilters, whose most important property is NIX */Dl x D, = (Mr/Dl)J/Dl, i.e., if we take ultrapower twice, it is like taking it once for the product. By this we get h+good, not A + +-good ultrafilters over p 2 A. We use independent functions to construct ultrafilters with various

properties; e.g.,

m. 0 01 INTRODUaTION 323

(1) For p 2 A, h regular, over p them is a regular ultrafilter which is A-good but not h + -good.

(2) If 2” 2 h = A%, then there is a regular ultrafilter D over p and At < w (t E I ) suoh that ntsI nt/D = h (the conditions h = P o s 2Y are neoeeeary (for an infinite A), see Exercise 2.10).

(3) If 2” 2 A, p 2 K ; h > K regular, then there is a regular ultrafilter D over p suoh that h = lcf(K, D) =d& mink: in {a E #ID: # / D t a < a for every a < K} there is a set unbounded from below of cardinality x}. We prove also that Thd is essentially “simpler” than the theory of

linear order. In the exeroises, we indicate why any two elementarily equivalent

models have isomorphic ultmproducts, and other results. In Section 4 we define Keisler’s order and prove some theorems on it.

We oould change the definition slightly, e.g., by omitting the requirement that D is an ultrafilter over A, and &ill get the same results. We also show that some ultraproducts are not A-compact.

Section 6 is the heavy section, utilizing the results from I1 on d-n- indiscernible sets and their dimemione, to determine how saturated are ultraproducts. We find which elementary olasses (of countable type) contain categorical pseudo-elementary claases. E.g. , we prove

THEOREM 0.1: If bf W a model of T , T countable and without the f.c.p., D an K, - imple te UUraJilter Over I , then M1/D iS ~~/D-eaturated.

THEOREM 0.2: If T i e countable, euperdable and without the f.c.p., then t h e i e a theory T I , T E TI, lTIl = 2u0, 8Wh that any uncountable model of T , h a a eaturated L( T)-red&.

Other theorems (6.1, VIII, 2.1) show this result is the best possible. Summing up our results we get the following picture for Keisler’s

Let order on countable theories.

H- = {T: T countable, without the f.c.p.}, Kscp = (5”: T#K,,,,, is countable and stable with the f.c.p.},

= { T : T countable, unstable K ~ ~ ( T ) < OO}, Kcdt = { T : T countable, Kodt(T) = GO,

T without the strict order property},

H,, = {T: T countable and with the strict order property}.

THEOREM 0.3: (1) If T,, TI are both Hmin[Kaop][Hmu] then they are @‘-equi~alent, i.e., T I @ Ta @ T l .

In Section 6 we find how compaot rn ultralimite.

PROBLEM 0.1: It would be very desirable to prove that (1A) T,, Ta E Ktnd implies TI, Ta are @-equivalent, (1B) T,, T , E Kdt (or we should aek &o whether K,,(T) = 00, K,,,(T) = 00) implies T,, Ta rn @- equhalent. This will complete the model-theoretio sham of investigating Keisler’s order for countable theories. For this it seems reasonable to try to find for T E KiDd a theory parallel to 11, Section 2 for stable theories.

PROBLEM 0.2: It would be desirable to replace in 0.3(2) the “consistent with ,’ by “provable form ,’. This is a problem in constructing ultrafilters. Another problem on ultrafilters is 6.1.

PROBLEM 0.3: On Keisler’s order for uncountable T see Shelah [Sh 721. We conjecture that for every T there are a countable T, and simple Ta (Definition 6.4) such that T’ @ T o [T’ @ T , and T’ @ Ta].

PROBLEM 0.4: We conjecture: if T is not @-minimal then T is not @,,,-minimal, and if M1/D is (21’I)+-compact, M is A-compact, then M’/D is A-compact (see 6.7).

In this chapter D will always denote a filter (or ultrafilter) over I (or sometimes J ) (For the definition and some theorems see Section 1 of the Appendix. )

VI.1. Reduced praduots and regular filters

DEWINITION 1.1 : Let D be ti filter over I and for each i E I let Mt be an L-model.

(1) We define an equivalence relation zD (or z for short) over ni,1 I Jft I *

Mi’s) :

f z g iff {i E I : f ( i ) = g(i)} E D.

(2) We define the L-model n , , M , / D (the reduced product of the

OH. VI, $ 1 3 RIDDUOED PRODUUTS AND REUULAR FILTER8 325

(A) Its universe is {f/ k : f E n,,, [Mi l } . (B) If R is en n-place predicate symbol in L then

RM = {(f1/#, . . .,fn/z): {i €1: (fl(i), . . .,f,,(i)) E RMd} E D}. (C) If P is an n-place function symbol in L then PM(fl , . . . ,fn) = f

where for every i E I , f ( i ) = PMi(f,(i), . . . ,fn(i)) and PM(f1/#, . . . ,fn/z)

(3) If D is an ultrafilter, the redud product is called an ultraproduct. If M , = N for every i E I, then we write N1/D for nt,g M,/D and call it a power instead of a product.

Remurk. As D is a filter, z really is an equivalence relation andf, z gl for 1 5 1 s n implies {i: ( f , ( i ) , . . . , f , , ( i ) ) E R'c} E D o {i: (gl(i), . . . ,

a well-defined L-model. Notice that for a term T , 7(f1, . . . ,fn)(i) 3:

= PM(f, , . . . ,fn)/#.

gn(i )) E RMg} E D a d PM(fly * * * , fn) z P"(g1, * * - 8 gn) 80 ntsl Mt/D is

4fl(i ), - ' * Y f n ( i )I* Notation. ( 1 ) If M , is an L-model for i E I, P # L, N = nrSI M t / D and (a,, P,) is an L u {P}-model for every i E I, then in an abuse of notation (N, PN) = n,,, (X;, P,)/D.

(2) We do not strictly distinguish between a E In,,, M,/DI and a representative of a in n,,, lM,l. We write (a[ i ] : i E I ) for the representative of a. Also (in this chapter) if ii = (aoy.. ., a,,-l) E In,, M,/DIy then 7i[iI = (ao[i] , . . . , a,".

(3) We identify a and <a: i E I ) and so regard Mr/D as an extension of M. The embedding a H (a: i E I ) is called natural.

Remark. Note that if La E L,, Mt is an Ll-model and Mf is the L,- reduct of Mt (for i E I), then n,,, M f / D is the La-reduct of n;,g M t / D .

THEOREM 1.1 (EoiYs Theorem): Let D be an dtrajlter over I . ( 1 ) M r / D is elementarily equivalent to M . (2) If M, is an L-model (for each i E I ) and cp(Z) E L, ii E In,,, M,/DI

n M t / D C cp[ii] i,# {i E I : M, C cp[ii[i]]} E D. then

ICl

( 3 ) The natural embedding of M into M1/D (defined by a H (a: i E I ) ) is elementary, so we can say M < W / D .

Proof. ( 1 ) Follows by (2), for sentences.

fail for filters). (2) Easy by induction on formulas (only for negation does the proof

(3) Follows by (2).

DEFINITION 1.2: (1) A basic Horn formula is a formula of the form A,<,, 'pi + 'p where p,, cp am atomic, n c w, or ~ ' p .

(2) The family of Horn formulae is the cloeure of the family of baaio Horn formulas under conjunction, exietenticll quantiihtion and universal quantification (but not disjunction or negation).

Remark. In Definition 2.1(1) we o m take n = 0, so every atomio formula is a basic Horn formula. Also, every negation of an atomic formula is a baeio Horn formula.

THEOREM 1.2: If p ( l ) i8 u H m forncukc, M = n,, MJD, iz E 1M1 and

{i E I : Mi C tp,P[i]]} E D,

t h iK b p[iz].

Proof. By induction on 'p.

DERTNITION 1.3: (1) The family {Xi: i c A} of subsets of I is regular if for w E A, n x, z 0 1 ~ 1 < N,.

f6Y

(2) The above-mentioned family regularizes D (which is a filter over I) if it is regular and X, E D for i < A.

(3) The filter D over I is A-regular if some {X,: i < A} regularizes it. (4) D is regular if it ia 111-regular. (6) D is A-incomplete if there are X, E D( i < a c A), nlSa X, = 0. (6) D is A-complete if for X, E D (i c a c A), X, E D.

LEMMA 1.3: (1) If D is A-regular, p 5 A, then D ii3 p-regular. (2) Tk$&r D i8 N,-btiX~m~Zete iff D i8 K,-reg&r. (3) D is not 111 +-regular. (4) Over every infinite I there is a IIl-regular filter, hence a regular

ultrafilter.

Proof. (1) Immediate. (2) Immediate. (3) If {X,: i < 111 +} c D is regular, each X,, being a member of D,

is non-empty, so choosej, E X,. Clearly for somej*, I{i c 111 + : j, = j*}l = 111 + but n {X,: j, = j*} 2 {j*} # 0, contradiction.

(4) Clearly it suffices to find a set J, I JI 3: II] over which there is a regular filter. So let: J = &,(I) = {w: w E I , lw] c No}, for W E J ,

Oa. VI, 0 13 REDUOED PRODUaTB AND REGULAR FILTERS 327

X, = {w: W E J , w c w } c J. The filter D generated by {X,: WEJ) (see Definition 1.1 and Theorem 1.1(1), (2) of the Appendix) is reg- ularized by {X{f,: i E 4. The second phrase follows by 1.1(4) of the Appendix.

Proof of the C e n e 8 8 Th-mrem (I, 1.1): From 1.1 and 1.3 we get a proof of the Compactness Theorem. Let T be a set of sentences, such that every finite t c T has a model Mt. Let J = BN0(T) and let D be the filter over J defined in the proof of 1.3(4), and D* an ultrafilter over J, extending D (exists by 1.1( 4) of the Appendix). Let M 2: n,, M,/D*, thenforevery#ET,{wEJ:M, C # } 2 {wEJ:#Ew} = X , , , E D E D*. Hence, by h g ’ s Theorem, M C #, so M is a model of T.

DEFINITION 1.4: A type is atomic if all its formulas are atomic.

THEOREM 1.4: (1) Let D be afilter over I . Then the following condition% are quid&.

(A) D i8 A+*. ( B ) For ewery L and every family Mt (i E I ) of L-models, every atomic

(C) For every M every atonzic type g over in W / D , lpl 5 A, is

(D) Condition (C) holds for M = (rSNo(A), r). (2) If in (1) D is an ultrafilter we can replace “atomic type” by “type”

type p aver 0 in each Mi, 1p1 5 A, i8 reat%zed in nisr MJD.

realized in M’/D (notice: p is in M iff it is in MIID).

in the umditifma.

Proof. (1) (A) * (B) Let N = nfsr Mi /D, {X,: a < A} regularize D , p = {v,(Z): a c A} be the type. For m y finite w E A let a,[<] E 1M,1 realize {v,(Z): a E w}. Now as {X,: a c A} is regular, for every i E I, w(i) = {a < A: i e X,} is finite. Define E IN1 by 8[i] = GwiJi]. Hence for a < A {i E I : Mi C rp,p[i]} 2 {i E I: a E w(i)} = {i E I : i E X,} = X , E D. Hence N C tpa[i3], so a realizes p. (B) * (C) Use (B) for M i = (My a)oslarl, for every i E I. (Clearly p is

a type in M.) (C) * (D) Holds by their definition. (D) * (A) Let N = M1/D and let p = {{a} E 2: a < A}, so clearly p

is finitely satisfiable in M y (hence in N ) . Suppose a E IN1 realizes p , and let for a < h

X, = {i E I : M C {a} E a[i]} .

328 SATURATION OR’ ULTRAPRODUCTS [a. m, 8 1

Then X, E D aa N C {a} c a, and for w c A, nUeu, Xu = { i ~ l : w c a[i]} , so clearly it is empty iff w is irhite.

(2) The same proof using LOPS Theorem.

DEFINITION 1.5: M is (A, d, m)-compact if every ( A , m)-typep over 1M1, in M , 1p1 < A, is realized in M . For d the set of atomic formulaa of L(M), we write at. Similarly the sets of quantifier free formulas, conjunctions of atomic formulaa, and formulaa of quantifier depth s n are denoted by qf, cnat, and qd,, respectively; i f m = 1 we omit it.

DEFINITION 1.6: (1) For models M,, Ml and an ordinal a, (L(M,) n L(Ml) 2 L) we define a game Gt(Ho, Ml) between the players I and I1 aa follows:

A t the @h move player I chooses I E {0,1} and afi E lMil, and then player I1 chooses a;-i E ~ M l - i ~ .

The play ends after a moves and then player I1 wins if for every atomic formula cp(z,,. . . , 2,) E L and ordinals p(O) , . . . , @(n) < a,

otherwise. (2) A strategy (of a player) is a sequence of functions f B @ < a)

which “tells” him what to do ( f B for the /?th move) depending only on the previous choices in the play. A winning strategy is a strategy such that in any play in which the player chooses according to it, he wins. A player wine in the game if he has a winning strategy.

J f o C ~ [ a & ) , - - - 9 $(,)I * C ~[a j (o ) , * - * 9 a;,,)], and player 1 wins

(3) We omit L if L = L(M,) = L(Ml).

LEMMA 1.5: (1) In the game Gz(Mo, Ml) nd bothpZaymu win.

lation among models.

L E L(l)] then he m’ne in G&l,(M,, Ml).

( 2 ) The relation “player I1 wine in Gz(M, N)” is an equivalence re-

(3) If player 11 [I] wine in Gi(Moy M,) , L(1) E L, /3 5 a [a 5 8,

(4) If M,, M , are ~sonwrph~c, then player I1 Wine in Ua(M,, Ml). ( 5 ) If M,, M , are elementarily equivalent, and A-eaturated, then player

I1 wine in G”M,, M I ) .

Proof. (1) Suppose both players win, then in some play, both use their winning strategy, but one of them must lose this play, contradiction.

(2) Combine the strategies to prove transitivity. Symmetry is trivial, and reflexivity follows by (4).

(3) Trivial.

OH. m, 8 11 REDUCED PRODUCTS AND REUULAR BILTEFtS 329

(4) If 8 : IM,l+ lMll is the isomorphism then player I1 chooses

(5) Player I1 has to choose at-' such that tp,({az: a zs; p), 0, 22,) = a;-1 so that g(a;) = a;.

tP*({d: a PI, 0, Ml).

Proof. Immediate.

THEOREM 1.7: The L*-mud.ele M,, M, are elementarily equivalent iff for every jinite L E L*, n < w, player II wine in G;(M,, Ml).

Proof. Let us define by induction on n, the sets F;,,,(L), F;,JL) of formulas ~(z,, , . . , z,,,-,) E L.

(i) F;,,,(L) is the set of atomic formulas y(s,, . . . , z,,,-,) E L. (ii) If F;,,,(L) is defined, F;,,,(L) is the set of formulas of the form

(iii) If F;,,,+,(L) is defined A {cp(Z)u(@'6r): cp(@ E F;,,,,(L)} for F c F;,,,(L).

e+l ,m(L) = { ( ~ ~ r n ) F ( ~ o , * - - 9 2,): dzo, * * * 9 z,) E %n+l(L)I*

Clearly any sentence in L is equivalent to a disjunction of some of the sentences in F;,, for every sufficiently large n (in fact for its quantifier depth), and this holds for any formula.

If M,, M, are elementarily equivalent, L c L* finite, n < W, a winning strategy for player I1 in Gft(M,, M,) is to preserve the satisfaction of

(*I M , C ?[at, . . . , a!-,] iff M, C cp[ua, . . . , a:-,] for every F(zO, * * - 9 z k - l ) E F;-k.k(L)-

For k = 0 this holds as M,, M, me elementarily equivalent; and by the definition of F;-k,k(L) it is easy to play so that (*) is preserved.

For the other direction, it suffices to prove that if ut, . . . , a!-,, a;, . . . , a;- have been chosen in a play in which player I1 uses a winning strategy for Gft(M,, M , ) then (*) holds (because then for any

330 SATURATION OF ULT'RDRODUOTS [CH. VI, 5 1

$ E (J I'l,o(L), L c_ L*, L finite, Mo C $ e Ml C $, and aa any sentence $ of L* is a sentence of some finite sublenguage, bl,, M, are elementarily equivalent). The proof is may, by induction on n - k.

THEOREM 1.8: If &I$', Mi are elementarily equivalent L-modeze for i E I, ILI S A, D a A-regeclarfilter over I , a < A + , thenplayer I1 wiw in Ga(Mo, M1), wlrere N o = no Mf'/D, M1 = n16x M i / D . (80 Mo, M1 are elementarily equivalent.)

Proof. The number of formulaa ' p (~~<~, , . . . , z4(,,)), E L, p(1) < a, is 5 A, 80 let {'pj(~sco,n, . . . , z,,~,,~,.,): j < A} be an enumeration of the set of thew formulaa (possibly with repetitions). Let {Xj: j < A} E D regularize D , and for t E I, let w(t) = {j: t E Xj}, (which is finite), let

h O . t ) , - - - Zv(Mt).t)} = u {%#(o.n,. . .} where y(0, t ) < - - - < y(n(t), t).

Let Lt be the minimal sublanguage of L such that vj E & for j E w(t). So by the previous theorem, player I1 wins in P&(t)+l(Mf, Mi) for every t. Now we shall describe the winning strategy of player I1 in GC(M0, W): when he haa to choose E lM1-il he chooses each aj-I[t] separately. If /? $ {y(k, t ) : k I n(t)}, he chooses a;-I[t] arbitrarily. If fl = y(k, t ) he imagines he is playing qt)+'(Yp, M i ) , that a~(,,,,[t], , . . , a:(&- l,o[t], a k - la[t] and a:(k,t,[t] have been chosen, and he chooses ai,;;.',,[t] by his winning strategy in GZt)+l(Mf, M;). It is eaay to check that this is a winning strategy for Gg(M0, Ml).

j€ur(t)

CONCLUSION 1.9: If M y N are elementarily equivalent, D ie a A-regular Jilter over I then M x / D i s (A , A , m)-wmpad iff N x / D i8 (A, A , m)-wmpa&.

LEMMA 1.10: If for each t E I player I1 win8 in Ga(Mf, Mt), D a jElter over I , then player I1 wins in Ga(n , , M f / D , n,,, M t / D ) .

Proof. In view of the proofs of 1.8, 1.9, we leave it to the reader.

CONCLUSION 1.11: If M,, N , are elementarily equivalent and h- saturated, D a N e r , then M = n,,, M t / D ie (A , A , m)-wmpact ifl N =

n,,, N t / D i8 (A, 4 m)*pa&.

Proof. By 1.6(6) player I1 wins in Gh(Mt, N,) . By 1.10 player I1 wins in GA(M, N ) . Henoe, if A > No, by l.6(4) M is (h ,A, m)-compact iff f l is (A , A, m)-compact. If h = KO, every model is (A , A , m)-compact; so we finish.

CH. m, 8 11 REDUCED PRODUCTS AND REGULAR FILTERS 33 1

Notation. If A,, i E I is a family of non-empty sets and D is a filter over I we define nt,, A,/D = (fl z : f E nier A,} where z is the equivalence relation from Definition 1.1. If A, is a ctlrdind A, for every i E I , we write nter &/D for IrIteI WDI and nier A, for Inw 4- $0 llrI:er WDII = ntsr ll~:ll/D.

If Dis afilter over I, J c I then D 1 J = {X n J : X E D}so when I - J $ D , D tJisnottrivid.

EXERCIBE 1.2: (1) If D is a filter over I, J E D then nt, M,/D z nteJ M t / D J (the isomorphism maps (art]: t E I ) / D to (a[t]: t E J ) / D ) ; 80 Titer &/D = nteJ &/D I J .

(2) If J c I, I - J$ D, D a p-regulrtr filter over I then D J ie p-regular.

EXERCIBE 1.3: (1) Show that if (M, P) = nteI (Mt , P J / D then

(2) Show that n,,, A,/D < nlol A, ; and {t : A, < p,} E D implies

( 3 ) Show that if Q E 1 A,/D > 0 then n,,, A,/D 2 n,,, A,/D, r J , for

IPI = nt,r IPtIID.

n t e r &/D s n t e r pt/D.

any filter D , 1 D, J E I.

THEOREM 1.12: (1) If D i s a p-reguZurjZter over I , & (t E I ) (injnite) cardinale, and h = nter &/D > 0, x = mink: {t E I: & s x} E D} then h 1 x".

(2) If in (1) p = IIl then h = f. (3) Moreover tkre are natural numbers nt mlr that nt,l nt/D 2 2",

and if D i s r e g h r ntsr nt/D = 21'1.

Proqf. (1) By Exercise 1.3(3) we can assume D is an ultrafilter (by extending the original filter to an ultrafilter containing {t: & > x,} for each x1 < x (by 1.1(2), (3) of the Appendix).

Case I: x > p. By Exercise 1.2(1) we can wume that for every t , p 5 & s x. Now we want to define models Mt (t E I) so that llMt 11 = At

332 SATURATION OF ULTRAPRODUUTB rm. w, 8 1

and f pairwise contradictory types will be realized in nt,, dd,/D. Let L consist of the one-place predicates Pa (i < x, a < p). We define Mt such that for each a, for i < & the Pi(Mt) are pairwise disjoint, for & 5 i < x Pk(iWt) = lM,l and for every finite w E p and function f : w+ X M , I= (3x)[/\,,,P{(")(x)]. It is quite easy to construct such a model :

lMtl = {cy: w c p, IwI < No, f: w + & a function},

Pk(Mt) = {c/" E lMtl: a E w, f (a) = i or i 2 &}.

By the definition of x, for every i < j < x, a < p

{t E I: M: I= -1(3%)(pk(%) A Pi(%))} 1 {t E I: li 1, lj I < &} E D,

hence M = M J D C 7(3s)[Pk(x) A Pi(z ) ] . Let for each r) E ' x , p,, = {e[al(z): a < p}, 80 the p, , '~ are pairwise contradictory; 60 it suffices to prove they are realized in M . But clearly each p,, is finitely satisfiable in each Mt, 80 by 1.4 it is realized in M.

Cme 11: x 5 p. By Exercise 1.3(2) it follows from 1.12(3). (2) h 2 xfi by (1) and h I xu as we can assume (again by Exercise

1/41)) A, 5 x for every t , so by Exercise 1.3(2), &/D 5 n:pr h: S x ~ r ~ =

(3) We use a method similar to the one used in (1). Let L consist of the one place predicates Pf, (1 = 0 ,1 , a < p), {X,: j < p} regularize D, and w(t) = { j : t E X,} which is finite. We define Mt such that llMtll = 2Iw(:)l, and denote this number by nt. We define it such that fo r j E w(t) , P:, Pf are complementary and for any finite w E p

f .

Mt I= A Po,(%) A A Pi(%)). crew crew(:) - w

For this let

]M*l = {cw: w G w(t)} ,

Pf,(M,) = {cw: 1 = 0 and u E w or I = 1 and a E w ( t ) - w or cc $ w(t ) } . Let M = n f s , M t / D , then P:(M), P i (M) are a partition of IMI, and for every w E p, p , = {Pf,(x): a < p and a E w e 1 = 0) is realized in M, 80 llMll 2 2", but 11M11 = ntel llMtll/D = n:EIn,/D. When D is regular we get equality as in (2).

CONCLUSION 1.13: If D is a regular filter over I , t 7 m hr/D = hir!.

OH. VI, 8 21 FILTERS AND COMPACTNESS OF PRODUCTS 333

EXERCI8E 1.4: Suppose the filter D is p-regular for every p < A, and Xis singular. Prove that D is A-regular. [Hint: If {Xf : i < p} regularizes D for each p < A, K = cf A < A,

A = 2 p(a), p(a) < A, then {Xf@) n Xi: a < K , i < p(a)} a<x

proves our aesertion.]

EXERCISE 1.5: Suppose A is a regular cardinal, prove there is a filter D which is p-regulw for each p < A, but is not A-regular. [Hint: D will be generated by {Xf : i < p < A}, for each p, {Xf : i < p} is regular, but for every a < w , and distinct pn < h (n < a), and finite sets w, c {i: i < p,} nnsa nlEtu. Xtm # 0. Use 1.4 of the Appendix to prow this is possible.]

QUESTION 1.6: In Exercise 1.6, if K 2 AWo clearly we can find such a D over K. Can we always find such a D over A ? (For A a successor it is trivial.)

EXERCISE 1.7: Prove that in 1.4(1) (and (2)) to the four equivalent conditions we can add:

(6) If F is a set of I A atomic formulas (in 5 A variables) which is finitely satisfiable in each Mt, then F is satisfiable in n,,, MJD.

(6) In (6) instead of atomic formulas we can allow Horn formulas.

EXERCIBE 1.8: Prove that for any ultrafilter D end cardinal A, D is A-incomplete iff D is not A-complete.

EXERCIAE 1.9: If M = (A, R,, . . .), M1/D = N = (A', Ri, . . .) and A E lMll, then N ; = ( M I , A , R,, . . . ) , /D = (M{/D, A', R;, . . .). (Itis convenient to use this for MI = (H(K), E, M), where H(K) is the family of sets hereditarily of power < K , M E B(K)).

VI.2. Goad Biters and compactness of reduced products

DEFINITION 2.1: (1) A filter D (over I) is A-good if for every p < A, every f: A&) + D which is monotonic [i.e., w c u f(u) c f (w) ] has a refinement g : flH0(p) --t D [i.e., g(w) E f (w) ] which is multiplicative [i.e., g(w u u) = g(w) n ~(u)] .

(2) D is good if it is 111 +-good.

Remark. (1) Clearly we can add “g is monotonic”, because multiplicity impliea monotonicity .

(2) We can delete the ctssumption “ f is m~n~tonic” becauee for any f : 8&) + D, let f * be defined by f * (w) = nUsw f (u). AS l{u: u E w}l = 21wl c KO, and D, being a filter, is closed under finite intersections, f *(w) E D, and any multiplicative refinement off *, is a multiplicative refinement of fi and f * is monotonic.

C W 2.1 : Euery j&r is Et,-good.

Prmf. Let f : Bb(w) --+ D. Define g(w) = n {f (u): mas u s max w}. As for emh w E ~ ~ ( w ) max w < u, and the number of u c {n: n 5 max w} is finite, clearly g(w) E D. The function g is multiplicative aa max(u U 20)

= max{max v, max w}.

Proof. (1) * (2) Let N = nter Y t / D , p be an (at, m)-type over IN1 in N , 1131 < Asp = {v,(Z; a,): a < p < A} (v, atomic). Let for w f (w) = {t E I : Mt t (33) A,o,,, tpa(P; BJt])}. As p is finitely satisfiable in N, some 8, E IN1 realims {cp,@; aa): a E w}, so X& = {t: Mt t cpa[Zw[t], a,$)]} E D, (for a E w ) hence naew X;w E D, but clearly it is a subset of f (w) , henm f ( w ) E D. As D is A-good, there is a multiplicative function g : El,&) + D

refining f. Let for t E I , ~ ( t ) = {a < p: t E g({a})}. NOW, for every finite w E w(t), a E w t E g({a}), hence by the multiplicativity of g, t E g(w); and aa g refinesf, t E f (w). Hence Mt t (32) A,,, q~,(3,7i,[t]), and &B this holds for any finite w c w(t), pt = {v,(Z; Za[t]): a ~ w ( t ) } is finitely satisfiable in Ht. As Mt is h-compact, there is 8[t] E lMtl which rsalizes pt , and let 8 = (. . . , 8[t], . . .)ter/D. Let us prove that 8 realizes p; for every a

{t: Mt C v,@[t], 7i,[t]]} 2 {t: a E w(t)} = {t: t E g({a})} = g({a}) E D

hence N t v,[& a,].

OH. w, 8 21 FILTERS AND OOMPAOTNESS OF PRODUCTS 335

(2) + (3) Trivial. (3) + (1) Let p < h andf : LYNo(p) 3 D be monotonic. For each t E I

we define a&] E ]Mi for a < p such that:

(*) for every finite w rzLYN0(p),

bl c (3z)( A z c a&] A P(z ) e t E f (w). aEw )

[If we want to define a,[t] for finitely many a's only, we can easily choose them in M,,. By the h-saturation of M we can choose all a,[t],

So let a, = (. . . , a,[t], . . .)ter/D and p = {z E a,: a < p} u {P(z)} . If p c p is finite, then for some finite w c p p E {z c a,:aEw}U

a < PI.

{P(z)} , and, by (*) {t: M t (3z)[ / \aew 2

= f ( w ) E D. %[tI A P(z)I} = {t: t Ef(W)}

Hence M t (3%) A p, so p is snitely satisfiable in N = Mf/D and by

Define g(w) = {t €1: Aasw c[t] c a&] A P(c[t])}, and it is easy to (3) it is realized in it, so let c E IN1 realize p.

check that g: LYwo(p) + D is multiplicative and refines f.

THEOREM 2.3: Let D be a$lter over I , h a cardinal > K O then tk follm'ng conditions are equivdent :

(1) D is &good and N , - i m p l e t e . (2) Por every family of L-models Mt (t E I ) , ntpf M J D is (A, at, m)-

(3) N = Mi/D ie (p+ , at)-compact for every p < h (M,,--asde$nedin co?wt.

2.2( 3)).

Proof. (2) =r (3) Trivial. (3) * (1) The set {{a} c z: a < p} is a set of formulas over lM,,l, and

it is snitely satisfiable in N,,, hence some c E IN1 realizes it. Let for a<Cc

X, = {t E I : M,, t {a} c c[t]}

so clearly X, E D, and if w c p is infinite, t E naEW X,, then M,, t {a} E c[t] for each a E w, hence w s c[t] E (M,,I contradiction. So {X,: a < p} regularizes D; hence, for any h-saturated My M,, < M , by 1.8 player IIwin~inaY+~(Mi/D, M'/D);and, by 1.6(4),asN~/Dis(p,a&)-compact also M x / D is (p, at)-compact. So by 2.2 D is h-good. As D is Ko-regular, it is 8,-incomplete, by 1.3(2).

(1) + (2) As in (3) * (1) it suffices to prove that D is p-regular for every p c A.

336 SATURATION OF ULTRAPRODUCTS [OH. VI, 8 2

CLAIM 2.4: If D ie pf-goOd and N,-incomplete then D is p-regular.

Proof. As D is X,-incomplete, there are X, E D for n < w , X, = 0. Let f : S,(p) + D be defined by f (w) = Xlwl , and let g : BN0(p) + D be multiplicative refinement off. Then for a < p, g({a}) E D, and for any infinite w E p, if t E naew g({a}), then for each n < w , choose w(n) E w, lw(n)l = 12, and then t E g({a}) = g(w(n)) c .X, , hence t E n,,<, X, = 0, so necessarily naew g({a}) = 0, hence {g({a}): a < p} regularizes D .

THEOREM 2.5: Let D be a A-good, K,-incomplete Jilter over I , D, a Jilter over I , D E D,. Let ILI < A, L = L(MP) = L(Mi) for t E I and M' = rite, MflD,. If NP, M i are elementarily equivalent for every t E I , then player I1 wine in GA(M0, M1).

Remark. Compare with 1.8, 1.10. We assume more and prove more. See Exercise 2.3.

Proof. We use the notation of Definition 1.6. Players 11's strategy is to play so that for every formula ~ ( q , . . . , z,) E L, and a, > - . - > a, {t E I : MP C ~ ( a : ~ [ t ] , . . . , aEn[t])} = {t E I : M i C ~ ( a t , [ t ] , . . ., ain[t])} mod D and therefore mod D, since D E D, (where X = Y mod D means

Suppose a:, a: have been chosen for i < a < A, and player I has chosen 1 E (0, 11, af E IM'I, and player I1 has to choose at-'. By the symmetry we can assume 1 = 0. Let @ be the set of formulas Q =

Q(Z, ql,. . . , xi*), Q EL, i,, . . ., in < a. Let @ = { Q ~ : f i . For w E SHo(p) we define

f (W) = {t E I : f i E W * t E x,9, and M i b (3%) A E€wnh(t)

' p ,9(X; ai,,8))[t], . . . )}. By the induction assumption

)> { t E 1: M: b (3%) A 'p8(Z; a;l,8)[t1, . . .

Eewnh(t)

= { t E I : M i k (32) A ~ ~ ( 2 ; ai,,,,[t], . . . t?ewnh(t)

CH. m, 8 21 FILTERS AND COMPACTNESS OF PRODUCTS 337

hence it is eaay to check that f (w) E D. So some g : S,,(p) + D is multiplicative and refines f .

Notice that @: t E g({/3})} c (Is: t EX,} which is finite. So for each t E I, define a:[t] E lM1l so that it realizes ( ~ ( 2 ; a:(1,8,[t], . . .): /3 E g(t)} and a: = (. . ., a;[t], . . .),,I/D. For the induction hypothesis, for each Q, we get only an inclusion ; by applying it also for -Q, we get equality mod D.

THEOREM 2.6: The following d i t i o n e on theJilter D over I , and the cardinal A > No are equivalent:

( 1 ) D iS A-good. (2) For every A-saturated nmkl M of Tora [the theory of deme linear

order I with no Jinite or last element], MI/D is ( A , at)-mpact (where

(3 ) For every A-saturated d l M of Tor,, and for every set A s IMI/DI which is linearly ordered by s , every atomic 1-type p over A in MIID, IpI < A, is realized in MIID.

Proof. The implication ( 1 ) =. (2) follows by 2.2 and (2) =- (3) is trivial; so assume (3), and we shall prove (1 ) .

CLAIM 2.7: Assume (3) from 2.6, and let N = ( I N [ , I ) be a A-saturated rooted tree (i.e., I is a partial order, and for every a E IN I, {b E IN I : b I a} is linearly ordered, and 0 I a for every a E IN/ ) ; 8uch that each a E IN1 hag 11 NII = 11 NI( immediate successors, and for every a, b E IN I there is is a nmxiwl c, c I a and c I b. Then

( 1 ) If N'/D C cf I c, for i < j < a, where u < A, then for some c, NI/D C c, I c for i < a.

< u, i(2) < j(2) < p, where u, p < A, then for some c, NI/D C c: I c I cs for i < a, j < p. Proof. We first note that if A is a linearly ordered subset of IM'/DI, M a model of Tor&, then if every (at, 1)-type over A in M 1 / D is realized in MIID, then for every m, 1 I m < w , every (cnat, m)-type over A in MI/D is realized in MIID.

Now let us choose for each a E IN I ; a A-saturated linear order <a on S, = {b E INI: b an immediate successor of a} (this is possible because IS,] = IlNll = llNll<A).DefineonA* = IN1 x (0, l}anorder ~ * : ( a , l ) I * (b, k) iff one of the following conditions holds:

M = (pq, I)).

(2) I f " / D ca1, I 4 1 , A C h , I CXa, A C X a , I c L , f o r i ( l ) < $1)

(1) a = b, 1 I k,

(2) u < by 1 = 0, (3) b < u,E = 1, (4) let c be the maximal element for which c s a, c s b; c < a, c < by

and a' cC b' where a', b' ES'~, a' 5 a, b' s b. It is easy to check that N, = (A*, s*) is B dense h e m order, with

firat and last elements and it is A-saturated. Hence it follows 888iIy by our hypothesis that if A E IN',/DI is linemly ordered by s*, then every atomic (n + 1)-type p over A in Ni/D, 1p1 < A, n < o is realized in N',/D.

Now define a function P. (1) For a E IN] , F(a) = (Po(a), Fl (a) ) where Po(a) = (a, 0), Fl(a) =

(a, 1) so Po(a), Fl(a) E lNll and for a E INr/DI,

~ ( a ) = << . . . . .~o(a[ t~) - - . >tJD, < - Pi WI) - * - > t d D ) E IWDI - (2) For atomic formulas 'p = p(z; y), define

m'p) = P('p)(%B,, = <ZOY 6, B = <YO*Yl)Y

m 5 9) = [zo s Yo A Yo 5 Y1 A Y1 5 511,

P ( z = Y) = 1% = Yo A 2 1 = YiI,

F ( y 5 2) = [Yo S 5 A 20 5 21 A z1 5 YJ.

(3) For an atomic formula 'p(z; a) define

W z ; 4) = P('p)(zo:O, z1; p ( a .

P ( p ) = {F('p@; 4): ' p k 4 EP}. (4) For a set p of atomic formulas 'p(z; a) let

Now it is eaay to check that for every such type p in N, p is realized in N iff F ( p ) is realized in N, . Hence by 2.8 below for each atomic type p in Nr/D, p is realized in N r / D iff P(p) is realized in N',/D. For each p as in Z.Q(A) or (B), F ( p ) is an atomic 2-type over some linearly ordered subset of JN',/DJ in N',/D, so F ( p ) is realized in ",/D, hence p is realized in Nr/D, hence we get our conclusion.

SATURATION OF ULTRAPRODUCTS

i.e.,

P(F~(Z', = # I ( $ , 61) ( I = 1,2) implies 9 1 = pa * #1 = $2.

(3) For every p E Dom F , p ia r e d i d in M ifl

P(p) = {P[p(Zl, a)]: y(Z1; a) E p }

ie r e d i d in N. Then if p = {pt(Z1;at): i < a} i s a set of atormi0 fotrnulae in Mr/D,

u n d f m w y i < a { t E I : p t ( P ; ~ i [ t ] ) ~ D o m E 3 ~ D t ~ p i a r ~ i d i n Mr /D iff P ( p ) = {P[yf(Z1; Q ] : i < a} ia r e d i d in Nr/D.

Prmf. Suppose E realizes ~ ( E E lMI/Dl); for every t E I let w(t) = {i < a: M C pt(8[t]; a#]) and pi(%'; a#]) E Dom E") E D and define P[t] E IN1 so that it realizes

{P[pl(Z1; qt])]: i E w(t)}.

It is easy to oheok that Z1 realizes F ( p ) . The other direotion is e a q too.

Continuation of the pmf of 2.6. Let M be any h-saturated model, IL(M)l s IlMll, and we shall prove that iKz/D is (A, at)-compaot. This is suffioient by 2.2. Let

r= {(po(2,a~),...,pm-l(2,a~-l)): m < o ,a t~ lJ f l r (~r~L(Jf ) ,

pf atomio, and Af C (32) A pt@, af)}.

Clearly II'J = J1MJ1, so there is a one-to-one function g from ]MI onto I'. Let us define some new relations and functions over lMl (aeauming

that they 4 L(N)). (0) c* wi l l be g'l(( )). (1) s : a s b o g(a) is an initial segment of g(b). ( 2 ) Fl: P ( a ) is an element of M realizing g(a). (3) Pe (for eaoh conjunotion p(z; @) of atomio formulas): PJb, a) o

-oh oonjunot of p(z; a) appears in g(b). (4) Qe (for eaoh oonjunotion p(z; j i ) of atomic formulas): Q,(b, a) o

cp(x;a) is consistent withg(b), i.e., g(b)^ (+o(x;d ' ) , . . . , pnpl(x; a"-') ~r where p(z; a) = At<,, #{(2; #).

(6) FZ (for eaoh p(z; 1) &B in (4): PZ(b, a) = c i fQ,(b, a) and g(c) =

g(b)^(#o(s; a'), . . . , #,,-l(s; a"-')) or +,(b, a) and c = b.

i e m

340 SA"RA!FIOW OR' ULTRAPRODUOTS [m. 5 2

(notice that in (6) if M C (34942; a), then M C (3z)[Af., qr(z; at) A

Let us define N = (My c*, s , Ply P,, &,, P:, P:),, and let N1 be a A-saturated elementary extension of N , [INl 11 = [INl 11 , and Ml be the L(M)-reduct of N, . Clearly Ml is A-saturated, hence by 1.11 M*/D is (A, at)-compaet iff M , = M i / D is (A, at)-compact, so it suffices to prove that M i / D is (A, at)-compact. Clearly Ma is the L(M)-reduct of N , = N',/D. It is easy to check that (lN1l, 5%) satisfies the requirement of claim 2.7, hence also the conclusion of that claim.

Let p = {tpf(z; af): i < a. < A} be a cnat-type in M , which is w.1.o.g. closed under conjunctions, and we will prove that it is realized in M,. We define by induction on i I a. elements cf E

d x ; a)l).

such that: (A) j < i * N , C C$ 5 cf . (B) For every fl < ao, Na C &(pa(cfs 3fi)- (C) If i = j + 1, N , C P,,(Cf, a,).

Case I: i = 0. Let co = c* (the root of the tree). Conditions (A), (C) are vacuous, as for condition (B), for every /3 N C (Vij)(Vz)[rps(z, 8) + &,&*, jj)] and this is a Horn sentence, so N , also satisfies it. As p is a type in Ma, N , C (32)9~&, Zfi) hence Na C QCp4(c*, a#).

Case 11: i + 1. Let cf+l = P:,(cfy Zf). For condition (A) notice that (i) N t (Vzyz)(x 5 y A y ls; z +z 5 z), (ii) N I= (VzP)(z s P:JXY P)),

and both are Horn sentences, hence Na satisfies them. By (ii) N , C cf s c , + ~ , and by (i) and the induction hypothesis j < i

As for Condition (B), for each fl < a. there is y = y(fl) < a. such that * N , Ccj 5 c:+1.

p,(z; PI = 'pv(z; Pi, Sa) = vfi(z; Pi) A vi(% 9,) and 8, = afi-4, (remember p is closed under conjunctions). Clearly

N C (VZ, zl, gl, ~,) [&, , (z; j j1 , P,) A 21 = P:Jz9 Pa) + &cpp(zly 8113 and as this is a Horn sentence Na satisfies it. As N , C &,&; a, si,) A cf +1 = q c , Y a clearly N2 c &,a[c:+l, %I.

OH. VI, 8 21 FILTERS AND COMPACTNESS OF PRODUCTS 34 1

As for Condition (C), N C (Vg, zl, z)[&,,(z, 8) A z, = P;Jz, 8) + P,Jzl, g)] so by condition (B) clearly N , C P,,(ct+,, at).

Case 111: i = 6 is a limit ordinal. By Claim 2.7( 1) and Condition (A) for j < 8, clearly there is do E lN,l such that j < 6 =r N , C c, 4 do. So do satisfies Condition (A) on c6, but not necessarily Condition (B). We now define by induction of j 4 a. elements d, E lN,l such that

(*)

We have already defined do; for limit j, the existence of d, follows by part (2) of Claim 2.7 (using (*) and Condition (A)). For j -+- 1 let a?,+, = Pz,(d,, a,), and we shall prove that (*) holds.

jl < j, 4 ao, y < 6 * N , C cy 4 dj2 A d5* I

Notice that N C ( v W ~ ) ( q , ( z 9 a, 2)

and this is a Horn sentence, hence N , satisfies it, hence N, C d,,, I d,, hence (as &'a is transitive) j, < j + 1 5 N , C d,+, I d,l. On the other hand,

N C ( ~ 8 N V Z l 9 Z,"l I 2, A &a& 8) --f 21 I F:,(z,, a11 and this is a Horn sentence, hence N, satisfies it.

As for y < 6 N , C cy 5 d, (by the induction hypothesis on j) and N , C &,,(cy, 8,) (by the induction hypothesis on i = 6) clearly N , C cy I P&(d,, a,), but P:,(d,, 8,) = d,+,. So we prove (*) f o r j + 1.

So we finish defining d,,j I ao, and let c6 = dao. Condition (A) holds by (*), Condition (C) is vacuous and Condition (B) holds because for eachp < ao,

N c (vg)(vzl,z)r&,,(z, 8) /t 21 5 z + QQJq(Z1r ?a1 and this is a Horn sentence, hence N , satisfies it, and N , I= duo s dB+,.

this is a Horn sentence, and as p is a type in M,, for some b E ]N,l

So we finish the definition of ct, i I ao. We shall prove that P1(cao) realizes p and thus finish the proof.

Now for each /I < a. N C (Vg, zl, z2)[Pe,(z1, 8 ) A z1 S 22 + P,,(z,, a)] and N C (Vz)(Vjj)[P&, 8) + tpB(P1(z), g)] and they are Horn sentences hence N , satisfies it, and N , C Pmp(cB+,, Gg) A cg+, I c,,, by Condition (C). Hence N , C rpg[P1(c,,), BB] for every /3 < ao.

Now N , C &e,,(dB + i s 3,) because N C (vz, 8, z)[~g(z, 8) +Q,p,(P:,(z, ji), 811;

N , t ag[b, @gI hence N , I= &,p(p:,(d,, %), ad.

THEOREM 2.9: Let h > No, D a Jilter over I . Then the following conditions are equivalent.

342 SATURATION OF ULTRAPRODUCTS [CH. 8 2

(1) D is A-good and K,-inmmplete. (2) For every model H of Tor,, M'/D is (A, at)-mpact. (3) NLlD is (A, at)-compact, where N , = <"'p, <), <--whit

order, for euch p < A.

Proqf. (1) + (2), (1) =+- (3) Hold by 2.3. (2) * (1) By 2.6 it s d c e s to prove that D is 8,-incomplete. As

(&, 5 )I/D is (A , at)-compact (&-the rationals) for some c E I(&, s ) I / D ~ , n 5 c for each n < w , and let

X , = {t E I : n < ~ [ t ] } . Clearly X, E D, (I X, = 0. n

(3) * (1) The proof is similar to that of 2.6, hence we leave it to the reader.

CONCLUSION 2.10: If A i.9 a singular Cardinal and thefilter D is A-good, t h D i8 A+-good.

Proqf. We use 2.6(3), so let A€ be a A+-saturated model of Tor,, A s IN1 = IM'/DI be linearly ordered by s , and p be an atomic 1-type over A in N , 1p1 s A, and we shall prove that p is realized in N , thus finishing. If for some a, (z = a) EP, a realizes p (because p is finitely satisfiable in N). SO let p = {U 5 Z: u E A,} U {X < a: u E As}. As A, c_ A, A, is linearly ordered by 5 , so it hm a cofinal sequence

{a,: i < pl}, pl a regular cardinal (so ( V u ~ A , ) ( 3 i < p,)(a s a,) and a, E Al). As p1 is regular, p1 < A. Similarly there are a' E Aa (i < pa) such that ( V a ~ A , ) ( 3 i < pa)(ai s a), pa regular. As M satisfies the Horn sentence saying 5 is transitive, an element realizes p if it realizes

p1 = {a, s x: i < pl} u {z 4 a,: i c pa}.

As lpll c A, and D is A-good, by 2.2 N realizes ply so we finish.

Rernurk. See Exercise 2.7.

Up to now we have dealt with filters and atomic compactness. By the following lemma, we do not need to deal separately with ultrafilters and compactness.

LEMMA 2.11 : Let D be an ultrafilter over I , A > No a cardinal.

A--pact. (1) For every M y MI/D is (A , &)-cumpact iff for every A€, W / D is

OH. 5 21 FILTERS AND UOMPAUl"Es8 OW PRODUaTS 343

( 2 ) For evey L, and L d l s Mty n:,, M J D is (A, ati)-wmpwt iff for every L and L-nwaeb Mt, ntsI Mt/D $8 h-wm?mct.

(3) A8 ( I ) [2] W for A d W t Z & & M[MJ

Proof. In all parts the if part is trivial. For the other direction, for each model iK define M* aa follows:

LW*) = {%a: 9m E L(M)}Y

(%3) w Places)

1M*1 = 1M1, R$*) = {a E IN!: M k tp[a]}.

By Logs Theorem it is easy to check that (ntsI M,/D)* = nt,I M r / D hence Mr/D is A-compact iff AZ*'/D is (A, at)-compact and nsI Mt/D is A-compact ii€ nt61 M:/D is (A, at)-compact, and iK is A-saturated iff M* is A-saturated. Hence the "only if" part follows easily.

THEOREM 2.12: 8uppo8e 2" = A + , IIl = A, D a good, K,incumplete, UltraCjElter over I ; M y N are elementarily equidnt nzodeb of mraidi ty <A+,andL(M) = L(N)ha%pomr < A . ThenMr/Dy Nr/Darei80mtwphic (and saturated).

Proof. By I, 1.11 every two elementarily equivalent A+-satursted models of power A+ are isomorphic. By Log's Theorem Mr/D, Nr /D am elementdy equivalent. So, by the symmetry, it suffices to prove that Mr/D haa cardinality s h + and is A + -saturated. Now

llMr/Dll < ~ ~ M ~ ~ ~ r ~ < ( A + ) A < 2"'" = A+

and by 2.3 and 2.11 Mr/D is A+-compact. As JL(M)I s A, HI/D is A+- saturated, so we finish.

THEOREM 2.13: If D & a good ultrafler over I , nter nt/D 2 KO, then nteI n J D = 2Irl.

Proof. I f D is 8,-complete, clearly for some n {t: n, = n} E D hence nt/D = n; hence D is XI-incomplete (by Exercise 1.8) so by 2.4 D

is regular. For each n let M , be the following model: M,, = (n + 2", P,, Bn>,

where P,,, B,, am one-place and two place relations respectively, P,, = (0, . . . , n- l} and R, is such that for every w E P, for some i, n < i < n+2", and for every k < n , k E w o R , ( k , i ) . Let for t ~ 1 , m(t) =

[log, ntl - 1, 80 ll~m(:)ll 5 %, but for every n {t: llJfln(:)ll 5 .} E

344 8A"RATION OF ULTRAPRODUCTS [OH. VI, 5 2

{t: nt s 2"+l} $ D. Hence M = (A, P, R ) = nteI M,,,,,)/D is an infinite model and is 111 +-saturated. So clearly JPI > IIJ (otherwise {P(z) A x # a : a ~ P } is omitted by M) and ]A1 2 21'1 (for let C E P, ICI = 111, then for everyB s C, p , = {R(a; x)if(aEB): ~ E C } is realized by some b,eA, so IAl 2 21'1 = 21'1. We can conclude

and the other inequality is Exercise 1.3(2).

EXERCISE 2.1: Show that to the list of equivalent conditions in 2.3 we can add:

(4) If p is an atomic 1-type in M = niEI M f / D of cardinality <A, then p is realized in M.

EXERClSE 2.2: Show that in 2.5 instead of assuming q, Mi are elementarily equivalent for every t E I it suffices to assume n t e r M!/D and rite, MilD are elementarily equivalent.

EXERCISE 2.3: Show that in 2.6 we can omit " D is 8,-incomplete" but must then demand that for each t E I, M;, M: are A-saturated and elementarily equivalent.

EXERCISE 2.4: Show that to the list of equivalent conditions in 2.9 we can add.

(4) (i) N:/D is (A, at)-compact for some p. (ii) D is p-regular for each p < A.

EXERCILYE 2.5: Let E be a family of subsets of I closed under finite intersection, and let E generate the filter D. Every monotonic f : SNo(p) -+ E has a refinement g : SNo(p) + E which is multiplicative iff D is p+-good.

EXERCISE 2.6: (1) Every A-complete filter is A+-good. (Hint: See 2.1.) (2) If D is p-complete for each p < A, A singular, then D is A + -

complete.

EXERCISE 2.7: I f A is regular 5 Ill, then over I there is a non-A+- complete filter D which is A-complete. Moreover D is p-good iff p S, A+ (Hint: D is generated by {X,: i < A}, X , decreasing, 0, X, = 0.)

CH. M, 5 31 CONSTRUOTINQ ULTRAFILTERS 345

EXERCISE 2.8: Suppose the filter D over I is p+-incomplete and A+-good. Show there am X, E D (i < A) such that the intersection of any p X1)s is empty.

EXERCISE 2.9: In Definition 2.1(1), when A = p$, it suffices to take I.1 = Po.

EXERCISE 2.10: Suppose D is an 24,-incomplete filter over I, and A = ntErnt/D 2 24,. Prove that P o = A. [Hint: Let M = (w, <, +, x , P). where P(x) = [GI, N = Mr/D, and n* = (. . . , nt,. . . ) /D . and for a E INI, 1.1 = l{b E INI: N t a < b}l. Then In*] = A, 1.1 2 KO =- 1.1 =

IP(a)l, and let nt = P(n*), nf+, = P(nf) , and for every sequenoe = (b,,; n < w) , bk < nf, let 1); = (n,*bo + n:b, + - . . + nzbk < x < ntob0 + - - . + nfb, + nz: k < w}

clearly p; is realized by some ai;, thus providing n k Inf I = AHo distinct elements of {b: b < n*}.]

EXERCISE 2.11: In Theorem 2.12 instead “D good” we can assume only “there is a good K,-incomplete filter D, G D.”

EXERCISE 2.12: Show that no 24,-incomplete filter over I is 1I1+ +-

good.

VI.3. Constructing ultrafilters

We shall show that over every cardinality there is a good ultrafilter. We define the product of ultrafilters, and find how regular and good the product is. Then we assume 2Wo > N, and MA (Martin’s axiom) to prove the existence of an ultrafilter D over w , such that for every N,- saturated model M of Find, M a / D is N,-saturated.

THEOREM 3.1 : Over any A there ia a good K,-inunnplete ultrajlter.

Proof, We first give a definition and prove some claims.

functions from A onto A. Meanwhile let D denote a filter over A, and 59’ denote a family of

DEFINITION 3.1 : 59’ is called independent mod D if for every n < w and distinct go, . . . , qn-, E 59’ and every jo, . . . , j , - , < h

{a < A: go(a) = jo, . . . , g,,-l(a) = jn-,} # 0 mod D

(and D is non-trivial).

CLAIM 3.2: Let Do be the@er over A generded by {A}. T h t h e ia a famay 8 of eardidity 2" which b id- mod Do.

Procf. This is a reetatement of 1.6 of the Appendix for a p d c u l m c&88.

CLAIM 3.3: Let Q be h u i ? e p e mod D, and 8 E A. Then for 81nne jinite W C 0,5% - 8' ia id- mod Dl or mod Day w h e DI[Da] ia thejElter generated by D U {IS)[D U { A - IS)].

Proof. If 8 is not independent mod D,, then for some n < w there are distinct go, . . . , g,,- , E Q and there are joy. . . ,j,,-, < A such that: W, = {a < A:go(a) =joy.. .,g,,-,(a) =j,,-,} = 0 mod D,, hence W, c A - 8 mod D (notice that n = 0 mems D, is trivial). Let 5%' = 8- {go, . . . , g,, - and assume 8' is not independent mod Day 80 there are m < w and ju, . . .,jm-l < A and distinct go, . . ., gm-' E Q' such that Wa = {a < A: go(.) = jO,. . ., #'"'''(a) = jm-'} = 0 mod Day hence Wa c 8 mod D. So W, n Wa = 0 mod D, contradicting the independence of 8 mod D.

CLAIM 3.4: 8-e Q is independent mod D, g E Q, 9' = B-{g}, and f :BNo(A) + D W momtonic. Then there is a f l e r D , D c D , and a mdipl icdve function f' ; tlN0(A) --t D' refining f m h that Q' W i d - pended mod D'.

Procf. Let {w,: a < A} be an enumeration of Lgn,,(A), let f'(w) = @: g@) = a, 8 E f (w,), w C w,} and D the filter generated by D u v({i}): i < A}. Clearly D' is non-trivial and f' is multiplicative, into D', and a refinement off.

Let us prove 3' is independent mod D , so let jo, . . . , j,,- < A, and go, . . . , gn-, E Q', the glYs being distinct and

W = (8: go(8) = j o , . . . Y !&I-,@) = it-,}. We must prove W # 0 mod D. For this it suilhes to prove that if w E&~(A) , W n f'(w) # 0 mod D. Let w = w,, 80

so we finish, &B 8 is independent mod D and f (w) E D.

OH. VI, 8 31 CONSTaUCTINa ULTRAFILTERS 347

Proof of 3.1. Let (8,: a < 2”, a > 0 even} be an enumeratiiga of (8: 8 G A}, and dfor: a < 2A, a odd} an enumeration of the functions f : &,(A) +- {S: 8 G A}, each appearing 2A times. Let Do be the filter over A generated by {A}, and So be independent mod Do and of cardinality 2 A .

Now we define by induction on a D,, g, such that: (i) ’3, is independent mod D,, (5) 1’3 - 5 K O + lal, (iii) for /3 < a, DB G D,, ’38 2 ’3,, (iv) if a is a limit ordinal D, = U,<, D,, ’3, = n8<, g,, (v) if a is even, a > 0, 8, E D,+, or A - 8, E D,+l , (vi) if a is odd, f , : &,(A) + D,, f a monotonic then there is a multi-

plicative refinement f: of f a , f: : &,(A) + I),+,. The induction is easy: for a = 0 we have defined, for a limit see (iv)

(using iii), for a + 1, a even, a > 0 use Claim 3.3 and for a + 1, a odd use Claim 3.4, and for a + 1, a = 0 let D , = {{a: n < g(a) < w}: n < w},

’3, = ’3,,-{g], where g E go. Now D,A is an ultrafilter by (v). I f f : &,(A) + DaA, then for some

a < 2A, f : &,(A) +- D,, and for some /3, a < /3 < 2”, f,3 = f , /3 is odd, so (when f is monotonic) there is a multiplicative refinement f’ off, f ’ : &,(A) + DB+1 E D+ SO D,A is a A + -good ultraater over h. AS D, DaA, D,A is N,-incomplete.

CONCLUSION 3.6: If My N are elementarily equivalent models of T , of cardinality 5 A + , I TI 5 A, 2A = A+, then for some ultrajlter D over A, MA/D g NA/D.

Proof. Immediate by 2.12 and 3.1.

DEFINITION 3.2: If D, is a filter over I, (1 = 1, 2) then D1 x D, is the family of sets 8 E I , z I , such that:

{i E I,: {j E I , : (j, i ) €8) E Dl} E D,.

LEMMA 3.6: (1) D1 x D, is ajlter over I , x I,.

jlter over I , x I,. (2) If D, is an ultrafilter over I , ( I = 1, 2) then D , x Da ie an dtra-

(3) I%i,j>er1 Jfi,j/Di x Da I L r P (nrer , Jfi,j/Di)/Da. (4) Mrixra/Dl x D, 2 (Mrl/DJa/Da.

Proof. Immediate.


LEMMA 3.7: (1) If D , OT Da i8 h-regular then D, x Da is h-regular.

incomplete. (2) D , x D, i8 h - g d ifl D , iS hyood; PO& that D , i8 81-

Prmf. (1) Immediate by 1.4(1)(A), (C) and 3.6(4). (2) Immediate by 2.3 and 3.6(4). If D, is h-good, for any h-saturated model M , Mri ra/Dl x D,

(M'l/Dl)'a/D, (by 3.6) is (A, at)-compact (by 2.3). Hence by 2.3 D , x D, is h-good. If D1 x Da is h-good we get the result similarly.

CONCLUSION 3.8: POT every h I p, there is a regular ultrafilter D Over

n,<, n,/D 2 No impliea n,<N n,/D 2 2A > A+. p Whkh is h+-good but not A+ +-good. If A+ < 2", D i8 not A+ +-good but

Prmf. Let D, be a regular ultrafilter over p, and D, a A+-good K,- incomplete ultrafilter over h (exists by 3.1) D, is not A+ +-good by 2.4 and 1.3(3). So D , x D, is an ultrafilter over p x h which is h+-good but not A + +-good. As IX x pI = p, we finish. The second part follows by 2.13.

LEMMA 3.9 (MA): There i8 an ultrajlter D Over w such that: If IP"nI I No, and M = n, < Ir) M,/D and p is an m-type in M of cardinality < and P(xo) A - A P(x,,,-,) ~ p , then p i8 realized in M .

Remark. On Martin's Axiom the rertder can consult, e.g., Jech [Je 741. It is consistent with ZFC + 2N0 = K, for any regular Eta, if ZFC is consistent and it implies h < 2No + 2" = ~ H o . As we use it only rarely, we do not elaborate.

Proof. It is sufficient to prove the lemma just for models M , such that

a odd} be an enumeration of {S: S E w } and {(p,, ( M t : n < w ) ) : a < 2N0, a even} be an enumeration of all pairs ( p , (M,: n < w ) ) where L = L(M,) has cardinality < 2N0, IIM,II < 2N0, and p is a set of cardinality < 2No of ~(3,7i) ( I @ ) = m, E E n,,<Ir) M a ) , and lPMnl < No; each pair appearing 2'0 times.

We define by indaction on a < 2n0, families E, of subsets of w , such that

(1) E, generates a non-trivial filter [E,] over w which is regular, (2) /3 < a + E6 E E, and for limit a, E , = u s <, E , and IE,l < 2H0, (3) for a odd, S, E E,,, or w - S, E E,+,,

llJfnII < 2'0, IUMn)J < 2'0. So let {S,: a <

CH. VI, 8 31 COWSTRUCTINQ ULTRAFJLTEE8 349

(4) for a even if for every tpl(Z, a,), . . ., tpn(P, a,,) €pa,

{i < W: Zt b (%)(p(Zo) A * * . A p(Zm-l)

A tpl(Zs %[iI) A * + * A Ts(z, an[iI))} E [#a]

then for some 6 E nieO M f , 6 = (bo, . . . , b, - I), for every tpl(Z; 8,) E pa

{i < w: Mf k P(bo[i]) A * * * A P(b,-,[i]) A p1(6[i], 81[i])} E [Ba+l].

Clearly if we succeed then [,?#,NO] is the ultrafilter we want: and there is no problem for a = 0 or a limit, or a odd (by 1.1 of the Appendix). So aasume a is even and pa, M: (i < w) satisfy the hypothesis of (4) (the other cases are trivial).

Let us define a set V of “forcing conditions” a forcing condition is a finite set w of equations xl[i] = a where I < m, i < w, a E PI; such that q[i] = a E w and zr[i] = a’ E w implies a = a’. V is a partially ordered by inclusion, and it is countable, hence there are no N, pairwise incompatible conditions. For eachq(E; G) = {A q : q C pa, q finite} and finite interseation S of members of Ea let

w, t p m a)) = {w E V : for some i E 8 and bo, . . . , b, - E P ( M f )

Mp c cp[bo,. . ., bm-,, tqi]], {ZO[i] = bo,. . ., X,-l[i] = b,-,} E u}.

Clearly each V(8, tp(Z; a)) is dense in V (by our hypothesis from (4)) and their number is <2% Hence by MA there is a generic V* E V , i.e., every two members of V* have a common upper bound, and V* n V(S, tp@; a)) # 0 for every tp(Z; a) EI);, S a finite intersection of members of Ea. Let bl E ni<O Mf, b,[i] = a iff q[i] = a E U {w: w E V*} ( I < m). 6 = (bo , . . . , b,-l) is well defined aa V* is generic. Let

E,,, = Ea U {{i < o: Mf C tp(6[i], a[i])}: cp(Z; a) €pa}.

[Ea+,] is a proper Glter as V* intersects each V(S, ~ ( f , a)) end the conclusion of (4) clearly holds.

THEOREM 3.10 ( M A ) : (1) There is a regular ultrafilter D Over w, 8uch that if M is a A-snturated &el of Ti,, (see 11, 4.8), A 5 2H0, then M m / D ie A-saturated too.

( 2 ) If h < 2’0, there is a regular ultraJilter D over h which is not N,- good such that for any model M of Tind, MAID is A+-saturated.

350 SATURATION OF ULTRAPBODUOTS [OH. VI, 0 3

Proof. (1) Let D be the ultrafilter from 3.9. If p is any 1-type over P / D , 1pI = p < A, let p = {q&; a=): a < p}, let A, = U {aa[i]: a < p} by 1.6 of the Appendix, and the elimination of quantifiera of Tina, and the p+-eaturation of bl, there is BI E 1611, lBil = KO, such tha& any non-algebraic finite type over A, is realized by an element h m BI. Let us expand M to Mi = ( M , P'a), P a = BI and apply 3.9.

(2) Let D, be a good K,-incornplete ultrafilter over A (exista by 3.1)) so D, is regular by 2.4, and Da the ultrafilter from (l), then D = D, x D, satisfies our demands (Dl x Do is regular by 3.7(1) and MAX"/D = (MA/D,)"/D,, iKA/Dl is A+-eaturated by 2.3 and 2.11 as D, is A+-good and K,-incomplete. So by (1) (MA/D1)"/D, is A+-saturated too).

Let 9 denote a family of functions g : A 3 A, g onto A, D a filter over A. Exercises 3.2-3.4 are from [Sh 7 1 ~ 1 .

DEFINITION 3.3: (g,, g2, D) is K-independent if whenever j, < A (5 < to < K) g' E g a (1 < n < o) and f,, f' E g1 me distinct ( 5 < to, 2 c n) then {a c A: fr(a) = j, for t < to, f'(a) = g'(a) for 1 < n} # 0 mod D.

EXERCISE 3.1 : There is a gl of cardinality 2A such that is K-independent, Do = {A}, provided that A = A<".

8, Do)

EXERCISE 3.2: If (g , ,a ,D) is K-independent, D = [El (the filter generated by E) , Yta is a family of functions from A into a, a < K , then there is '3; G g,, Igl - yP;( s 111 + I g s l i- K , such that ('3;, g a , D) is K-independent .

EXERCISE 3.3: If (g,, 0, D) is K-independent, S c h then for some 9; c g, Ig, - gil < K and (g,, 0, D') is K-independent, where D' = [D u {s)] or D' = [ D u {A - A!?}].

EXERCISE 3.4,: Assume (gl, 0, D) is K-independent, D = [El. Assume also that Mi (i < A) is an L-model, lPMiI s x < K, aB,,, E n ~ < ~ Mi forB < Po < 2", 1 s rn s n(B).AssurneqBELand{qs(z,YB,,,. . . , Y ~ , , , ( ~ ) ) : /3 c Po) is closed under conjunctions, and for every B < B0

OH. VI, 8 33 OONSTRVOTWO WLTRAR'ILTIRS 35 1

EXERCILYE 3.5: (1) There is an ultrafilter D over A suoh that, if h = A<",then:

(i) If M, N are elementarily equivalent and of cardinality C K ,

then MA/D z NA/D. (5) If IlMll < K, 2Y s 2A then MA/D is p+-saturated.

($) If IpMtl s KO < K, 2' s 2A, p a l-tme in n { < ~ H{/D, P ( X 0 )

lpl s p then p is realized in nfeA M,/D. (iv) If 2" s P, 2% = A, M a model of Thd of mrdinality sA, then

(v) If IlHilJ 5 x < K, MI, M' are elementarily equivalent where isomorphio (Z = 1,2) provided that

M"/D is p+-satuahd.

M' = ntxA M!/D then M', M' IL(M1)1 s A.

Remark. Notice the following difference between the proof of Exeroise 3.6 aketohed by the preceeding exercises, and the proof of 3.1. In the proof of 3.1 the funations in g are used as partitions of A, whereaa in the proof of Exeroise 3.6(1) they are used as elements in the ultra- produote.

EXERCI8E 3.5: (2) In Exeroise 3.6(1) (and Exercise 3.1-4) instead of atesting with the filter Do = {A}, we can stwt with any K-complete filter D, over h provided the oonolusion of Exeroise 3.1 holds. (In Exeroises 3.2 and 3.4 [D, u El replace [El, and in Exeroise 3.6 D, c D.] Of oourse, we c)an star t also with [D, u El, provided that IEI < 2", and the ooncluaion of Exeroise 3.1 holds. If there me A, c A, A, # mod D,, A, n A, = 0 (for i < j c A) then the c#>nclusion of Exeroise 3.1 holds.

EXERCIHE 3.6: Let f be a 2-place function from 8%(p) to subset of I , let D be a p + -good filter over I , and X , (i < p) be subsets of I . Suppose

(1) For m y 8, t Ergwo(P),

x, n n ( I - x,) c f(8, t ) mod D. te8 tct

(2) If 8, t €8&), 8 A t # 8 thenf(8, t ) = 8. (3) f(0,O) = 1.

352 SATURAlTON OF ULTRAPRODUOTS [a. m, 8 3

(4) I f 8 , t , WE&&), 8 n t = 8, 8 U t C U, then

f(8, t ) u (f(81, ti): 8 C 81 C U, t E ti c U, 81 u = U}.

Then there am subsets Y1 (i < p) of I such that: (A) Y1 = X 1 mod D for each i c p. (B) For every 8, t E &&),

n y1 n n ( I - yl) E f(8, t ) . 168 1e:

On Exercises 3.6 and 3.7 see [Sh 72~1.

EXERCIBE 3.7: Prove that the following conditions on the filter D over I and the cardinality A > No are equivalent.

(1) For every set of L-models Mt ( t E I ) n,, M,/D is A-compact. (2) D is K,-incomplete, A-good and B(D) is A-saturated, where B(D)

is the Boolean algebra of the subsets of I mod D. (Hint: Use the Fefer- man-Vaught theorem; see e.g. [CK 731 and Exercise 3.6.)

EXERCI8E 3.8: In Exercise 3.7 we can replace A-compact and A- saturated by (A,d)-compact for d the set of Z,, (or n,) formulas (A C,-formula is a formula of the form #(#) = (3Zl)(VZ2). . . ‘p(Zl, . . . , Z,,, g), ‘p quantifier free; a lI,,-formula is the negation of a C,-formula).

EXERCI8E 3.9: Prove that B(D) is isomorphic to M i / D where M, is the Boolean algebra with two elements.

EXERCI8E 3.10: Show that in general, in 2.3(3) we cannot replace “for any p < A” by “for some p < A”. (Hint: Use Exercise 3.5.)

EXERCIHE 3.11: (1) Show that for any A-good filter D, B(D) is (A, at)- compact (See Exercise 3.12).

(2) Suppose M , is a ( A + , at)-compact Boolean algebra, of cardinality A+, and 2A = A+. Prove that there is an K,-incomplete good filter D over A such that B(D) is isomorphic to M,.

(3) We can suppose M , is a compact Boolean algebra of cardinality 2A, and (Vp < ZA))(2J’ s 2”), and get the same conclusion.

Remark. In (3) the existence of M, implies (Vp < 2”(2” s 2A) by VIII, 4.7. Similarly in (2), it implies 2“ = A+.

[Proof: (2), (3). Similar to the proof of 3.1. In the ath step we have a filter D, = [E,], lEal s A + lal, a sub-

algebra N , of No, llNIIll s 1.1 + H,, an embedding Ha of N , to the

Boolean algebra of the subsets of I such that 0 # a E lMol implies &(a) # 0 mod D,, and a family 9, of functions from h to A, such that for every distinct g,EY, and j, < h (I < n < w ) and a € Val, a # 0 {i < h:go( i ) =jo, . . . ,g , , - l ( i ) =j,,,}nH(a) # OmodDaandlS- s h + 1.1. (Clearly Nu, Ha, E, increaae with a, and S, deareaees with it, eaoh of them oontinuously. In the end D = D,n, H = H 2 ~ . )

We have steps of t h kinds: Kind I: We want to include b in the domain of H . Choose g E 9,,

= {i: g ( i ) = 0}, and let Nu+, be generated by Nu and b, S,+, = S, - (s) and

B,+, = E, u ( I - Ha+,@,) n H,+,(a): a E Nu, a n b, = 0

and b, is b or its complement}. Kind 11: We want to refine the monotonic funofion f : S&) + D,.

We do it exactly aa in 3.1. Kind 111: We want to decide the fate of S c h (that is, w e want that

for some a E No+, , 8 = Ha+,@) mod Let here A denote a function: A = {(gf,jf): i < io} where io < A+ +

1.1 + and gf E S are distinct. For such A let P be the filter generated

~h = B, u {&+(a> = jf: a < A}: i < io}

S h = g, - {gf: i < io}.

by Eh9

and

Cme (i): For some a E Nu and A, H&) = S mod Dh. We then dehe: Nu+, = Na,H,+l = H,,9,+l = gh,E,+, = Eh(andofcoureeD,+, = [E, + ,I). It is eaay to check our demands hold.

Cme (ii): Not (i), but there are a, b EN,, A such that H,(a) E 8 c H,(b) mod Dh, and b - a is an atom of Nu (i.e., (Vc E N,)(b - a E c v (b - a) n c = 0)). We then define Nu+, = Nu, So+, = gh, I#,+, = Eh u { A - (IS - Ha@))}. The only way in which our demands may fail is that for some A,, A E A,, and c EN,, c # 0 such that H,(c) G S - Ha@) mod P'. But then Ha@) n Ha@) = 0 mod D', hence c n a = 0; but also H,(c) E H J b ) mod P', hence c c b; so as b - a is an atom of Nu, c = b - a. Now it follows that Ha@) = Ha@) U H,(c) = S mod P', so caae (i) holds, contradiction.

Cue (iii): Not (ii) or (ii) but for some a E Nu and A, H(a) E S mod Dh, and for every b EN^, h' : h E hl, H,(b) c S mod Dh' implies b E a (in N,, of course). We then choose S,+,, B,+, as in 0&88 (ii). The

OONSTRVOTWO WLTRAR'ILTIRS

only way in which our demands may fail is that for some C E N ~ , c # 0 and hl, h E hl, the following holds:

H(c) s 8 - HJa) mod P', but then again c n a = 0, and H(c u a) c S mod P', so c u a c a, henoe c = 0, contrediotion.

Cme (iv): Not (i), (ii) or (iii), but for some b E Nu, h: S c H(b) mod P, a n d i c h 1 , c ~ N a , 8 c H ( c ) m o d P ' i m p l i e e b c c .

The p m f is similar to that of c&88 (iii); this time Ea+, = Eh U

Cme (v): None of the previous ones. Choose h such that if c E Nu, c f 0, and 8, E {S, h - 8), B c I , , and Ha@) n S1 = 0 mod Dh' then Ha@) n 8, = 8 mod P. Now letp = (8 c b: b E N , , S c Ha@) mod P} u {a E x: a E Nu, &(a) E S mod P}. It is easy to check p is a type in

compect, some c E M,, r d i z e s p. D e h e Nu+, aa the subalgebra of N o generatedbyIN,I u {c},Ha+,eXtendHa,Ha+,(c) = 8,andE,+, = Eh. We leave the reader the checking.]

{A - (H(b) - S)).

Nu, and s llNall + No s A, p is atomic, and N is (A+,&)-

EXERCISE 3.12: Show that the Boolean algebra Mo is (A,&)- compact iff for every a, b < A, a,, bt E No satisfying No C a,,,, s a,,, s bj(,) s a$(,) for i(1) < i ( 2 ) < a, j(1) < j(2) < there is c E Mo, Mo k a, I; c s bjfori < a,j < P.

EXERCISE 3.13: Suppoee M,, is a Boolean algebra of cardin&ty 2". Then for some regular, goad filter D over A, Mo is elementarily embeddable into B(D), and B(D) is p-saturated, where p = mink: 2' > 2").

PROBLEN 3.14: For any A, characterize the possible B(D) for D a filter over I.

EXERCISE 3.15: Prove that for any n and A, there is a filter D over h such that B(D) is ( A + , C,,)-compact but not (A+, l7,,)-comprtct, and vice vem. (Hint for Exercises 3.13 and 3.15: The proof is like Exercise 3.11, but Nu (a < 2A) is not predetermined.)

EXERCIBE 3.16: Define a h-good filter in a Boolean algebra just as in Definition 2.1. Prove the parallel of Exercise 3.6. Show that if B, axe Boolean algebras, D, a filter in B,, Ir : 8, --+ 8, a homomorphism onto h'l(1) = D,, and D,, D, are h-good, then the filter h-l(D,) is &good. [Hint: Let g : S,&) + h-l(DS), p < No; so hg : S&) --+ D2, so it h a

a multiplioative refinement g1 : B&) + D, and choose X , E B,, h(X,) = g,({i}). Hence X , E g(e) mod D,. Define g1 : B,&) --+ D, bY

g1(8) = u { 1 - (n X , - g(t)) : t .}, S' : ~~;EOW -+ D, t8t

a multiplimtive refinement of gl, and g*: 8,&) --t h-l(D,) defined by p(8 ) = n,., X , n fl(8) is aa regarded.]

EXERCISE 3.17: Supp~ee p s x = x<= < 2n, and ahow thare is a regdar, K - p o d filter over p generated by a frtmily of sx sets. (Hint: Use 3.1, and show that any regular, good ultrafilter over p haa such subfi lh, using Exeroise 2.6.)

EXERCIBE 3.18: Suppose E is a family of subsets of A, IEI s A dosed under finite intersections, 0 $ E , and D is a A-regular filter over I.

(1) There is a function B : [El --+ D suoh that: (i) for A, B E [El, B(A n B) = G(A) n G(B), A c B + G(A) c U(B), (ii) if A, €[El (i < S),

n A, = 0 Q(A,) = 0. l < d f>d

( 2 ) [El is p-good [c1-1~gulm] iff [{Q(A): A E g] is p-good b-regulas3 (for eeoh p). (Hint: (2) ;See Exaroiee 2.8.)

EXERCIBE 3.19: If D is A-regular, A<= = A, then there is a K-good, A-regular filter D, s D.

DEWINITION 3.4: D is a, uniform filter if all members have the same Oardindity (it is IIl when D is a filter over I ) .

EXERCIBE 3.20: (1) A ultrafilter D over A is d o r m ii€ where @ = {A E A: I A - 41 < A} which is auniform filter.

E D

(2) If D is a uniform ultrafilter D over A, of[(A, < )"D] > A. (3) If D is a regular ultrafilter over A, 6, limit, then

r

EXERCIBE 3.21: Iffor some ultrafilter D over A, Do c D, Do a filter and o ~ [ ( K , < )"/Dl = p, then there rn functionsfr : A --f K, (i < p) suoh that for no g : A - K for every i < p ( K , c ) ~ / D , Cfr < g.


356 SATUI&ILTION OF ULTRAPRODUUTS [a. VI, B 3

EXERCIBE 3.22: If Do is a filter over A and there are functionf, : A 3 K

(i < p) such that for every g : A -+ K for some i < p ( K , < )"Do C g < f,, then for every ultrafilter D over A, extending Do cf[(K, < )"D] s p. If also i < a < p implies ( K , <)"Do Cf, < fa, then cf[(K, <)"/Dl = p.

Remark. The family of ft's as in Exercise 3.22 are called a (Do, p)-scale for K. On independence results conoerning scales see, e.g. [He 741. The existence of an 2b-s~ale is weaker than MA.

The following exercises a m an improvement of Theorem 3.10.

EXERCISE 3.23: Suppose D is A-regular M = nt,rMt/D, A c ]MI, IAI S A then there are finite Pt c lMtl such that A s PM where W, PM) = I I t a r (Mt, p t ) /D.

EXERCIBE 3.24: Suppose Dan ultrafilter over I, llMtll s A, A c 1M1,

(1) If IAI < cf[(h, <)'/D], then for some P, s lMtl, lPtl < A, and

(2) If A = Ha+,, IAI < cf[(N,+,, <)'/Dl for 0 s 2 s n, then for

n t e r Jft/D.

A c PM where (My PM) = n,,I (Mt, P,)/D.

some P, c lM,l, lPtl < K, and A E P', where

(Jf, PM) = n (Mt, PJD.

Remurk. Now a combination of Exercise 3.6 and 3.24 gives results like 3.10 (for the existence of such filters the existence of scales is sufficient, see Exercise 3.22).

t€t

EXERCISE 3.25: Suppose A = 2" < 2"+, and some p+-complete filter D, over A satisfies Exercise 3.1 and a (Dl , A + +)-scale for p+ exists. Then

(1) over A there is a regular and good ultrafilter D, such that for every A+ +-saturated model of Thd, MA/D is A + +-saturated;

(2) over A + there is a regular not good ultrafilter D, such that for every model M of Tina, M"+/D is A+ +-saturated.

EXERCISE 3.26: (1) If A = KO and MA holds, M , N are elementarily equivalent IL(M)I 5 A, 11M11 + IlNll s A then for some regular ultrafilter D over A, MAID = NX/D.

(2) If, e.g., IlMll + llNll + IL(M)I s p + , M , N elementarily equivalent and the assumption of Exercise 3.26 holds, then for some regular ultrafilter D over h M A / D = NA/D provided that A+ + = 2".

CONJECTURE 3.26: (1) In Exercise 3.26(1) we can eliminate MA. (2) It is consistent with ZFC, that for A = KO for every regular

ultrafilter D over A+, of@+, <)"+/D = A + + , cf(h, <)"+ID > A + + (thus Exerciee 3.6(1) is essentially best possible). We may repltloe A, A+ by A + , 2".

DEFINITION 3.6: For an ultrafilter D (over I) and reguIar cadinal K ;

1Cf(K, D) is the smallest cardinality A such that: there is a subset of {a E d / D : d / D C a < a for each a < K }

which is unbounded from below, and has cardinality A.

Remarke. (1) We do not distinguish between K and the model (K, <) (till the end of this section).

(2) Clwly lCf( K, D) is an W t e regular cardinal or 1. (3) Till the end of thh section, an ultrafilter is over I if not mentioned

otherwise.

CLAIM3.11:(1)IfK > III~~@?',IOf(K, D) = 1;inf&hf(K, D ) = 1 iff D ie cf ~damndingly complete (8ee De$nition 6.1).

(2 ) d / D k not ( K + lcf(K, D ) ) + - w t . (3) If D ie a A-good N,-inco))zlplete ultrajilter and K < A, then lcf(K, D) 2 A.

Procf. Immediate.

THEOREM 3.12: For every p = pNo 5 2", and regular K, No < K s p there is a regular ultrajEZter D over A such that lcf(Ko, D) = K and

THEOREM 3.13: (1) Suppose No = A0 < A, < - * * < A,, = A+ each 4 i8

regular and hI +1 s p l s 2", pl regular (for 1 < n). Then for 801126 regular A1-good ultra$lter D over A, lcf(K, D) = pl whenever A, 5 K < A1+,. I n

(2) 80 in particular for every h and regular p s A, thre i8 a regular fact D i8 not A?-good.

ultraMer over A which ia p-good but not p+-good.

THEOREM 3.14: Stqpme in the previowr thorem that A, = K, and k < w , xl = xp (1 $ k) are given xo < x1 < * a - < xk. = 2A and xf (i < 2, 1 < k or i = 0, 1 = k) are regular m r d i d >No, X! s xi, X I 5 X I . Let

Jl = (a E oA/D: 1.1 = xl}


358

where

1.1 = n a [ i ] / D = l{b E d / D : o A / D C b < a}] , f < A

then we can aizmme in addition, thad (i) for each a E oA/D, 1.1 ie jinite or E hl: 1 s k},

(ii) for I < k cf(J,) = x: (Jl is order as a subset), (iii) for 1 < k cf (J:) = xe (JF the inverse of JJ.

DEFINITION 3.6: (1) In Definitions 3.1 and 3.3 we allow the functions in 9, g1 resp., not be neclessarily onto A, and then demmdj, E Range@,),

(2) For a family 9 with domain A let (FI for finite interseotion): FI(9) = {h: h a funotion, Dom h c 9, lDom hl < No and h( f ) E

For hEFI(9) let A, = {i < A: f ~ D o m h implies f(i) = h(f)}.

j c E RtmgeVt) reSP*

Rengedf)}.

FI,(Y) = {A,,: h E FI(9)).

CLAIM 3.15: (1) All relevant krnmaa remain true ~ d r the new Dejinition 3.6: 3.2, 3.3, 3.4, Exerci8es 3.1,2,3 and 4. E.g., 3.2 becomee if I J,l s A for i < 2”, then there ia a f m i l y (fi: i < 2”}, fc fr.m A onto Jt, independent mod{ A}. (2) If 9 is i- mod D, for a < 6 , D , (a < 8 ) an ilacrerreing

(3) If D i8 a@er over A, 9 idpendent mod D, then there i s a d d 8 7 Of $&3’8 Over I t h 9 6 ~ d p e n d e n t mod ua<6 D,.

jEJter D* 2 D, modulo which 9 is id - (i.e., there is no D1 # D*, and D1 2 D* such ticat Q itt idpendent mod Dl).

crnd I1 -C 1, iff Ah, E AhI. (We &Card the w e IRange f I = 1.) (4) 9 is idpendent mod D iff A,, # 0 mod D for every h E FI(9);

DEFTNITION 3.7: (1) In a Boolean algebra, a partifion is a maximal set of pairwise disjoint non-zero elements. An element is baaed on a partition W, if b E W impties b c a or b n a = 0; the element is supported by a set W of elements if it is baeed on some partition W1 E W (notice if W is a partition, a is baaed on W 8 W supports a). A set W is dense iffor every non-zero a there is in W b c a. (2) If D is a f lhr over I, instead of speaking on B( D), AID, . . . , we

speak on {A: A s I), A , . . . (and then everything is mod D.) (3) CC(B), for a Boolean algebra B, is the minimal regular cardinal

A 2 No such that every partition of B hae omdimlity < A .

CLAIM 3.16: I n a Bodean algebra (1) aiebaaedtm W i f f l -aiebaeedon W, (2) if q ie baaed on Wl (Z = 1,2) then a1 n a,, a, u a,, a, - a, are

Basedon{cnd:ctsW1,cEc W a , c n & # O}wh&hieapadition, (3) for any eet W of ehnente olosed under finite intereectiorce 8uch that

everyaiz WbeJonstoapwtition E W,theeetofelenzenteitswpporbiea d+brcz, a d id& aU e k m m h of W , (4) if a , b are supported by Wand c E W =+ [c E a o c c b] then a = b .

CLAIM 3.17: Buppoae D ie a maxidjEuer Over I modulo which 9 ie i-.

(1) FI,(S) ie h e mod D; hence euch ehnent ie twppted by it. (2) Fw each f E a, U-l(j): j E Range f } is a pddtbn mod D. (3) If An E FI($), Dom hn(n < w) are pairtoke cEi.$Xnt, A E I, and

A n A,, = 0 mod D then A = 0 mod D. (4) Led 9 be the diejoint union of Yr,, 9 a , A E I ie eu- by

FI8(g1), h E FI(EB), A = h1 U ha, h1 E FI(gJ. If A, c A mod D then A,, c A mod D.

(6) CC(B(D)) = KO iff for dyfini te ly many f E B ie IRange(f)l > 1, and for 7u) one IRange(f)j = No. Othertoke CC(D) ie the fir8t regular A > No ezlol, that f E 9 =s IRange(f)l < A. Moreover, if p L h ie regular, A, # Omod D for i < p, then there ie 8 s p, 181 = p euch t7mt for n < w, and dietilcct i(Z) €8, nl<,, A,(!, # 0 mod D.

(6) IB(D)I s 191tre + 2 ' " ~ h K = CC(B(D)).

Prmf.(l)OtherwieeforsomeA E I , A # €imodDandA, $ A m o d D for eaoh h E F I ( ~ ) hence Ah A (I - A) # 0mod D. Hence 9 is independent mod[Du{I - A}], oontradioting the maximality of D (aa A # 0 mod D).

(2) Clearly the f - l ( j ) ' s are pairwiee disjoint mod D. Suppose A n f-'(j) = 0 mod D for eaoh j E Range(f), but A # 0 mod D. By (1) for some h E FI(O) A, s A mod D, and olearly for some h,, h E hl E

FI(Q), f E Dom hl. So also A,, c A mod D; but let h,(f) = jo, so A n A,, E A nf-l(j0) = 0 mod D, mntradiotion.

(3) Suppoee A # 8 mod D heme by (1) for some Is E FI(g), Ah E A mod D; for all but hitely many n's, Dom h n Dom hn = 0, so h U hn E FI(9). NOW Ahuh,, # 0 mod D and Aruh, E A mod D, and A n &,ha E A n A,, = 0 mod D mntrdotion.

(4) Let A be based on the partition {Ahl: g < g(O)}, h, E FI(9,). Now, A h E A m o d D , hence A n A h , P 0 m o d D ~ 8 , , n 8 , = 0 r n o d D


360 SATURATION OF ULTBdPtlODUCTS [m. 8 3

hence (m g1 u 9a is independent mod D) A n Ahr = 0 mod D =- Ah1 n A,, = 0 mod D hence (otherwise Ah1 - A contradicts the maximality of the partition) Ah1 E A mad D.

are distinct, [Range f,,l > 1, j: # j: E

Range f,,, and h,, is defied by h,,(j',) = j p for 1 < n and h,,( f,,) = j:, then h,, E FI( 9) and A,JD (n < o) are KO pairwise disjoint non-zero elements of B(D). By 3.17(2) f E 9 3 ]Range f I < CC(B(D)), hence the first phrase follows, and the second follows from the third.

So let p L h be regular, A, # 0 mod D for i < p. By 3.17(1) there are h,EFI(Q) such tht A,,, c A, mod D. By its definition h > No, hence p > 8, and we msume p is regular; and Dom hi is finite. So by 1.4 of the Appendix for some n < w end 8, E p, IS1l = p, for every i # j E S I D o m h i n D o m h , = cfo,...,fn-l}. As (Rangeft\ < h s p, there are 8 c 8, and j, E Range fi (1 < n) such that 181 = p and for every i €8, hi(fo) = joy h,(fl) = jl, . . ., = j,,-,. Clearly 8 is m required.

(6) By 3.16(4) each partition W supports s 2IwI elements. By 3.17(1) every element of B(D) is supported by {Ah/D: h E FI(9)). By 3.17(6) IFI(9)I 5 I 91 + K, and every partition of B( D) has cardinality < K.

Collecting those faots the result JB(D)J s Jgl*x + 2<= becomes obvious.

(6) Notice th t if f,, E

LEMMA 3.18: 8 u w e D a maximal filter mod& which g* u g is independent, 9*, 9 are disj~id; f E 9* u 9 implies IRange(f)l < cf a: of a > No, 9 = uB<a g8, g8 increas&ng and let 9 8 = g - g8. Swppo8e ale0 that the $&ere Di(i < a) satisfy D, s D, for i < j < a anal:

(i) Di is generated by D and sets &upported (mod D) by

F I p * u 9,).

(ii) g* V gE is independent mod D,. (E) Di is rrmxhal with re8peCt to (i) and (G) .

(1) D* = Ut<aD1 is a maximalJilter modulo which 3* is independent. (2) If 9* is empty, D* is an ul.tra@er and (ii) is ~atisjied whewer D,

(3) If D; satiejies (i) and (ii) we can extend it to afilter satiefving (i),

( 4 ) Iff€@ then (fl(t)/DB:tERsnge (f)) is a partition in B(D,).

T b

is m-trivial and satisfie (i).

(ii) and (iii).

OH. m, 8 31 OONSTRUOTINQ ULTBAFILTIDRS 361

Proof. Part (3) is trivial (aa the family of D’s satisfying (i) and (ii) is closed under amending chains), and part (2) first statement follows by (1). The second statement of (2) holds easily.

(1) As 9* is independent mod D, for each i, cleady it is independent mod D*. So we have to prove only the maximality. So let A E I , A # 0 mod B*; now for some S+ c 9, 19’1 < CC(B(D)), and A is supported by FI,(9+ u 9*) mod D; 80 for some y < a 9+ c g, (as cf a > IRange ( f ) l for f E 9* U 9, and cf a > No, by 3.17(6) cf a L CC(B(D)) hence of a > IS+ I). Now [D, u {I - A}] is a filter properly extending D,, and satisfies (i), hence by (iii) should fd to satisfy (ii). That is for 8ome h E FI(9* u 9,) Ah !z A mod D,, hence (by (ii) and 3.16(2)), for some B E D,, BID is supported by FI,(9* u 9,) and Ah n B C_ A mod D (by 3.16(2)) SO Ah S A U (A - B) mod D. h t h. = k1 u W’, W1 E FI(S*), ha E FI(9’). By 3.17 (4) (where D, A U (A - B), 9* u 9,, 9, stands for D, A , 9, Sl).Ahi s A u (A - B) mod D hence A,1 E A modD,. As this holds for every A # $4 modD*, D* is a maximal filter modulo which 9* is independent.

(4) Suppose feYp is a counterexample. Then for some A E 1, A # F, modDp and for every t~ Range(f) A n f ’ ( t ) = pI modDB so for some B, supported by FI,(Y* U Y@), A n f ’ ( t ) c B, modD and I - B,€Dp. By 3.17(2) there is a partition (A ,$ /D: i < 5) of B(D) on which A is based; by a hypothesis 5 < cfa and w.1.o.g. for some 5 < f [ , for i < 5, A,* c A modD, and for i 2 5(i < 5) Ah1 n A = F, mod D. So for i < 5, A,( n f’(t) E B, modD. Let hi,o = hi r (Y* u g’), so as B, is supported by FI,(Y* U Yp) andfEW = ~-9?’, by 3.17(4) Aht E B, modD. Define A* = Ui..cAht,or now:

h - s t ly , A c A* modD [as for i < 5 A,* G A,i,O c A* so A -A* is disjoint to Ahi(i < 5) and for i 2 5 (but < E ) ( A -A*) fl A,* E A n A,, = Y, modD, so A-A* = Y, modD would contradict the choice

Secondly, A* is supported by FI,(Y* U YP) [for this it suffices to prove that A*/D is the union of A,JD(i < 5) in B(D) , i.e. that if h € F l ( Y * U Y), A , n A , = 9 modD for every i < 5, then A , n A* = Y, modD; but A , n x,t,o = fj modD iff A , n A h t o = fj iff h U is not a function, so this is obvious. More formally use a maximal antichain G { A , : ~ E F I ( Y * U Yp), h extends some ht,o or h u hi,o is not a function for every i < 01.

Thirdly, for each t E Range (f) A* _C B, mod D [otherwise for some ~ E F I ( Y * U 9) A , c (A*-B,) modD, then w.1.o.g. h extends some h,,,, i < 6 (by the proof of “secondly ”) and we get contradiction to

of <A,*; i < 5)l.

362 SATURATIOX OB ULTBhPaODUaTS [=. VI, 9 3

A,,,o c B, modD]. But I - B t ~ D B , hence I-A* ED!, hence I -A ED@, a contradiction.

Remark. We have shown that B(D) is complete (this holds generally when for some 8, D is maximal such that Y is independent modD).

CLAIM 3.19: ( 1 ) Let Y be in dependent modD, and (f-'(t)/D: t E Range f) is a partition of B(D) for every f E 8 [which holds if D is m a z i b l over which Y is independent but also if D, Y are Da, YB from 3.18 (see 3.18(4))]. If g : I + K , and KI/DI=u < g/D for every a < K, and j~ $9, then d / D t f/D < g/D (hence for every ultrafilter D* 2 D, K'/D* I= f/D* < g/D*).

(2) 8 w p p e CC(B(D)) s K , K, p regular, f, : I --+ K for i < p; and KIIDCa <&ID < f!/D,foreverya < K , j &ID. Then for every dtrafier D* 1 D, lof(K, D) = p (and thi8 iS exempli&?d by the f,/D*'8).

(3) Part o w bZd8 for [D u {Ah}] when A E FI(B), f $ Dom A (k for [ D u {A}] when A # 0 mod D).

Proof. (1) Suppose not, then A = {t E I : f ( t ) 2 g(t)} # 0 mod D, hence by a hypothesis for some j, < K A n fl(j,,) # $j modD, hence

contradioting KIID k j o < g/D. (2) Otherwise there is g/D*(g: I --+ K ) contradioting it. Let A, =

{t: a > g(t)} and define induotively a(i) < ~ ( i < K ) tm follows: a(0) = 0, a(8) = Ufeba(i) , a(i + 1) the first a > a ( i ) suoh that A, - ANi) # 0 mod D. If a( i ) is defined for every i < K , - A,)/D: i < K }

contradiots CC(B(D)) s K , hence for some i, a(;) is defined but not a(i + 1). Define gl E I -t K, by

then gl contradiots the hypothe . (3) Etlay.

CLAIM 3.20: 8wppwe 9 is a nun-empty family of facnctions olrto w, D a d d j 2 t e r modulo which Y is i n & p e .

( 1 ) I f f E Q, t h nt, ( 1 + f ( t ) ) /D s lQINo for any draJilter D* 2 D, nt61 f(t)/D* = ]{a: o l /D Ca < f /D)] s ]ZSPINo.

(2) I f f e Q,, gl : I + w, wl/D t gl/D s f / D ( I = 0, 1) a d for every n g;l(n) = g; '(n) mod D t h wl/D C go/D = gl/D.

P m f . (1) The phase affer " hence " is trivial, so suppoee nt, (1 + f ( t ) ) /D > p =da IQINo, so there me g, : I + w, g,(t) s f ( t ) suoh that {t: g,(t) # g,(t)) # 0 mod D for i < j < p+. The number of possible sequences (g i l (n ) /D: n < w ) is s p (it is s IB(D)INo; but by 3.17(6) CC(B(D)) = N,, hence by 3.17(6) I B(D)I s Po, but pH0 = p). So w.1.o.g. gi l (n ) /D = gyli(n/D) for i < j < p+ but we get a contradiction by part (2) .

(2) Suppose not, so A = {t E I : go(t) # gl(t)) # 0 mod D. By 3.17(2) there is n < w suoh that A n f-l(n) # 0 mod D. Let for 1, k s n A,,, = {t E A: f ( t ) = n, go(t) = k, gl(t) = 1) so clearly A n f -l(n) = U,,, rn Ak,,, hence for some k, E A,,, # 0 mod D. As A,,, s A, olearly k # I ; it is also olem that A,,, r gcl(k) and A,,, r gi l ( l ) ; hence g o l (k) n g i l ( 1 ) # 0 mod D, but g; ' ( k ) is disjoint to gcl(E) and g; ' (1) = g; ' ( 1 ) mod D, contradiotion.

CLAIM 3.2 1 : #uppose Q*, 9, g6y W', a, D,, D* are ars in Claim 3.18. Buppose f u r t h e m e tlrat f E g* l h g e ( f ) I < K ; K > KO or Q* = 0, K = N o ; ~ i 8 r e g u l a r ; f 6 ~ ~ 6 + l - ~ 6 a n d fOTeVe?'yf< K,{t:[< f 4 ( t ) E K ) ~ DB + I'

Then for every ultraJlter Dl 2 D*, l c f (~ , Dl) = of a and thia is exemptifid by f6/D1 ( p < a) and d / D 1 t f6/D1 < fy/D1 for y < )8 < a.

Proof. By 3.19(1) d /D* b f4/D* < fy/D* for y < < a, and by 3.19(2) (for K > No) and 3.18(2) (for K = No) it suffices to prove

(*I If g : I + K , and for every f < K , d /D* t f < g/D* then for some 43 < a d / D S t= g/D* > fs/D*.

Suppose g falsifies (*) then for m y [ < K , B, = {t: [ < g(t)} E D* hence for some a(& < a, Be E DNO. As of a > K , a(*) = supt,, a([ ) < a, so Be E DM.) for every f , SO {t: [ < g(t)) E D,(.) for every [, henoe by 3.19(1) { t : f,,,,(t) < g(t)) = ImodD,,,,, hence { t : f,(,,(t) < g(t)) = I mod D*, oontradiotion.

CLAIM 3.22: If I J,I s h for j < jo s 2Qhen there are functions f, from h d o J,, and a regular A+ -good@er D, such tlrat Q is independent mod D, and D is a d m a l euchjiaer.


364 SATURATION OF ULTRAPRODUCTS [a. VI, 0 3

Proof. By 3.16(1) and 3.2 there are functions f j ( j < j,) from h onto J , and g, from X onto X (for i < 2”) such that Q* u Q is independent mod{A}, where Q* = uj:j < jo}, ’3 = {g,: i < 2”). Let Do be a maximal filter modulo which Q* u 9 is independent {H,: 0 < i < 2“} a list of all functions from Sw0(A) into (8: 8 s A}, each appearing 2” times. We define D, as in 3.18 (letting B, = uj: j < i}). We use f o to define a regular D,. I f Range (H,) c D,, we work aa in Claim 3.3, using f, to get D’ z D, and extend it to Dt+, by 3.18(3), then Ut D, is the required flter.

CLAIM 3.23: Let D be a maximal Jilter d& which 3 i a independent, p = CC(B(D)), forinJinitelymany f E B IRange f I > 1 and D* 2 D an ultraflter. Then D* ie not pi-good.

Proof. Choose distinct f , E g and thenj,, E Range f , such that f ; l ( j , , ) 4 D* (possible as IRange fnl > 1). Let A, = n,<,(A - f i l ( j n ) ) E D* and define g : SN0(p) --t D* by g ( 4 = A,,,. We suppose 9, : &&) + D* refine g and is multiplicative, and get a contradiction.

As gl({a}) E D* (for a < p), g,({a}) f 0 mod D. Hence for some ha E F I ( S ) Ah. E gl({a}) mod D. By 3.17(6) p > No, and it is regular by definition; so for some S c p, 181 = p and for some k Dom ha n u,: n < w} -C df,: n < k} for every a €8. By 3.17(6) for some 8, c S, I8,l = p, for any n < w , and distinct a(l) ~ 8 , ( 1 < n) nl<, Ahacl, + 0 mod D. Choose 8 = (a(2): 1 s k}, a(2) €8, distinct so g(8) E I - f<l( jk) , hence 91(4 s I - f iV jk ) .

N O W ~ , ( ~ ) = n z S k g l w w = nlrk Ah.(l, = Ah,whereh = (by the choice of the a(1)’s and S,, h E FI(Q)); but f k 4 Dom k,,, (by the choice of 8) hence f k 4 Dom k, so extend k to k, E FI(’3) by defining hl(fk) = j,. Cleady we get Ah, E Ah c g1(8); but we had proved g1(8) E I - fG1(jk), hence gi(8) is disjoint to Ah, -c fk( jk) , contradiction.

CLAIM 3.24: 8uppoae D i8 an N,-inmmplete &mal f l e r Over I d u b which g u g* ie independed, 9 u Q* a family of functione from I onto o, B* = df: i < S}, of S > KO. DeJine D1 a8 theplter generated by

(B) A: = { t : g(t) < f ,( t)} when g : I --+ w, and for ewery n < w, g-’(n) i8mqpo9de&byFI,(SU{fj:j < i ) ) ,andforanyh~FI(Su{f~:j < i}) for ~cnm? n A, n {t: g(t) 5 n} # 0 mod D. Then

(A) t k d W 8 O f D,

(1) g &I independent mod D,. (2) For any UltraJilter D* 2 D,. (i) ox/D* C f,/D* < f,/D* for j < i < 8 ,

OH. VI, 5 31 CONSTRUCTING ULTBAF'ILTEBB 365

(ii) for any a E w'/D*, for 8ome i < 6, d /D* C a < fJD* or fm 8 m

g : I + W , d /D* C g/D* = a and o'/D C n < g/D for every n < w.

Proof. (1) Note that by this we prove D, is non-trivial (take A the empty function). So suppose gl, i(Z) (1 < n) as in (B), A E FI(S), and we have to prove

Ah n n A;,) # 0 mod D. I<n

Let {j(k): k < m} = {i(Z): Z < n},j(k) < j ( k + 1) (note an ordinal may appear several times in the list i(Z) (1 < n)). We now define by induction on k 5 m A, E FI(S u S*), such that (letj(m) = 6):

(i) Dom A, c 9 u { f j : j < j(k)}, (ii) A, = A; A, (k I; m) is increasing, (iii) if i(2) s j(k) then for some unique nl,

Ah, E gi'(n1) mod D,

(iv) Ak+l(fj,k)) is the minimal natural number n, n > nI whenever i(Z) = j(k), 1 < n.

For doing this it suffices to prove that: if A1 E FI('3 u {ff: j < j*}), i(Z) = j*, then for some ha, n*, A' E ha, ha EFI(S u {fj: j < j*}) and Aha c g;'(n*) mod D. But &8 i(l), g, satisfy (B) for some nl, Ahi n {t E I : g I ( t ) 5 n'} # 0 mod D, hence for some n*, Ahi n g;'(n*) # 0 mod D. NOW Ah', gil(n*) ~ I W supported by FI,(S U {fj: j < j*}), hence (by 3.16(2)) also Ah1 n gi'(n*) is, hence there is ha E FI(S U {tf: j < j*}), Ah* E Ahl n g;-'(n*) mod D. so we can define the hk's.

NOW it is trivial to check that Ahm E Ah n f l l c n A$:) hence Ah n n # 0 mod D.

(2) (i) Clearly fi : I + w for every n f i ' (n ) is supported by FI,(Gf,}), hence by FI,(S U dfor: a < j}) and by 3.17(2) A # 0 mod D implies that for Borne n A n f;l(n) # 0 mod D. Hence A:, is one of the genera- tore of D,, BO {t E I : fj(t) c fr(t)} = A:, E D' c D*.

(ii) Suppose a = f/D*, and let {A,: a < a,} be a maximal family of pairwise disjoint subsets of I mod D, such that A, # 0 mod D, but for every n < W, {t E I : n 2 f ( t ) } n A, = 0 mod D. As CC(B(D)) = N, (by 3.17(6)) a, < wl, 80 w.1.o.g. a, 5 o. If a0 < w, let A = UI<,,, A,; and if a, = o, let A = (A, - {t: f ( t ) 5 2)). In any case for each n < W , Z < a, A n f - l ( n ) = 0 mod D, and A, c A mod D, hence df-l(n)/D: n < W } u {AID} is a partition of B(D). Clearly there is

< 6 such that the f-l(n)'s (n < o) and A are supported by w9 u G.: j < 18)).

366 SATWaTION OF ULTBaPaODUCTS cm. m, 0 3

Case (i). A # D*. Define

Cletlrly A: is one of the generators of D, (by 3.16( 2)), hence wr/ D* C a =

Case (ii). A E D*. As D is 8,-incomplete, then are B, E D, nn<, B, f /D* = g/D* < fs/D*.

= 0; w.1.o.g. B,+, E B,,. Define

Then wr/D* C a = g/D*, and wl/D C n < g/D for every n < W .

Proof of Tlwore~n 3.12. Let D, g be such that 4 is a family of p functions from h onto w , independent mod D, D a maximal filter over h modulo which '3 is independent, and D is regular (by 3.22). Let '3 = (fi: i < p ~ } (pK-ordinal product) = {A: i < p}. We define Dl (i < p ~ ) as in 3.18 (for g* = 8) such that A," = { t : n <ft(t)}~Dt+l for each n < w . We have to prove only that D' = [D, u {A!: n < w}] satisfies (i) and (ii). D is not trivial aa AF+l E A? and At 2 f i ' (n) # 0 mod D, and as A: is bamd on { f i l (n) : n < w ) E F18('3,+,), (i) holds. Hence by 3.18(2) (ii) holds for D' and by 3.18(3) we can define D,+, properly. By 3.18(2) D* = U, D, is an ultraflter.

By 3.21 lcf(w, D*) = cf(pK) = K , and fB/D*(p < p ~ ) exemplify this. Now if wA/D* Cn < g /D for every n < w, let i ( * ) be such that

oA/D* C fic.,/D* < g/D*, SO

n g ( i ) / D * = Idf/D*: oA/D* C f /D* < g/D*}l :<A

2 Idf,/D*:i(*) < i < p ~ } 1 = p

(for i < j o A / D C f j /D* < A/D*, henae fJD* # f,/D*). On the other hand n,<&a)/D* s ph = p by 3.20(1).

Proof of Theorem 3.13. Let D be a maximal filter modulo which 59 is independent, D regular (eee 3.22) suoh that 9 9 UIen 9,, 9; = cf:: A, s K < A, + ,, K regular, i c 2 5 ~ ~ ) (ordinal multiplication) wheref: is onto K.

We define by downward induotion, filters D, (Z 5 n) such that

(ii) D, is a maximal filter modulo which UL<, qL is independent, (iii) for any ultrafilter D* extending D,, A, s K <

(i) 0, = D, D,,, c 0 , s

K regular lcf(K, D*) = pi.

If D1+, is defined, we define DlS1 (i < 2 ” ~ ~ ) &B in 3.18 (with ULeI yPk, ub: A, s K < At+, , K regular, i < b}, 2“ p, for 9*, yPB, a reap.) such that {t E I: f 5 f i ( t ) } E D1,l+l for each f < K. Let D, = Uf D,,,, so (i) and (ii) holds by 3.18, and by 3.21 (iii) holds. For 1 = 0, for even i ’e we do &B above, and for odd i ’ s we immitate the proof of 3.22 (and 3.1) to gat a hl-good ult rdter Do. Now Do is not ht-good by 3.23 and 3.17(6).

Proof of Theorem 3.14. Combine the previous proofs, but use also Claim 3.24.

THEOREM 3.26: For every reguhr A s IIl, there W a regular ultrajilter D Over I such thud

(i) D W h-good but not A+-good, (ii) lof(K, D) = Afor every K < A, (iii) every function f : 8,&1) + D, IRange ( f ) l < A, can be r e j d to

a multiplicative one.

Proof. Let 9 = Ua<A ga, every function j E 9, is from I onto a, 3 is independent modDO and 19,l = 2A.

We now define by induction on a filters D, such that (a) a <

(c) D, is a maximal filter modulo which Uas < A gB is independent, (d) for each K I 111 and a < h for some f~ ga : K I / D , + ~ + i < f /Da+l

(e) D, is lal+-good We leave the detaile to the readers.

impliee 0, E DE, (b) Do E Do,

for every i < K.

EXERCISE 3.27: Generalize Claims 3.16-3.24, when we replace FI,(S) be any family S of subsets I , ‘Y independent modD” by “S n Dc = 0” (i.e., Dc disjoint toS, Dc = {A E I : A - A E D}) [Hint: (1) 3.17(6), (6), 3.10(1), 3.20 (aasume CC(B(D))) 3.21, 3.22, 3.23 and 3.24 (see 3J8) are left to the d e r . (2) 3.16(2), (3), (4), 3.17(1) and 3.19(2) are trivial. (3) 3.17(2) becomes: if B1 E S, A # B E S1 A - A n B E D; A 3

8 - 8’ * (3B E 8,)(3C E 8)(C E A n B), then 8, is a partition. (4) 3.17(3)becornes:ifrS1 E S , ~ ~ ~ ( V A E S ) ( ( ~ B , C E S ) ( C E A n B )

and ( V B E S ~ ) (A n B = 0 mod D) then A = 0 mod D. c 6) 3.17(4) becomes: if D is & maximal filter diejoint to {{B n A,: B E A, h E PI(9)} , B,, B, supported by 8, h E F I ( 9 ) and B, n A, E Bo mod D then B, E B2 mod D.

\

\


368 SATURATION OF ULTRAPRODUCTS [CH. VI, 6 3

(6) In 3.18 replace Yl,, FI,(9f1 u 9*))} by B,, {B n A,: B el?’, h E FI(9*)} (9* c 9) mp., and wume KO + CC(B(D)) < cf a.]

EXERCIBE 3.28: (1) Let B be a Boolean algebra, CC(B) 5 A then B is (A , at)-compaot iff B is complete.

(2) if A = CC(B(D)), D a A-good filter, then B(D) is complete.

EXERCIBE 3.29: Suppose B is a complete Boolean algebra, CC(B) 5 A, IlBll 5 2A. Then for some regular filter over A, B B(D). [Rint: If llBll 5 A, this is easy by Exercise 3.27 (on 3.22) and Exercise 3.28. Otherwise let f i be from A onto A, D a maximal filter modulo which ui: i < 2A} is independent; let B, be the subalgebra of B(D) generated by f i l ( j ) / D , h,, a homomorphism from 5, onto B, which we extend step by step, and get h* : B(D) + B, and let D* = {A: I (A/D) = l).]

EXERCIBE 3.30: Suppose Di (i < A) filters over I , D a filter over A, and let D* = {A: {i < A: A E D,} E D}.

(1) D* is a filter; and if p 2 CC(B(D,)), p*-+ (p)! (e.g., p* = (2<Y)+, p 2 A, see 2.6 of the Appendix) then p* 2 CC(B(D*)). If D, D, am ultrdters, then D* is an ultrafilter too (note D is an ultrafilter iff CC(B(D)) = 1, so we octn take p* = 1).

(2) Suppose A < K , p and K , p are regular, f a : I --+ K for a < p and for some A E D, for every i E A the following holds: .’IDi C y < f,/D, < f,/Di for every y < K, /3 < a f,/Di. Then for any ultrafilter D’ 2 D*, h f ( K , D) = p.

(3) Ifp; = n t e r nt/Di then n i c h

(4) Suppose each 0, is a maximal filter modulo whioh B is independent, p 2 CC(B(D,)) (see 3.17(6)), x regular > 2” and (Vx, x x ) (x:” < x) . Then x 2 CC(B(D)). [Hint: (1) If A, # 0 mod D(a < x ) let f (a, 8) = min{i: A,, A , # 0 mod Q, A, n A, = 0 mod Di} and see Def- nition 2.1 of the Appendix.

2 ntsr n t P *

(2) Use 3.19(2). (4) Suppose A, # 0 mod D(a < x ) are pairwise disjoint mod D. As

x > 2A, w.1.o.g. S = {i: A, # 0 mod 0,) does not depend on a. By 2.8 of the Appendix and the proof of 3.17(6) w.1.o.g. for each i ~ l ? , and distinct ~ ( l ) (I c n),

*

A,cl) # 0 mod D,. Hence A, n A, 0 mod D, contradiction. The proof gives more than required.] J EXERCISE 3.31: (1) Suppose for each a < A, K,, pa are regular cardin-

OH. VI, 5 31 CONSTRUUTINO ULTRAFILTERS 369

als > A , K~ < pa, and K~ # K~ for a # j?, III 2 zarh K ~ , pa I 21'1. Then for some regular ultrafilter D over I, lcf(Ka, D) = pa. (Hint: Use Exercise 3.30 and the proof of 3.13.)

(2) In (1) if ga(9) is a family of 21'1 functions from I onto K,(A), Ua ga u 9 is independent mod Do, then there is a filter D z Do, g is independent mod D, such that for every ultrafilter Do 2 D and a < A, lcf(Ka, Do) = pa. [Hint: (2) By the proof of 3.13, for every finite w E A there is a maximal filter D, modulo which '3 is independent, and for a E w f t / D (t < pa) exemplifies lcf(Ka, Dl) = pa for any ultrafilter D1 2 D,, and then use Exercise 3.30(2).]

EXERCISE 3.32 (G.C.H.): Suppose K < plC 5 21'1, regular for each K ES; K ES implies K 5 I I1 is regular, and for inaccessible A {K ES: K < A} is bounded below A. Then for some regular ultrafilter D over I, lcf(K, D) = plC for each K E S. [Hint: Let {fg: a < 21'1, K ES} is independent mod Do, Do regular, and Range (f,") = K . For regular x, 8, c { K ES: x s K} , let

(*)$, there is a filter D = D(S,, x ) 2 Do such that (i) {f,": a < 21'1, K < x} is independent mod D,

(ii) for every ultrafilter D* 2 D, K ES,, 1Cf(K, D*) = p x and this is exemplified byf,"/D*(a < pa) .

We prove it by induction on the order type of S, (for x a successor of a singular, use x+ and then add x+, if necessary).]

DEFINITION 3.8: For a model M and a complete Boolean algebra B and ultrafilter D of it, we define the Boolean ultrapower N = M('B)/D as follows (the sup exists by the completeness):

(i) On the set {f: Dom f is a partition of B, Range f E M } we define an equivalence relation x : f, x fa iff sup{a, n a,: a, E Rangef, ( I = 1 , 2) f,(a,) = f,(a,)} E D. By 3.16(2) x is an equivalence relation, and for everyf,, . . . , f, there are f: x f, which have the same domain.

Now IN1 will be the set of equivalence classes of x and

RN = {(f,/x,...,f,/x):

(similarly for J").

370 SATURATION OF ULTRAPRODUCTS [CH. VI, $ 4

EXERCISE 3.34: Prove that Boolean ultrapower is a particulaz case of limit-ultrapower from Definition 4.2, hence the parallel of Exercise 4.10 holds.

EXERCISE 3.35: Let D be a filter over I , 5 = B(D), K, = CC(B), D, 2 D an ultrafilter, and D* an ultrafilter of 5, D* = { A / D : A E Dl}. We define a function H: w(m)/D* + wl/D, as follows: i f f / z E w(S)/D*, let Domf = {a,: n c a,, I; w } (as C C ( 5 ) = Kl), let a, = A,/D and w.1.o.g. I = Un<o A,, A , n A,,, = 0 for n # m (for let Ah = I - U n , O A , , = A, - U l s n A ; ) and let H ( f ) = g/D, where g(l) = n t+ t E A,. Prove

(1) H is well-defined as is an elementary embedding. (2) The range of H is an initial segment w’/D1.

Remark. By this, the problem on “what can be ateI n,/D: n, c w}”

can be reduced to a problem of Boolean ultrapowers.

EXERCISE 3.36: In 3.19(2) replace the a < K byf’ (j c K ) .

EXERCISE 3.37: Suppose g is a family of functions from I into the ordinals, and D is a maximal filter over I modulo which Q is independent. Let p = CC(B(D)), and suppose D is A-good. Then for any ultrafilter D, 2 D, and regular K , x c A satisfying K , x 2 p + 1’31 + and: K > 2X or x > 2’F, or K = x are weakly compact wI/D, has no ( K , x)- Dedekind cut (see VII, Definition l.lO(B)).

EXERCI8E 3.38: Use Exercise 3.37, Theorem 3.25 and Exercise 3.30 to investigate the possible { ( K , x): d / D has a ( K , X)-Dedekind cut}. Phrase the open problems.

QUESTION 3.39: Investigate the problems from Exercise 3.38.

VI.4. Keisler’s order

DEFM~TION 4.1: (1) For every A Keisler’s QA-order on theories is defined aa follows: T , @A T, if when M, is a model of T, (1 = 1,2) and D a regular ultrafilter over A, the A+-compactness of HG/D implies the A+-compactness of M:/D.

(2) Keisler’s order @ on theories is defined aa follows: TI @ Ta if for every A T , Ta.

OH. VI, 8 41 EEISLER'S ORDER 37 1

(3) T, and T, are @.,-equivalent (@-equivalent) if T, @., Ta @., T1 (Ti @ Ta @ Ti).

LEMMA 4.1: (1) T7w relations @.,, @ are transitive and re$hAve. (2) @.,- and @-equivalence are equivalence relations.

Proof. (1) Easy to check the transitivity by the definition. The reflexivity follows by 1.9.

(2) Follows from (1).

Remark. So naturally, @ (@J is an order on its equivalence classes.

LEMMA 4.2: (1) T ie @.,-&TTuz~ (i.e., TI @A T for every T , ) if fw every model M of T and regular ultraJilter D over A,

M A / D i8 A+-COm$MGt + D is A+-g&.

(2) T is @.,-minimal (Le., T @., TI for every T,) iff for every model

(3) T is @-maximal [-minimal] iff it is @.,-WWX~TTUZ~ [-minimal] for M of T and regular ultra$lter D over A, M A / D ie A+-mpact.

every A.

Proof. (1) Let M , be a model of T,, M a model of T, and D a regular ultrafilter over A. If T satisfies the condition mentioned in (l), and M A / D is A+-compact, then D is A+-good, hence by 2.3 and 2.11 M t / D is A+-compact.

As this holds for every M,, D; TI @., T, so T is @.,-maximal. Suppose now T is @.,-maximal, D a regular ultrafilter over A and

M a model of T and assume M A / D is A+-compact. By the @,-maximality of T, Th(Mh)@., T [ M A from 2.2(3)] hence M i / D is A+-compact. By 2.3 this implies that D is A+-good.

(2) If T satisfies the condition, clearly it is @.,-minimal. If T is @.,-minimal, then T @., T, [the theory of (A, =)I. But for every regular ultrafilter D over A, II(A, =)h/DII = AA/D = 2A > A, hence (A , =)"D is A+-compact. So clearly T satisfies the condition.

(3) Immediate.

THEOREM 4.3: Any theory with the strict-order property is @-mtwiml.

Proof. Just like 2.6 and then use 2.11 and 4.2. (Alternatively use Exercise 4.6.)


THEOREM 4.4: Let M be a moclel of an urntable theory T , D an ultrafilter over I and suppose that tkre are finite cardinah m,, i E I , a h that No s n,, m,/D < 2A. Then N = M'/D is not A+-wmpact.

Proof. By 11, 4.7 T has the strict order property or the independence property.

Case I. T has the strict order property. So there is a formula ~ ( 2 , 8) which is a partial order, and not reflexive, and for every n there are q, . . . , ?i&l E M such that ktp[sir, Let P, = {ZP: 1 < m,}, ( N , P) = nipr (N, P,); so by assumption IPI < 2A. Notice that each (M, P,), and hence (N, P), satisfies the sentence (VZ)[(3g)[P(g) A (VZ)(P(Z) 3

Suppose N is A+-compact, then we can define ti,, 5, E P 7 E A 2 2 by

(i) N t= ~[si,, 5,] and for every n

F(2, 2 ) = q(8, 2))l v (VY)(P(B) + 1q(% a))] . induction on Z(q) such that:

(N, P) k (360,. . . ,Zn)( A 9 ( Z l , Zl+l)) A sin = 2, A b, = Zn A A P(ZJ), -

1 en 1 en

(ii) 7 <i v implies N I= Y[Z,, a,] A &,, 5,] A &,-<o), si,-<l~]. Clearly the Z,'s are pairwise distinct and EP, and their number is s lPl < 2A, but also 2 = 2A, contradiction.

Case 11. T has the independence property. So some y(x; 8) has the independence property. Hence we can find sequences ti: E IN1 (1 < K )

so that k(3z)[AlCk ~(z; ?$)l-"lEW)] for each w c k, so let bk E IMI, be such that

'A F(bk,; Zr)i(lew).

Let p = min{n,,, nJD: n, < w for i E I and n,,, nJD 2 No} < P, P = l?rer nilD.

For every ~ E I define k ( i ) = [log,n,] if n, # 1, k ( i ) = 1 if n, = 1. Let P, = {a;(,): 1 < k ( i ) ) , &, = {b","): w c k( i ) } , so clearly n,/2 - 1 I 1Qf1 s 12,. Let ( N , P, &) = I L r ( M , Pi, QO, 80 clearly IS1 = Riel ISil/D s p, but clearly !PI s 101 and for every 1 < w

1Ck

{ i d : lP,l 2 1 ) 2 { i d : n , 2 2')ED

(as n n , / D 2 No). Hence IPI is infinite; hence by p's definition =

/PI = p. Let A1 = min(A, p) and choose I s P, 111 = Al. By the definition of the Pi's, clearly for every J E I, pr = {y(z; si)ii(dor): si E I) is

OH. VI, 5 41 KEISLER’S ORDER 373

a type in N. Now by the definition of the Qr, if pJ is realized in N, it is realized by some element of Q. Hence the number of types pJ, which are realized in N is I IQl = p (because J, # J, implies no element realizes both pJl and pJJ. On the other hand the number of such types is I{J: J c I>I = 21’1 = 2”. Clearly 2” > p, and by hypothesis and definition of p, 2A > p; hence 2.,1 > p. So for some J G I N omits pJ, and as IpJI = A, 5 A, N is not A+-compact.

EXERCISE] 4.1: Prove that if M = ntsI M,/D is a model of an unstable theory, D an ultrafilter over I , KO I nfoI m,/D < 2.,, then M is not A+ -compact.

THEOREM 4.6: Let T be stable and urith the jnite cover property. Let M be a model of T and D an ultrafilter over I ; if KO I p = nfsI m,/D then M’/D is not p+-mpact.

Proof. Let q(z, y; Z) be as in 11, 4.4, i.e., for every E E IMI q(z, y ; a) is an equivalence relation and for n < w , q(x, y ; E,J has ~n but < No equivalence classes. Let Neq(E, M) be the number of equivalence classes of q(z, y; 5 ) in M. Clearly for every family of models M, i E I in which q(z, y; 2) is always an equivalence relation

Choose E[i] as En[rl where n[ i ] = max{Z: Neq(E,, M ) I m,} if such exists, n[ i ] = 1 otherwise. Clearly No I Neq(C, MI/D) I nfo ,m, /D, so we finish.

EXERCIBE 4.2: Prove the parallel to Exercise 4.1 for Theorem 4.6.

LEMMA 4.6: If T is @A-minimul, p 5 A, then T is @.,-minimal.

Proof. Assume T is not @.,-minimal. By 4.2(2) there is a regular ultrafilter on p, and a model ill of T such that M@/D is not p+-compact. Let D, be a regular ultrafilter on A, D, = D, x D, I = A x p, so D, is a regular ultrafilter on I , 111 = A, and M1/D, z (MA/D,)#/D. By 1.1 M = MA/D, so by 1.9 and 2.11 M p / D is p+-compact iff (MA/D,)U/D is p+-compact so MI/D, is not p+-compact, hence not A+-compact. So T is not @.,-minimal. Contradiction.


THEOREM 4.7: If Krdt(T) = 00, D ie a regukcr dtraji2ter over I , M ie an IIl++-eaturated model of T, t h n Mr/D is not IIl++-cornl)ccct (eee Definition 111, 7.2).

Proof. If T has the strict order property the theorem follows immediately from 4.3, 4.2, 2.4 and 1.3. Next aasume K ~ ~ ( T ) = 00 ; w.1.o.g. we can assume (see 111, Definition 7.3, 111, Exercise 7.4 and 1.11) 11M11 > p = aA+ where IIl = A and there is a formula cp(Z;g) and a{ E \MI, B < A, a < p , I @ { ) = Z(g) such that for every f l { c p ( f ; a:): a < p } is 2-contradictory, and for every 7 E " p {q@; atlE,): < A} is consistent, realized by En. Let P = {a:: /3 < A, a < p}, Q = {En; q E "}. We define equivalence relations E(E, g) , Z(Z) = I (g) = I @ ) as follows E(iZ2, iZ2) iff p1 = pa; -,E(E, ?I{) if 17 $ P , E(E, a) if 8 , a $ P . IZ has X equivalence classes in P(M) and IA+ equivalence classes in P(Mr/D) each one of cardinality z p . Let Jf, i < A+ be distinct equivalence classes of E, then for every sequence (a,: a, E J,, i < A + ) , {cp(E; at): i < A+} is consistent, and if M r / D is A++-compact it is realized in &(Mr/D). p < p + s n,<,,,+ lJfl s 1Q(M'/D1 = IQ(M)Ir/D s pA = p , contradiction. So we are left with the case K,,,(T) < 00, ~ ~ ~ ~ ( 2 ' ) = 00, hence by 111, 7.11 and 1.11 we can assume there are a formula p(Z; g) and a,, E IMI (7 E 'p)Z(G,,) = Z(J) such that if 7, v E > p are <-incompatible then M C 7(3z)[cp(Z; a,) A ~ ( 2 ; a,)] and for each 7 E E,, E 1M1 realizes {cp(Z; a,,,,,): n < w}. Let P = {an: r ) ~ @ > p } , Q = {En: r) EOUCL), E = {(a,,, a,): for some n < w, 7, v E "p}, and < ={(sin, a,): Z(7) < I(v);

7, v E >p}, El = {(a,,-cf), an-<j)): i, j p}. Let Nl = ( N , PN, QN, EN, EF, < N ) = Mi/D, MI = (N, P,Q, E , El, <): so as Disregular there are P E PN, N , l=iP < for a ,A we define E PN such that N , k E@W, zi((W)) and the @'a am distinct, and N , C (3Z)(&Z; a*) A cp(Z, 7i"1°), N , C E,(F^<f), ZJ-<j>). We do it by induction on Z(7). For Z(7) = 0, Z(7) = a + 1 there are no problems. For Z(7) = 6 limit let p,, = {y(Z; ZIa): a < 6); clearly it is consistent, 80 if it is not realieed, M1/D is not even h+-oompaot, 80 we finish. So assume i9 E ldfl realizes pny but

Mi l= (E)(W[Q(Z) A (V#)(P(#) A (P@; 8) 4 d Z ; 811

(by the requirements on the p(Z; 4 ) ' s ) 80 we 0811 wmme 8 E v, and the rest is easy. Similarly for every q E "+p there is P E QN recllizing p,,, but

s lQMIr/D s pA = p; the pnys am pairwise explicitly contra- diotory and their number is pA* > p; contradiction.

OH. VI, Q 4) KEISLER~S ORDER 375

EXERCISE 4.3: (1) Show that in 4.7 it suffices to assume D is uniform. (2) Show that by 4.7, there is a model Mo of T, 11 Mo 11 = 1 I1 + , such

that Mo < M implies Mr/D is not I I I ++-compact.

DEBTNITION 4.2: Let M be a model, D a filter over I ; for a E I Mx/DI, eq(a) = ((8, t) E I x I: a[8] = a[t]} and for Q a filter over I x I , MLJ Q is the submodel of Mx/D whose set of elemente is {a E lMx/D1 : eq(a) E a). We always aasume {(t, t): t E I) E Q.

DEBWITTON 4.3: T1 @* T, if for every set I , XI-incomplete ultrafilter D over I , filter Q over I x I , cardinal A, and ( A + + (I1 +)-saturated models MI, Ma of T,, T, respectively: if MGDlG is A + -compact then M',,lQ is A+-compact.

DEFINITION 4.4: Let Q) = {vr(Z; 2'): y < a}, Q), = {#,(g; 2,): y < p} be indexed sets of formulas (possibly with repetitions) from L(T,) and L(T,) respectively; Z(Z) = ml, l(g) = ma. (a1, m,) 5 (a,, ma) if there is q E such that for every model M , of T, and By E lMll, y < a, there are a model Ma of T, and 6~ E 1 Ma], y < /3 such that: for every w c a {g@; BY): y E w} is consistent with M, iff {#,,c,l(g; 8nCrl): y E w} is consistent with Ma. We write (@,, m,) 5 (Qi,, ma) by q.

Remarke. (1) Clearly by the compactness theorem (a,, m,) 5 (a,, ma) by 3 iff for every finite Q) E @,, (@, m,) < (a,, m,) by q r w, where W, = {y < a: py(E; Ir) E @}.

(2) In Dehition 4.4 we can take M,, Ma as k e d A-universal models (for large enough A).

(3) We write Q) E L (in this section only) to mean @ is an indexed set of formulrts, possibly with repetitions, from L.

EXERCISE 4.4: (1) If (Q),, m,) 5 (Q),, m,), @l E Q),, 0, E Oa then (@lS m1) 5 ( P s ma).

(2) If O1[P] is the olosure of @, [a,] under conjunction and disjunction, then (a,, m,) s (a,, ma) implies (@I, m,) 5 (P, ma) (why do we not eay "negation " 2)

(3) If (@lS ml) 5 (@a9 and (@a9 (@a9 %) then (@IS ml) ('J's, 7%).

EXERCISE 4.5: If for every 0, E L(T,), 1@,1 = A, there is Q), E L(T,) and % < o such that (Q),, 1) s (Q),, ma) then T, @, TI.


DEFINITION 4.6: Let @,, §,, m,, ma be as in Definition 4.4 (@,, m,) S* (Oat ma) if there is 7 E ap such that for every model M, of T , and Sir; E 1 M1 1 y < p, n < W , there are a model Ma of T , and 6; E 1 Mal, y < 8, n < w such that: for every w c a x o {tp,(E; a;) : (y, n) E W } is consistent with M, 8 { # , , E y , ( ~ ; 6:[~1): (y, n ) E W } is consistent with Ma. We write (Qs,, m,) r * (@,, ma) by q.

EXERCISE 4.6: ( 1 ) I n Definition 4.6, we can replace o by any a 2 o.

(2) ( @ I , m,) s* (@a, ma) by 7 implies (@I, m,) 5 ( 0 2 , ma) by 7. (3) (@I, mi) I* (@a, ma) implies (01, mi) 4 (@a, ma). (4) If @,, @' contain the same formulas (with a different number of

repetitions) then

( 6 ) If @l G @,, Q2 E Oa then (a1, m,) S* (Oa, m,) implies ( @ I , m,) s* (Ga, ma).

(8) (@,, m,) S* (@,, m,) (by the identity map).

EXERCIflE 4.7: The following statements about T,, T , are equivalent. ( 1 ) For every c L(Tl) there are 0, G L(T,) and ma < w such

that (@,; 1) s* (@,, ma). (2) For every §, E L(T,), s (T,I + 1T,1 + there are @, E

L(T,) and ma such that (@,, 1 ) s (@,, ma). (3) For every @, s L(Tl) there are 6, c L(T,) and ma such that

(@I, 1 ) (@a, ma). (4) For every @, c L(Tl), 1@,1 s IT,I there are 0, c L(T,) and

ma such that (@,, 1 ) s * (@,, ma). (6) Let 0, be the set of formulas tp(x; jj) E L(T,) (clearly )§,) =

I T1l). There are 0, E L(T,), ma such that (@,, 1) s* (@,, ma).

EXERCIflE 4.8: ( 1 ) If 0, is the set of all formulas in L(Tl), and for some @, c L(T,), ma < w, (Go, 1) s* (@,, ma) then T , @* T,.

( 2 ) In fact it suffices to demand that there are a,, i < i,, such that: if,M, is a non-A+-compact model of T,, then there is a type p over M,, p = ipa(z, 7ia): a < a, < A+}, such that for some i < i, every tp,(x, jja) E

0,; and there are @,,, E L(T,), ma,, < o such that (QZ,, 1 ) r* (@a,is ma,{)-

EXERCILSE 4.9: Suppose M, is a A-universal model of T , M, is a A + - compact model of T , and D an ultrafilter over A. If N',/D is A + -compact

CH. VI, 8 41 KEISLER'S ORDER 377

then M i / D is A+-compact. If D is uniform it suffices to demand MI is ( < A)-universal, M2 is A-compact.

EXERCISE 4.10: For M, I , D, Q as in Definition 4.2, D an ultrafilter; M',la is elementarily equivalent to M, moreover a H (. . . , a, . . . )te,/D is an elementary embedding of M into M',]a. Generalize 1.1.

DEFINITION 4.6: A complete theory T is simple if

and one other two-place predicate E ; (1) L( T) contains only one-place predicates, the equality sign = ,

(2) for every model M of T, EM is an equivalence relation over 1M1; (3) there is a model M of T such that for every ~ E W I , [aIM is

infinite where

[aIM = {b E M : M k bEa, and for every predicate P(x) of L(T)

M I. P(a) = P(b)};

(4) there is a model M of T such that for every a E 1M1, there are infinitely many b E ( M ( from Merent E-equivalence classes which realize the same type over 0.

EXERCISE 4.11: Let T be a simple theory. Show that:

an automorphism of M.

formulas of the following forms:

(1) If M is a model of T, a E IMI, then any permutation of is

(2) Every formula of L( T) is equivalent to a Boolean combination of

(9 x = Y, (ii) xEy,

(iii) P(z) , (iv) ( ~ Y ) [ x E Y A A j < n pj(Y) A A j < m l p j ( ~ ) l . ( 3 ) T is stable in every A 2 2ITI, so T is superstable.

EXERCISE 4.12: Suppose M is a non-A+-compact model of a simple theory T. Show that M omits a type p, which is of one of the following forms:

(i) p = {xEa} u {P,(x)n[C1: 5 < to I min(A, IT])}

(ii) p = { P , ( X ) " ~ ~ : u {x # c,: 5 < l o s A},

< c0 I min(A, ITI)} u 1p0 u {TxEc~: 5 50 5 A},

378 SATURATION OF ULTBbPBODUCTS [CH. VI, 8 4

where po consists of formulas of the fourth form from the previous exercise and of negations of such formulas, and r] is a sequence of zeroes and ones.

EXERCISE 4.13: If M is a A-compact model of a simple theory T, and N = MblG (see Definition 4.2) is IT1 +-compact, show that N is h-compad. (In fact it is ALIU-compact.)

EXERCISE 4.14: Show that: (1) A simple countable theory is @-minimal. (2) If N is a model of simple theory T, D a I TI + -good ultrafilter on p,

then Nu/D is Kg/D-compact. Hence if D is p-regular, W / D is 2,- compact.

EXERCISE 4.15: Show that for every theory T, and cardinal h there is a simple theory T2 such that Tl @A T, @A T,. If I TII I A then also lTal I A. Moreover if D is a A-regular ultrafilter over p, AZ, a model of TI, M , a model of T2 then MU,/D is A+-compact iff M $ / D is A+-compact. (Hint: See Exercise 4.5.)

EXERCISE 4.16: Show that for every set {Tc: &! < to} of theories there is a least upper bound for each of the orderings @, @A. Its cardinality is 5 zc I Tel .

EXERCISE 4.17: (1) Show that for every A there is a simple theory Tj,, lThl = h such that TA is @A-maximal. Hence if A < p, TA @ T, but not T, @ TA. So thare is an (uncountable) theory which is not @- minimal nor @-maximal.

(2) Show that if there is a countable theory which is not @-minimal nor @-maximal (see 5.9) then there are @-incomparable theories.

EXERCISE 4.18: Prove case I of the proof of 4.4 by 4.3.

EXERCISE 4.19: (1) Suppose xlnp(T) > l I l+ , iK an IIl++-saturatd model of T, and D a regulm ultrafilter over I . Prove that M r / D is not III ++-compact (see Theorem 4.7).

(2) Prove the parctllel of Exercise 4.3.

QUESrTION 4.20: Can we in Exercise 4.19 replace K ~ ~ ( T) by xcdt(T) ?

CH. 8 61 SATURATION OF ULTRAPOWERS 379

EXERCILIE 4.21: Show that T1 T i , Ta.

Ta for any complete theories

THEOREM 4.8: For any model M of an umtable theory T and 8,- incomlplete ultrafilter D Over I, K = lcf(w, D), MIlD is not ~ + - ~ ~ m p a c t .

Proof. Choose an unstable formula ~ ( z ; g) and sequences q (1 c n < o) such that {rp(x, $ ) u ( l > k ) : 1 < n} is consistent for every k < n < w. Choose a distinct b, E: 1M1 and let: PM = {b,: n < w},

< M = {(bk, b,): k , < n < W}

and Ffd (1 c l ( @ ) two place functions such that 7i; = (Po@,, b,,), PI(&, kn) , * * a ) . Let

N1 = ( N , PN, <*, F r y . . .) = (My PM <My Ff, . . .)'ID,

and c, E PN (i < K ) be such that N , C b, c c, c c, for n c w, i < j c K

and for no c E PN, N C b, c c c c, for every n c w, i c K. Let

I, = {Tv(x, Fo(bny GO), . . .): n < O} u { ~ ( x ; Po(c,, GO), . . .): O < i < K}.

Clearly p is finitely satisfiable, but N omit it (if a realizes it, then the formula

P(Y) A (VZ"(4 z Y + ?+, FO(YY CO)F,(Y, CO), - - *)I define a bounded subset of P with no last element).

VI.5. Saturation of ultrapowera and categoricity of pseudo-elementary classes

THEOREM 6.1 : Let T be a countable theory, MI a rnocEel of T for every i E I, and D an K,-incomp&e ultraJilter mer I . Let N = nlEI MJD.

(1) If T does not have tk f.c.p., A = Ki/D, then N is A-saturated. (2) If T is stable and hua the f.c.p., then N is A-saturated, but not

(3 ) If T does not have the f.c.p., each Aft is p-saturated, and A = pl/D,

(4) FOT everyjinite A s L(T) let &(A) = min{lpl: p is a A-I-type over

A+-saturated, where A = min{n,,, nJD: nlEI nllD 2 Xo}.

then N is A-saturated.

lM,l in MI which is omitted by Mt} and

t L(T), [A1 < H,}.

380 SATURATION OF ULTRAPRODUCTS [m. VI, $ 6

Let A be an (injnite) cardinal w h tlrat A = nisr Ai/D for ~ome A:, i E I, wkre for every jEnite A C_ L(T), {i: A: s &(A)} E D. Then if T aoe.9 nd liaVe t k f.c.p., N i8 A-saturated, but not (A*)'-Sat~rate&.

Remark. (1) Clearly the results, except (a), m the best possible. For example in (l), if we choose the M i aa aountable models, then IlNll = K',/D = A, Eo N is not A+-saturated. On (4) see Exercise 6.10. (2) Instead of demanding T countable, we can require D to be I TI + -

good. See Exercise 6.2.

Procf. Notice that as T is countable, for every model H of T and cardinality K > KO, M is wcompact iff M is K-saturated.

Now in part (2), N is not A+-saturated by Theorem 4.6. Similarly we can prove in part (4) that N is not (A*)+-saturated. So it remains to prove that in all the parts N is A-saturated.

N is XI-saturated by 2.1, 2.3 and 2.11. By 111, 3.3, 3.9 and 3.10, aa T is countable and stable, it suffices to prove: if {c:: i < o} c IN1 is an indiscernible set, then it can be extended in N to an indiscernible set of cardinality A. For every i E I let us choose a family Sf of subsets of lM,l such that:

(i) IS:l = 1 1 ~ : 1 1 , (ii) every finite subset of IHtl belongs to St, (iii) for every finite A c L(T), n < o, if w E 8, is ad-n-indiscernible

set, 0 s p s 11H,ll, and there is a A-n-indiscernible set w', w E w' c lMtl, Iw'I = p, then there is wN eSi, Iw"I = p, w E w" c lMtl, and wN is ad-n-indiscernible set. Let /Mil = {af:j < IIH,II},St = {wi:j < llH,ll}. Let us define the relation E{ on lMil: E' = {(a;, ah): a: E wh}. We shall write z E y instead of ~(z, y). In the language L = L(T) u {E}, clearly, for every finite A c L(T), n < o, there is a formula 'g,,Jz) meaning {y: y E z} is a A-n-indiscernible set.

Now for every i E I we define Pi according to the part of the theorem we want to prove;

in (1): F = {a:: lwhl z H,,},

in (3): F = {a:: 1w:I 2 p}; in (2): F = {a;: a < 11M,11} = [Mil;

in (4): Pf = {a;: lwtl 2 hi); where the A' are defined so that InlE, X/Dl = A, and for every finite A E L(T) {i: A: s & ( A ) } € D.

Now the following hold:

OH. 8 61 SATURATION OF ULTRAPOWERS 381

(*I For every finite d c L(T), n < w, there is m = m ( d , n) < w, such that the set of i's for which the following holds belongs to D : For every A-n-indiscernible set w;, Iw;l 2 my there is a A-n-indiscernible set wi, wk E wj and a: E P .

(**)

Let us prove it. In part (2) it is trivial. In the other parts T does not have the f.c.p. so in part (1) it follows from 11, 4.6(3) and in parts (3) and (4) from 11, 4.6(2). Notice that except in part (4) (**) holds for every i.

Now clearly (**) is equivalent to a first-order sentence in L' = L u {E} u {P} . Let N' = ( N , E ~ , PN) = nfo, (Mf, cf, P ) / D . N' is 8,- saturated (by 2.1, 2.3 and 2.11). By (*), clearly the sentences corresponding to (**) (for all finite A , n, m) are satisfied by N'. Remember it suffices to prove that {cf: i < w } can be extended in N to an indiscernible set of cardinality A. As {cf: i < w } is an indiscernible set, it is a d-n-indiscernible set for every A, n. Hence every finite subset of p = {c, E x: i < w} u {rpd,,,(x): d E L ( T ) , lAl < No, n < w } u {P(x)} is satisfied in N', hence p is satisfied in N' say by b. As N' C qd.,,(b) for every A, n, clearly w = {u E I NI : N' C u E b} is an indiscernible set, and of course {cf: i < w } G w. As N' != P[b], and lwl 2 I{cf: i < w}I = 8,, clearly IwI 2 h (the check for each part is easy). So we have proved the theorem.

It would be more satisfactory if in 5.1(4) h = A*. (This is possible if V i E I , M , = M ) . For this it suffices to prove

CONJECTURE 5.1: Let Let ( J , <) = (p, <)'ID. (<-the natural order on ordinals.) For a E IJI, let 1.1 = l{b E IJI: b < .}I. Suppose a,, E IJI for n < w , la,,l = lao[. Then there is a E I JI, u 5 a,, for every n < w , and 1.1 = la,[ (when D is an 8,-incomplete ultrafilter). See Exercise 5.11 and 5.12.

THEOREM 5.2: Let M be a h-compact model of T , D an ultrujlter over I , IT1 5 111, N = M'/D. If N is (21'I)+-saturated then N is hI/D-saturated.

Rernurh. (1) For countable T this theorem follows from 4.4, 4.6 and 5.1(3).

(2) Here the proof works also for D an 8,-complete ultrafilter. (3) See Exercise 6.13.

382 SATURATION OF ULTRAPRODUOTS [OH. VI, 8 6

Proof. As N in (21rl)+-saturated, by 4.4 and 4.6 T is stable and without the f.c.p., and so clearly every infinite indiscernible set can be extended to one of cardinality 2 (21rl)+. By 111, 3.10 (remembering that by 111, 3.3, K ( T ) 5 I TI + s 111 +), it suffices to prove that:

If I is an indiscernible set in N, 111 2 (21rl)+, then there is an indiscernible set J, 11 n JI 2 KO, lJl 2 JAr/Dl. Let {aB: fl < (2111)+} E I. The following statement will be proved later.

(*) There is an infinite w c (21'1) + such that for every i E I, {aD[i]: fl E w} is an indiscernible set in M.

We can assume A > I TI , as otherwise the conclusion of the theorem is trivial. For every i E I let P' be a maximal indiscernible set such that {a,[i]:flEw} E P c IMI.AsMisA-compact,A > ITI,clearlyIP'I 5 A. Let (N, P) = nfeI (My P ) / D . Clearly ]PI = miel lPfl 2 Ax/D. Now for every finite A G L( T), n < w , the statement " P is a A-n-indiscernible set " is elementary, hence P is an indiscernible set. So {aB: fl E w} E

P E INI, hence IPn 11 2 l {aB:f l~w}I 2 KO. So P satisfies the requirement for J. So we need only prove (*).

As T is stable, by 11, 2.13, 1231 I 21'1 implies IS(B)I I (21rl)lTl = 2Ir1. It is also clear that for B, c IN], lBtl 5 21'1, for every t €1;

Define for j 5 l I l+ , sets wj c (21'1)+ by induction: (i) wo = 0, wd = u j < d wj for a limit ordinal 8, and w, G wj for

i < j 5 Ill+. (ii) Let w, be defined. Then for every fl < (2Ir1)+ there is a unique

y E w , + ~ such that: for every i E I, a6[i] , a,[i] realize the same type in M over {a,[i]: j E w,}. Clearly for every j , Iw,I I 21'1. Choose a. < (2Iz1)+, a,,$~,,~+. For every a < lIl+, let j (a ) be the ordinal such that for every i €1, a,,[i], aj(,,[i] realize the same type in M over {a,[i]: j E w,} and j ( a ) E w,+ Clearly for every i, a I 5 5 5' < 111 + , aj(o[i] , a,(o[i] realize the same type in M over {aj,,,[i]: y < a}. By 11, 2.18 for every i, there is y ( i ) < 111' such that {ajc,,[i]: y ( i ) I a < l I l+} is an indiscernible set. Let yo = sup,,,y(i), w = {j(a): yo I a < 111+}. Clearly this is the w required in (*).

I!Xt,rfl(Bt)( = n t e r ISCBt>l I (2"')''' = 2'"-

Remark. We can in fact find such w of c d i d i t y (21'1) + . See Exercise I, 2.9.

DEFINJTION 6.1 : A model Y is complete if for every n < w, every n-axy relation on and every n-ary function from IN1 to is the inter-

CH. VI, 5 61 SATURATION OF ULTRAPOWERS 383

pretation of some predicate symbol or function symbol, respectively, from L(M).

Remark. Clearly for every set A there is a complete model M , ]MI = A, IL(M)I = K O + 21-41. DEFINITION 6.2: PC( T,, T) is the class of L( T)-reducts of models of T,, of cardinality 2 lTll, where T is complete, T s T, and T, has infinite models.

The restriction “of cardinality 2 ITl] ” is technical, and without it the class is denoted by PC*(T,, T).

THEOREM 6.3: If T is countable, superstable, and doap not h u e the

categorical in every cardinality 2 2No. f.c.p., then there i8 TI, T E TI, ITl] = 2 4 8Wh that PC(T1, T) i8

Procf. Let M be a countable model of T. We expand M to a complete model M,, IL(M,)I = 2% Let L, be the language of M , and T, the theory of Ml ( ia , the set of sentences from L, that M , satisfies). Clearly T, contains its Skolem functions (see Definition VII, 1.1).

Let N , be any uncountable model of T,, and let N be the reduct of N , to L(T). It suffioes to prove that N is saturated (as by I, 1.11 every two saturated models of the same complete theory which are of the same cardinality are isomorphic). So let p be any 1-type in N, 1p1 < IlNll, and it suffices to prove that p is realized in N.

Let p1 be any extension of p to a complete type over INI, and let

{ai: i < w}, and let c,, i c w be individual constants in L, such that cfdi 3: a,. Clearly there is ao E IN,], ao # $1 for i < w. Define A = {FNl[ii, ao]: P is a function symbol in L,}. Clearly the submodel Nf of N,, INfI = A, is an elementary submodel of N , (by the definition of T, and the Tarski-Vaught test I , 1.2 or VII, 1.2). Let N* be the reduct of Nf to L(T). Clearly N* is an elementq submodel of N.

(*)

So let q be a countable type in Nf and we must prove it is realized in Nf. Let q = {ql(z; a&, . . . , ak): i < w}. As every a: E A, for some Ff , E L,, a: = Fi, jp, ao]. So by substituting

we get q = {I,~~(Z; ii, ao): i < w}. Remembering that IMI = {a1: i -c w},

Q ( z , ~ ) E ~ , be such that R [ q ( ~ ; i i ) , L , a ] = R(pl ,L ,a) . Let 1M1 =

We now show Nf is N,-compact hence N* is K,-mturated.


C ~ I = a,, M, is complete; it is clear that there is a function symbol Q in L, such that for every a,,, 6, bo from lMl, QMi(a,,, 5, bo) realizes {t,h,(x; 6, bo); i < m} for the maximal possible m I n. Clearly for every n

M , c (VWY)[ (A Y z ci A (32) A t,h,(x, 2)) -+ A t,h:(Q(y, @, 211- l<n :<n f e n

As a0 # $'* for i < w, clearly QN*(ao,a, ao) realizes q. So we have proved (*). As N* is N,-saturated, by 111, 1.2 and 111, 2.16, we can find B c

"*I, IBI = KO, a E B, such thatp, 1 B is stationary, and we can define b, E IN*I for i < w , such that b, realizes p1 1 (B U {bj: j < i}). By 111, 1.10 Av({b,: i < a}, INI) = p l , so if I is an indiscernible set in N, b, E I for i < 0, then d(z, E) ~ p , , implies {b E I: N C 4 [ b , El} is finite. So clearly it suffices to prove that {b,: i < w} can be extended in N (not N*) to an indiscernible set of casdina;lity llNll (because thenall but s I p I + KO elements of the set will realize p).

Let 8 be a family of subsets of 1M1 such that:

(2) Every finite subset of 1M1 belongs to 8. (3) If I is a finite A-n-indiscernible subset of 1M1, (A a finite subset of

L), and I can be extended to an infinite d-n-indiscernible set in M, then there is such an extension which belongs to 8.

Let 8 = &: i < a}, and noting JM) = (a,: i < w}, let EMI = {(a,, a,): a, E I,}, PMi = {a,: lIjl = No}, where E, P belongs to L,, and let P E L, be such that for every a, E P i , PMi(x, aj) is a function from I, onto ]MI; we write x E y instead of E(Z, y). Clearly for every finite A c L( T), n < w, there is a formula vd,,,(x) in L, saying that {y: y E z} is ad-n-indiscernible set. Let q = {pld,,,(z):A c L(T), n < w, lAl < No} u {b, E x: i < w } u {P(z)}. It suffices to prove that q is consistent with Nf. Since then, aa Nf is N,-compact, q is realized by some element b EN?. Hence I = {c E N , : N, t c ~ b } is an indiscernible set (as Nf C vd,,,[b], Nf is an elementary submodel of N,) . Clearly b, E I for

Now in order to prove that q is consistent with N;f it suffices to prove that every fmita subset of it is consistent. By I, 2.3(3) instead of a finite number of q-~~, , , (x) we need take only one. So it suffices to prove the consistency of q' = {P(x ) , q.~~., ,(x)} u {b, E z: i < m < w}. By 11, 4.6(3) for every finite A, n there is T = r (A, n) < w , such that: i f m 2 r, {bo , . . . , b,} is a A-n-indiscernible set in M, then there is an infinite A-n-indiscernible set in M which extends {bo, . . . , b,,,}. So for r 2 r (d , n)

(1) 181 = KO.

i < 0. Also 111 = llNll aa N , I= P[b] (using P i ( % , b)).

OH. VI, § 51 SATURATION OF ULTRAPOWERS 385

M1 C ( V Z N V Y ~ * * * Yr)[ (A Yt + ~j A qd.n(Z) A A ~1 Z) 1 < f i s r

--+ (jy)(qd..(y) A P(Y) A A tsr Y; E Y ) ] .

This clearly implies the consistency of q', as {bi: i < w } is an indiscernible set (in L(T)) and for every c,, . . . , cn E N , there is c E N , such that N , C: (VZ)(Z E c A?-, z = cf).

Remarks. ( 1 ) TI has no models in any uncountable cardinal < 2'0 and is categorical in KO.

( 2 ) By 5.1(2), VIII, 1.7 and VIII, 2.1 the theorem is the best possible. The following theorems have similar proofs so we omit them.

THEOREM 5.4: (1) If T is countable, withut the f.c.p., and stable in KO (i.e., totally transcendental) then there is T,, T G T,, ITl! = No, m h that PC(T,, T ) is categorical in every h 2 KO, and every model in it is saturated.

( 2 ) If T has the f.c.p., is countable, and stable in KO, h 5 2N0 then there is T,, T G T,, lTll = h such that PC(T,, T ) is categorical in h and every model in it of cardinality h is saturated.

THEOREM 5.5: If T is countable and superstable, then there is T,, T C T,, lTll = 2No such that PC(T,, T) is categorical in 2%, and every model in it of cardinality 2w0 is saturated.

Remark: We use the following fact: if M, is a complete model which expands ( w , c), N, is an uncountable model of the theory of M,, a s INl[, [{b E lN1l: b < a)I 2 KO, then I { ~ E lN1l: b c a}I 2 2% We leave it as an exercise.

THEOREM 5.6: Let M be a model of a countable and superstable theory T , M, a complete expansion of M , N E PC(Th(M,), T). Then N is 2~0-saturated.

THEOREM 5.7: If T is not @,-minimal, then it is rwt @,,-minimal for every p 2 min{21TI, A}.

Proof. If p 2 h the conclusion follows by 4.6 so we can assume h > p 2 2ITI; and by 4.6 again it suffices to prove the theorem for the case p = 2ITI. So let h > p = 2ITl, T is @,-minimal but not @*-minimal.

386 SATTJRJ~TION OF ULTRAPRODUCTS [CH. VI, f 6

As T is not @,-minimal, by 4.2(2) there is a regular ultrafilter D over A such that for every model N of T, NA/D is not A+-compact. Let M , be a A+-saturated model of T, M < M,, llMll = (TI.

Suppose first M A / D is not IT) +-compact. Then there is A c ]MA/D1, IAI I; IT], such that MA/D omits a type over A. W.1.o.g. (by D s regularity) there is an equivalence relation, eq, over A, eq G A x A, such that for every a E A, eq c {(i, j) E h x A: a[i] = a[ j ] } and eq has IT1 equivalence classes. Let N be the submodel of NA/D defined by IN1 = {a E IM’/DI: eq c {(i,j): a[i] = arj]}}. Then N is also not ITI+-compact and clearly for some ultrafilter D, over IT1 N is isomorphic to NITI/D,, so T is not @lTl-minimal hence not @,,-minimal (as we can make D, regular).

Assume nowMA/D is ITI+-saturated. By 1.8,111, 3.10 there is an indiscernible set I = {a,:n < o} in M A / D , dim(I,Mt/D) < A. Without loss of generality there is an equivalence relation eq over h with I; IT1 equivalence classes such that eq E {(i,j): a,,[i] = a,[ j ]} for every n < o. Clearly {a E wt/DI : eq G {(i,j) : a[i] = ao]}} is the universe of an elementary submodel N of M t / D and I c m. It is also clear that for some ultrafilter D, which we can assume is regular, over I TI, N and MTI/D, are isomorphic. As Mo is A+-saturated, h > 2ITI it suffices to prove N is not (21TI)+-saturated. If it were, it would be A+-saturated by 5.2, hence A 2 dim(I,Mt/D) 2 dim(1,N) 2 A+. Con- tradiction.

Now we shall try to deduce some results on @.

THEOREM 6.8: Let T be countable. (1) T is @-minimal iff T does not have the f.c.p. (2) For h 2 2Mo T is @,-minimal iff T does not have the f.c.p. ( 3 ) I f No < h < 2N0, T is @,-minimal i,lff T ia stable.

Proof. (1) By 4.2(3) and.Exercise 4.21, this follows from (2) and (3). (2) We use the criterion in 4.2(2); i.e., T is @,-minimal if for any

model N of T and regular ultrafilter D over A, MA/D is A+-compact. Suppose T does not have the f.c.p., N is a model of T and D is a regular ultrafilter over A. By 5.1(1) MA/D is K,”/D-compact and by 1.12 N$/D = N,” 2 A + , hence MAID is A+-compact.

Suppose now T does have the f.c.p.; by 3.12 there is a regular ultrafilter D over A and n, < w (i < A) such that nl<,n, /D =‘2h (aa

OH. VI, 5 61 SATURATION OF ULTRAPOWERS 387

2M0 5 A) and by 4.6, for any model M of T, YA/D is not (2h)+-compact hence not A + -cornpaat.

(3) Similarly. For M a model of T, T stable; if D is a regular ultrafilter over A, by 6.1(2) M A / D is p-compact, where p = minm,,, n,/D: nfeA n,/D 2 KO}. But by 1.3( l), (2) D is 24,-incomplete, hexme by Exercise 2.10 pMo = p, hence p 2 2%, hence p > A, hence M A / D is A + -compact.

Now let T be unstable, by 3.12 there is a regular ultrafilter D over A such that lof(No, D ) = K,, hence by 4.8 M A / D is not K,-compeaf, but K, 5 A, so it is not A+-compact.

THEOREM 6.9: Anwng countable t W e a (1) the t h i e a with& the f.0.p. fm an equivalence class, (2) the stable them&?% with the f.c.p. fm an equivahce c h 8 ,

(3) the t-ea with the etrict order property are all equivalent, (4) if T,, Ta and T , are a~ in (l), (2) and (3) rqectively, and T

8&8* n i ~ ~ of them, t h T , @ Ta @ Tad @ T @ T,, but not T , @ T,, and not Thd @ Ta. If, e.g., M A + zn0 > K, holds then not T , @ Thd; m m e r K , ~ ~ ( T ) = 00 impliea not T @ Thd.

Remarks. (1 ) The theorem holds for @A instead of @ for any h 2 2no; and for No < h < 2Ho, the only difference is that all the stable theories become equivdent .

(2) For (4) see also Exercise 3.26.

Proof. Let T,, Ta, T , and T be as in (a), with respective models M,, Ma, bl, and M and ikfhd a model of Thd.

(1) This is the previous theorem. (2) M?JD is A+-compact iff

6.1 (2)) hence they me all equivalent. (3) This is 4.3. (4) T , @ T , is clear from the proof of 6.8 and 6.9(2), T , @ Thd is

clear by the proof of 6.8(2) and 4.8 (as nf<A n,/D 2 KO * n(<A n,/D 2 lcf(Ko, D)). Now Thd @ T can be proved by Exercise 4.6, and T @ T , follows by 4.3. NOW T , @ T , fails by 6.8, Tina @ T2 fails, as by 3.12, for every A there is a regular ultrafilter D over A, n,,A n,/D 2 KO == nteA n,/D = 2A, lof(Ko, D ) = N,, SO M$/D is A+-compact by 6.1(2), but M&,/D is not N,-compact by 4.8. Now aaaume MA + 2Mo > K,. For not T, @ Thd use 3.10(2) (by 4.3 and 4.2(1) Mh/D is A+- compact iff D is A+-good). For not T @ Tad when K&t) = 00, note

n,/D 2 KO * nieA n,/D > A (by

388 SATUBATION OF ULTRAPXODUClTS [m. 0 5

that in 3.10(2) D = D, x D, SO MA/D = (MA/D,)a/D, is not compact by 4.7.

EXERCIBE 5.2: Show that if we replace “ T is countable” by “ D is I TI +-good ” in the hypothesis of Theorem 6.1 , the conclusions still hold.

EXERCIBE 5.3: Let M be a model of a countable and superstable theory T , D an 8,-incomplete ultralilter over I , U a filter over I x I , such that 11MLlUll > No, MLlU # M. Show that if T does not have the f.0.p. and M is h-compact, then MLlU is ALlB-compact. (See Definition 4.2.)

EXERCIBE 5.4: Suppose E(Z, g; 2) is aa in II,4.4, M a countable model of a stable theory T which has the f.c.p., D an N,-incomplete ultrafilter over I and U a filter over I x I . n, = I{Z/E(E, #; a): E E 1M]}1, P = {n,: E E IMl}. Show that there is E E IMLlUl n, = A, iff there is 8 E I(w + 1, < , P)LlUl, such that P(8) and

EXERCIBE 5.5: Let M be a model of a stable theory T which haa the f.c.p. D an K,-incomplete ultrafilter over I , U a filter over I x I , and let A c L(T) be finite. Let p be a A-1-type over N = MLlG which is omitted by N; but every q c p, 141 < lpl is realized in N; and lpl is regular. Show there is 8€(11itf11 + 1, <)&la such that 1231 =

p: CllJfll + 1, 4blU l=t < 811.

EXERCISE 5.6: Let M be a model, M, a complete expansion of M y T = Th(M), T , = Th(M,). Then {MLIU: D an ultrafilter over I , U a filter over I x I) is equal to PC,(T,, T ) .

EXERCIBE 5.7: Let T be countable and complete. Then: for some countable T,, T E T , and PC(T,, T ) is categorical in No iff ID(T)I = No. (Hint: Like 6.3.)

EXERCISE 5.8: Use Exercise 1.9 to get a better ( 2 ) presentation of the proofs of 6.1 and 6.3 (aee next exercise).

OH. VI, 8 51 SATURATION OF ULTRAPOWERS 389

EXERCISE 5.9: Let L E L,, P E L, a one place predicate, and L,- model M , . Let M: be the submodel of the L-reduct of M I with universe P(M, ) . Then for every theory T, c L,, { M f ; M , C T,, M , an L,- model} is PC(T2, Ta) for some Ta (you can use VII, 1.1 and VII, 1.2).

EXERCISE 5.10: In Theorem 5.1(4) let p = &I 11M,11, and A(d) E pr /D be (. . . , &(A), . . . ) t6j /D and let A** = I{a E pr/D: for every finite d c L, pr /D C a < A(d)}]. Prove that N is A**-wturated. [Hint: Use Exercise 1.9 and 6.8. There is (in N ; ) an order of type “ w ” of the formulas of L (including non-standard ones) and for every “set” I c N there is maximal n = nI such that I is tp,-indiscernible for every k < ni, It suffices to prove that each such I can be extended to an indescernible set of cardinality A**. We define by induction on a E N i , ii, E N such that nr, is maximal where I, = I u {?is I f i < a}. Then I u {aa: N ; C a < A(d) for each finite A } is ae required.]

EXERCISE 5.11: The regular ultrafilter D over I satisfies the conclusion of conjecture 5.1 when (J < ) C w 5 a, for each n iff lcf(K, D) > No, K s IIl where K = lcf(Ko, D).

Remark: The conditions (J, < ) C w s a, are satisfied in the case we are interested in. [Hint: Let a, = (. . ., %*, . . . ) J D , and w.1.o.g. a: is a cardinal (see Exercise 1.1) and let X, = mink: { t : x < a:} 4 D}; clearly No 5 x,+, 5 x,, so w.1.o.g. X, = xo; and by 1.12 la,l = XI’’. If { t : qn = x} E D for infinitely many n’s, a = (. . . , x, . . . ) / D prove the assertion. So w.1.o.g. 4 c x for each n, t . Necessarily x is a limit cardinal, cfx s IIl. Let b, = (. . ., l$, . . .)ts,/D (j < K ) exemplify lcf(N,, D) = K ; and c, = (. . ., . . . ) t ,I/D; then (J, <) Cc, s a, A c, < c,fori <j < Kandlet{dEJ:(J, <) Cd 5 a,foreachn} = U,<k {d E J : ( J , <) C d c c,}. So cf x = K. If a,, D fails the conjecture I c , ~ < xl’lfor each j hence for some A < x, (J, <) C c, < A, and it is easy to show Icf(K) = 8,; and the other direction should be eaey.]

EXERCIBE 5.12: Show that for good K,-incomplete ultrafilter D conjecture 6.1 holds. [Hint: If (J, < ) C a, < w for some n, by 2.3 there is c, (J, < ) C n < c < a, for each n, and use 2.13. Otherwise by 2.3. K I IIl * lcf(x, D) > 111 and uae Exercise 5.11.1

EXERCISE 5.13: Suppose that for some n,, p = nlEInt/D > No and N = M1/D is ( I TI + p) +-saturated, and M ia A-compact. Then N ie


AI/D saturated. [Hint: As in 5.2, let A > [TI, T without the f.c.p., I E N indiscernible, 111 = ([TI + p)+. There are finite Pt E 1M1 such that IP"I I p, 11 n PNI 2 KO, where (N, P") = n t , i ( M y Pt ) /D. There is 7i E I such that stp(a, u I") E Av(1, N), and Qt = It E 1M1 an indiscernible set over u Pt based on stp(a[t], U Pt), and (M, P, &) = nts, ( M , P,, Qt)/D. Then Q is indiscernible, /&I = K/D, Av(Q, N ) =

AV(1, w.1

VI.6. Saturation of ultrslimits

For every model M and ultrafilter D over I the natural elementary embedding of M into Mr/D, is defined by a H (a: i E I ) / D . Hence we can look at M'/D m an elementary extension of M ; and so we get an elementary chain of models by repeatedly taking ultrapowers and taking unions at limit stages. For simplicity, all the ultrapowers will be with the same ultrafilter, which will be assumed Kl-incomplete.

DEFINITION 6.1 : Let M be a model, D an N,-incomplete ultrafilter over I. We defhe the ultralimit, UL(M, D, a) by induction on a, so that for /3 < a, UL(M, D, /3) is an elementary submodel of UL(M, D, a).

(1) UL(M, D, 0 ) = M . (2) For a a limit ordinal UL(M, D, a) = UBSK UL(M, D, p). (3) For a = /3 + 1, UL(M, D, a) will be isomorphic to UL(M, D, @)'/D,

and for each a E IUL(M, D, /3)1, the isomorphism ElB takes (a: i E I ) / D E IUL(M, D, /3)'/Dl to a. So UL(M, D , /3) < UL(M, D, a).

Notation. As M and D are fixed for most of this section we let M , = UL(M, D, a) and FB be the isomorphism mentioned in (3) of the above definition. Clearly we can msume that for every a, /3, UL(M, D, a + 8) = UL(M,, D,/3). We write T for Th(M). We shall try to find how compact the ultralimits am, for various properties of the ordinal, the ultrafilter, and the theory of the model. As MK+, is isomorphic to ML/D, we shall restrict ourselves to M6 for limit ordinals 8.

THEOREM 6.1: If cf 8 2 p, and for every A < p, D is A-regular, then Md i8 p-COTn$XXCt.

Proof. Let p be a type in Md of cardinality < p. Then clearly p is a type in M B for some /3 < 6. As D is ~p~-regular, p is realized in M B + by 1.4,

OH. VI, f 61 SATUIZATION OF ULTRALIMITS 391

hence p is realized in M,. So every type in i t i d of cardinality < p is realized in Md; hence Md is p-compact.

Proof. M , is K,-compact by 2.1, 2.3 and 2.11 (remember D is K,- incomplete). As T is unstable, by 11, 2.13 and 11,2.2; there is a formula ~ ( z ; g) and sequences a,, a,,. . . a,, . . . from M , (an of the length of g) such that: for every m < w, {cp(z; 7in)M(nrm): n < w} is consistent with M , . As cf 6 = p, let 8 = U,<” u(j), wherej < i < p implies 1 < u ( j ) < u ( i ) < 8. We shall now define by induction on j sequences such that :

(1) lMa(j)+Il,~’# l%j)l. (2) qj = {7cp(z; a,): n < w} U {~(z; a’)} is not rertlized by any

element of Ma(j). (3) For every m < w, ppJ” = {cp(z; iE,)if(“Zm): n < w} u {cp(z; 8): i s j)

is consistent (with Mw+l) . If we succeed in defining the $’a then clearly by (3) p = {-,cp(z; a,):

n < w} u {cp(z; Z): i < p} is consistent (with M,), because every finite subset of p is a subtype of ppJ” for some m, j. But i f p is realized in Hd, then it is realized in M B for some /3 < 6, and there is j < of 6, < u(j) < 8. Hence p is realized in Ma(,), in contradiction to (2). Hence p is a consistent type in M,, which M , omits, and IpI = KO + p < p+. So Md is not p + -compact.

It remains only to define $, assuming 8 has been defined for i c j. As D is K,-incomplete there are I,, E D, I,,, s I,, I , = I , on<, I, = 0. Let us define 6j E lMk(,JDl: if i E I, - In+l, then @[i] = a,, 8’ = (6’[i]: i E I ) / D , and

Let us check conditions ( l ) , (2) and (3) are satisfied. Clearly 8 E

lMa(,)+,1. Now for any n < w, {i €1: 6j[i] = a,} = I , - I ,+ , 4 D so $ 4 lMml. So (1) is satisfied.

For proving (2), suppose, for somej < p, c, E lMa(,,] realizes pi. Then {i E I: Mw) b cp[P&(c,)[i], 8j[i]]} E D, that is, {i E I: M,, b ~[c , , P[i]]} E D. Hence for some n < w Ma(,) k cp[cj, a,], so cj does not realize qj , contradiction, hence

(2) holds; and (3) has a similar proof.

= Ba(j)(8j).

DEFINITION 6.2: p ( D ) is the least infinite cardinal p such that D is p-descendingly complete, that is, p ( D ) is the least infinite cardinality p


such that I , E D, a < p, and a c /I * I , c I , implies no<,, I , # 0 (equivalently na<, I , E D).

Remark. If D is K-regular then K c p ( D ) ; also p ( D ) I ] I ] + . Note also that p(D) is a regular cardinal.

Proof. Let p be a type in Md, 1p1 < p, and we shall prove that p is realized in Md, thus proving the theorem.

As /p i < p I cf 6, p is a type in Ma for some a < 6. Let lpl = K,. We shall prove by induction on y I p that:

(*I

As /I I K, = 1p1 < p 5 of 6, a + /I + 1 < 8, hence by proving (*)we shall prove that p is realized in Ma.

Suppose we have proved (*) for every y1 < y. Hence every subtype of p of cardinality < K, is realized in M a + , (remember every model is 24,-compact, hence every finite subtype of p is realized in Ma). Clearly we can assume w.1.o.g. that p is a 1-type. So let q = {g~,(z; a,): j < H,} be any subtype of p of cardinality K,; we shall prove q is realized in Ma+,+ , . By the induction hypothesis, for every j < K,, there is c, E

lMa+,1 which realizes {p,(z; ZJ: i < j } . As K, I IpI < p I p ( D ) there is a decreasing sequence I,, j < K , , I , E D, n,<H, I , = 0, with I, = I . Let us define C E l (Ma+y) l /D[ : if i ~ r \ , < , I , - I j then c[i] = cj (clearly c is well-defined). Now F,+,(c) E IMa+,+,l realizes q, as for every j c H,, {i E I: Ma+, C 9~[c[iI, 2 Uf>j (n ,<cI , - I?) = Ij+1 E D. So q is realized in Ma + , + , ; so p is realized in i i f d .

DEFINITION 6.3: A model N strongly omits a type p (in it) if no subtype of p of cardinality I p I is realized in N.

LEMMA 6.4: (1) If M strongly omitep, IpI = p(D) , then M , also strongly omits p .

(2) if Ma strongly omits p , 1p1 = p(D) , and a < 8, then M , also strongly omita p .

( 3 ) In (1) and (2) instead of I p I = p( D), its suflcea to aesurne that there are no I , E D for /3 < lpl, /I < y - I , c I,, n,<,,, I , = 0; and lpl is regular.

Every subtype of p of cardinality I K , is realized in M a + y + 1-

OH. VI, § 61 SATURATION OF ULTRALIMITS 393

Proof. We shall prove that (1) as (2) and (3) have similar proofs. Suppose (1) fails, so c1 E jMll realizes q G p , 1pI = lpl. Let c1 = Po@), p = {(p,(z; a,): I < lpl}. So clearly for every 8 < 191 = 1131 {i E I : 211 k tp,[c[i]; a,]} E D. It is also clear that for every i E I, q(i) = {(p,(z; Be): M C (p,[c[i], a,]} is a subtype of q, hence of p, which is realized in M ; hence Iq(i)l < 1p1. As lpl = p(D) is regular, for every i E I there is a bound ((i) < 1p1 to (5 : M b(p,[c[i];a,]}. Let for 5 < 1p1, I , = {i E I : [(i) 2 l}. Clearly I,, 5 < IpI is a decreasing sequence, and by the definition of [(i), n,<IpI I, = 0. In addition each I, E D as I, = {i E I : [(i) 2 t;} 2 {i E I : M != (p,[c[i]; S]} E D. So we get a contradiction to the definition of p( D).

THEOREM 6.5: If T is urntable, 6 2 p(D) , then M , is not p(D)+- compact. (Moreover there is a type in of cardinality p ( D ) which M , StrO?& OW&%.)

Proof. As it is similar to the proofs of 6.2 and 6.7 we omit it.

CONCLUSION 6.6: If T i8 unstable, p = min(cf 6, p(D) ) , then M , is maximally p-compact. ( i e . , p-compact but not p+-compact).

Proof. Immediate by 6.2, 6.3 and 6.5.

THEOREM 6.7: If K ( T ) > p = min(cf 6, p(D) ) , T is stable, then 116, is maximally p-compact.

Proof. By 6.3, M , is p-compact so we need to prove only that M , is not p+-compact. By hypothesis T satisfies p < K(T) , so there are A,, 4 I p , ~ E~W,,), Q&, Sc), andai,,n, (I < p, n < w); such that: S < 5 * A, c A,; for every 5 < p, {i&: n < cu} is an indiscernible set over A,, ii,,,, ~ p . Clearly it suffices to prove the theorem for the case L = L(27) is the minimal language containing all the formulas (p,(x; S,); so ILI I p. Choose a([) < 6 for 4 < cf 6 such that 6 = Uc<cfd .(I).

Now we define by induction on 5 I p an increasing sequence of (6, Ma)-elementary mappings H, and ordinals !(I) I 6 such that:

(1) The domain of H, is U,<, Range (8,,o-Z,,1). (2) The range of H, is included in IMB(,)+ll, /I(&!) 2 a(g).

(3) If 5' < 5 < p then p(6) < p(5) < 6, and for every C E IMB(,)I,

E A,+l and -(pe(x; 7i,,o), (p(x;

M , C n[c, ~ , + 1 ( ~ , , 0 ) 1 = R[C, HC+l(~,.l)l.

394 SATURATION OF ULTRAPRODUUTS [OH. VI, 5 6

Let H , be the empty function, p(0) = a(0). For f a limit ordinal,

He = U C < ~ H C , rB(O = max(a(€)i U<<c B(O) if € < p, and @(p) = Uf<& KO. CIf 6 < p, B(€) < 8 bf3caum p I; Of 8.1

Suppose H,, p ( t ) are defined, 6 < p, and we shall define HIE+,, p(e + 1). We first show:

(*I There is /? c 8 such that we can extend €2, to a (6, X8)-

elementmy mapping H* from Dom H, u u {?it,,: n < w} hto a6,.

If p = No, this is true, aa for every N, NxlD is N,-compact, so /3 = /I(& + 1 will suffice. So assume p > No. We now define, by induction on n, an increasing sequence of (a, Md)-elementary mappings Hn from Dom H , U u {ae,,,,: m < n} into Md. If we have defined H", and cannot define Eln+', we can deduce Md is not p+-oompad [as it omits {'p(Z; H"(I7)): 'p E L, I7 E Dom Hn, t &iC,,, E l } ] and so the conclusion of the theorem holds. So we can assume H" is defined for every n and let H* = Un<oP Hn. Clearly H* is a (Q, Md)-elementctry mapping with the appropriate domain into Mdy Range H , c M#(e)y B(1) < 8, of 8 2 p and H* is into h f 8 for some /3 < 8. So we have proved (a). But we cannot define H , + , = H*, as ( 3 ) may fail.

Define /3(( + 1) = max(/3, a(&). As D is K,-incomplete there me I, E D, n < w, I , = I , m < n s I , c I,,,, I, = 0. Define

fined as follows: if i E I, - In+,, E[i] = H*(iLE,nn?iC,n+l). It is easy to verify that He+, , p(f + 1) satisfy the induction conditions (by 11, 2.20).

He+l(at,Onac.l) lMER+1)+1I &8 p8(t+1)@) where EM8(C)+l'/D is de-

Now P = {'p,("Y H,+l(a,,o)) l'p,(", H,+,(B,,,)): 5 < p} is a consistent type in ikf6(u), and it is strongly omitted by ME(,,). As Md = UL(ME,), D, 8 - / l(p)), by 6.4 Md strongly omits p, and this means ikfd is not p + -compact.

It is natural to conjecture that if K(T) s min(pCC(D), cf a), and a, /3 < S * a + p < 6, then M is UL(X,, D, &)-saturated [uL(X,, D, 8) is the cardinality of UL(M, D, 8) for every countable dl] which would generalize 6.1(1). But this is not true. T may be superstable (~(5") = No) or even simple (see Definition 0.4) while M or M , strongly omits a type of cardinality p( D ) . However :

THEOREM 6.8: Supporre K ( T ) s min(p(D), cf a), D is a ITI-regdar ultra$lter; a,$ < 6 * a+$ < 6. Then Ma is A-saturated, where A =

UL(% D, 8).

CH. VI, # 61 SATURATION OF ULTRALIMITS 395

Remarks. (1) For every 6, there are S,, S such that 6, = S,+S; a, B < 6 =+ a + < 6 and UL(itf, D, 6,) = uL(Md,, D, 6). so the restriction on 6 is natural.

(2) Clearly A > I 2'1, so it s&ces to prove Md is A-compact.

Proof. Let p be a 1-type in i t f d , 1p1 < A. It suffices to prove p is realized in M,. Let q be any extension of p in S(lMd)).

Notice that if IBI < K(T) s of 6, B E 1Md1, then for Borne a < 6, B G lMal. Hence by 111, 3.2.there is a < 6 and B E lMal such that q does not fork over B, and by 111, 2.16 and I11 4.18, q 1 1Ma1 is stationary. So by 111, 4.2 there is a set B clHal, IBI s 12'1 such that for every 'p R(q 'p, 'p, 2) = R[(q 1 B) 1 'p, 'p, 21. Now we can define a,, for n < w such that:

(1) a,, realizes q 1 (B u {am: m < a}), (2) if 8 > ws a n E IJfo+n+ll*

This is possible since D is 12'1-regular, by 1.4(2). Hence {a,,: a < w} is an indiscernible set and q = Av({a,,: n < w}, Ma). suppose for a moment 6 > o. Let P = {a,,: n < w } E lMa+al (ae a < 6, o < 6; a + w < 6). Let (Ma, Pd) = UL((M,+,, P), D, 6) (remember 6 = a + o + 6). Clearly Pd extends P and is an indiscernible set. So 'p(z, 6) E p implies 'p(z, 6) E q implies {a: a E Pd, I= -+z, 6)) is finite. So all except lpl- 8, < h members of Pd realizep. As lPdl = UL(Ho, D, 6) = A, the theorem follows. So we remain only with the case 6 = O. But then we can define the an's simultaneously in Ma + I and the proof goes in the same way.

The following exercises will deal with the problem of categoricity in logics with generalized quantifiers.

Assumption: Let T be a complete theory in the logic L(Q1, ..., Q") (where ifg, is a formula so is (Q'xg,)). Let P,, . . . , P , be unary predicates in L , VxP,(x)€T and let for any cardinals ,ul < < , u m , A l , ..., A,, K(p, , ..., A,;..) = { M : M an L-model, that satisfies every $ET when (Qzx)O

is interpreted as "there are a t least A, x's such that B(x), and

Suppose further that ,uF = ,ul, each A, is regular and (Vx < A,)x'o <

Suppose (*)K(p,, . . . , A,, . . .) has a unique model up to isomorphism.

IRI = P 'Y '

4 *

396 SATURATION OF ULTRAPRODUOTS [m. VI, § 6

EXERCISE 6.1 : We can assume w.1.o.g. that for any formula q(Z ) in I,(()', ...,Q"), forsomepredicateR, (VZ) [~ (Z ) = R ( 2 ) l ~ T . L e t T'bethe set of first order sentences of T.

EXERCISE 6.2: Prove T' is superstable. [Hint : It is well known (see [CK 731) that K(pl , ..., A,, ...) is closed under ultraproducts for ultrafilters over w . Let M belong to K(p, , ..., A,, ...) (exists by a hypothesis), and let D be a non-principal ultrafilter over w . So M , gf M"/D and easily also M , gP UL(M, D, w ) belong to the class K ( p , , . . . , A,, . . .), hence by a hypothesis they are isomorphic. Now by 2.1, 2.3, 2.11(1), M , is K,-compact. By 6.5 if T is not superstableM, is not K,-compact. We conclude T is superstable.] We now define:

W = {(q(z, g), $(g), A,) :p, $ first-order formulas, (Vg)($(g)

= 1 (QLx)rp(x, g ) ) belong to T'} U {(Pz(z) , z = x,p;) : I = 1 , m}.

EXERCISE 6.3 : Suppose T is stable (but not necessarily (*)). Prove there i sMEK(p , , ..., A,, ...) which is theL-reduct ofM,, IL(M,)( = JL(, M , the Skolem Hull of UIZ:nI,,I, indiscernible &,,I,. [Hint: See VII, §2,5 and use V, 6.10.1

Now we apply the results of V, 96 to analyze the (n+m)-tuples of cardinals for which we get categoricity.

EzampZe. Let N = ("2, + N , P:),<" where : P r = (7 E "2 : ~ ( n ) = 0}, + is a two-place operation, 7 + N v = p iff for every n r](n) + v(n) -p(n) is even. Let M = (WI, P M , Q M , E M , + N , G N , H N , F M , P f , . . . , P r , . . . ), < w

where : P" = N , Q M = N x N x N , EM = {(a,,b,,c,),<a2,b,,~,)):a,,~,,~l,~2,~~,~~~~,~l = az,b, = b2}, + H N , G N are one-place functions from QM to P', H N ( ( a , b, c)) = b, @"((a, b , c ) ) = a,

P M is a partial two-place function, FM((a , b, c,), (a, b, c z ) ) = c1 + N ~ Z ,

P r = { ( a , b , c ) E N x N x N : c E P t ) .

is a ,two-place function from PM to P', = + N ,

CHAPTER VII

CONSTRUCTION OF MODELS

VII.0. Introduction

The character of this chapter is different from that of II-V as extending the theory does not change much; hence it is natural to use theories with Skolem functions, that is, when we want to construct models of T, we choose T,, T z T,, ITl! = ITI, and T, hrte Skolem functions; we construct models of T, and go back to their L( T)-reducts. The importance of T, having Skolem functions is that: (1) we can find such T, for every T; (2) every model of T can be expanded to a model of T,; (3) if M1 is a model of T,, every submodel of M1 is an elementary submodel. (This is done in 1.1-1.4).

Saturation is a very useful device, but unfortunately if A < A<", T does not necessarily have a K-saturated model of cardinality A. We have constructed in I, 1.7 a saturated model by an elementary chain Mf of models of T of cardinality A such that in Mf + , we realize rte many types over Mi as we can (remembering the cardinality restriction). We generalize this process (in Definition 1.6) and prove (in 1.7) that for many x's we get some weakening of saturation, e.g., of the form: if p is an m-type over M , 1p1 = A , and every q E p, IqI < p is realized in M then some r z p , Irl = x is realized in M. Notice that when A'" = A we get a K-compact model (when the length of the chain has cofinality Z K , of course); if (Vp < K,)(p<" < K,), 8,'" > K,, and h = our results become weaker as p increases, and we get none when ,9 2 y = K, > K,. However for the theory of dense linear order we can get something (see Exercise 1.7) and a generalization of this is utilized in VIII, 3.2. (There are also other applications of Section 1 to VIII, Section 2, and 3, but we do not need them in the main presentation.)

From one point of view a saturated model realizes many types, but from a deeper point of view it realizes few. Let TP(S, M) be the set of

397

398 CONSTRUCTION OF MODELS [OH. VII, f 0

types realized in M over ii, TP,(M) = {TP(ii, M ) : I = (a‘: i < K ) ,

a, E [MI} . Then clearly for a A-saturated model M of a countable theory ( K < A), TP,(M) has cardinality I ; ~ ” + I * I , but for a not necessarily A-saturated model, it is possible that ITP,(M)l = 2a“. We deal with these problems in 1.10-1.12 and prove, e.g.,

THEOREM 0.1: If p < 8, =- pK < X,, t9 .c K + , T countuble, 11M11 s Na+B then M hua an elementary extension N of cardinality Xcr+B for which ITP,(N)I = 2”.

In Sections 2 and 3 we deal with Ehrenfeucht-Mostowski models, generalizations in two directions, and various applications; which are preparatory for VIII.

For a theory T , with Skolem functions, every set A is a set of generators of a model, and it is natural to look for “nice”, “uniform”, such sets. The clmsical example is a free algebra in which every permutation of the set of (the free) generators can be extended to an automorphism of the model. But for any extension of the theory of dense linear order we cannot have such sets. However, by Ramsey’s theorem and compactness we can find models of Tl generated by indiscernible sequences, so the order of the generators is important. Clearly the elementary type @ of the sequence and the order type I determine the model up to isomorphism, so we denote it by EN1(I , @), and its L( 2’)-reduct by EM(I , 0).

Its importance is that we can easily find orders I with specific properties, and often the model inherits them; e.g., stability (of the model) and some of the generalizations of saturativity from Section 1. An application is

THEOREM 0.2: B”0r stable T with the f.c.p., lTll = No, I ( X a y T, , T ) 2 214.

Unfortunately this is true only for cardinalities > I T, I, so when we try to construct models of cardinality lT,I (e.g., in VIII) we have to generalize this construction. We first show we can assume w.1.o.g. that in L(Tl) there are only countably many non-logical symbols which are not individual constants, then we form a model N from individual aonstants, and define EM1(I, N 1 ) for any order I, and elementary extension Nf of N. Now we can determine properties of E M 1 ( I , P ) by those of I and W .

OH. VII, $01 INTRODUCTION 399

At first sight, we cannot expect anything better than an indiscernible sequence of generators. However, we sometimes know of a set of elements satisfying some conditions, and we want to find such a set as homogeneous as possible, which still satisfies those conditions. For proving this we always need the proper generalization of Ramsey's Theorem. In particular if T is not superstable there are v1 E L(T), and sin (for q E such that for q E Ow, v E O'w, vl&&,, a,] iff v Q q. For dealing with such situations we d e h e the indiscernibility of {G8: 8 E I> for a model I : the type of a8,co, -. , . , us(n-l) - depends only on the atomic type of (s(O), . . . , s(n - 1)) in I . The general treatment appears in Section 2 and the specific cases in Section 3. We deal there with theories with the f.c.p., and indiscernible sets indexed by trees. An important case, which does not completely fit with the general case, is uncountable D(T) for countable T,. What we get is {an: q E "2) so that the an's realize distinct types from D( T), and for every cp(Zo, . . . , Z,,,-l) E

L(T,) for every n 2 n(v) , if q1 1 n(l < m) are distinct, the satisfaction of v(an0,. . .) depends only on (ql 1 n: 1 < m) and a. For uncountable T's see a generalization VIII, 1.10.

The reader can relax while reading Section &almost no background is needed. Our main aim is

THEOREM 0.3: If h 2 2", p 2 2N0, T, countable (for simplicity), then T, has a p-universal model stable in p, of cardinality A.

If I is a well-ordered set of cardinality A, M = EM1(I , 0) is stable in every cardinality 2 ITl] (see 2.9). So if D is a good ultraflter over K,

M"/ D is K + -saturated, hence K + -universal, but it still is stable in every cardinality h = A". So assuming G.C.H., we can prove the theorem for a successor p. However, without it, we need h e r methods.

Remember that (by 111, 4.10) a type p does not fork over IN1 iff p is finitely satisfiable in !MI (though p is not necessarily over IMI). For not necessarily stable theories the second condition is still meaningful, and we can prove for it the extension property, and that it is satisfied by any type over IMI. So if M , (i < A) is a list of models of TI, containing, up to isomorphism, every one of cardinality s p , we can find Mi z M , such that for each E ]Mi l , tp@, U,<, M;) is finitely satisfiable in No. Let llMoII = ITJ, if T1 has Skolem functions utw generates a model of TI. For ,u 2 221T11 the model is ,u-stable ; but we work more and get it for ,u 2 2IT1l. By such a construction we prove our theorem. Notice that by such a construction we can get a sequence of indiscernibles.

400 CONSTBUCTION OF MODELB [CH. WI, $ 1

In the last section we deal with Morley numbers, i.e., Hanf numbers for omitting types.

PROBLEM 0.1: It will be interesting to find new kinds of indiscernibility. This may require proving new partition theorems or trying to find an application of a known partition theorem, e.g., Laver's generalization of Galvin's theorem 7 4 [7]& (i.e., 9 + [7]R(,,,)).

VII.1. Skolem functions and generalizations of saturativity

DEFINITION 1.1: (1) Lo has Skolem functions in T , ( = T , has Skolem functions for Lo) if for every formula p(z; 9) E Lo there is in T , a sentence of the form

(Vi7)[(34&; i7) + 'p(P(!i); @)I (P a function symbol.) Such a sentence is called a Skolem sentence of 'p = 'p(z; g) and P is called a Skolem function of 'p(z; 0).

( 2 ) T , has Skolem functions if it has Skolem functions for its language L, = L(T,).

THEOREM 1.1: (1) For every lamgwe L there is a lamgwge Lc1) I> L and a theory T(l) = T(l)(L) in L(l) such that:

(A) T(l) has flkokm functions for L. (B) Every L-model can be expanded to an L(l)-model of T").

(2) For every language L there is a language Lag z L and t h y TsK = TsK(L) in LsK 8ucJL thut:

(A) TsK has rSkolem functions. (B) Every L-model can be expanded to an LSK-model of Tag. (C) ITSKI = ILsKI = ILI.

(C) IL(1'I = p 1 q = ILI.

Condition (A) is satisfied by the dehition of 2'"). If M is an L- model we expand it to an L(l)-model as follows: we define $(a) as any element b of M srttisfying M C 'p[b, a] if M C (~s)~J(s, a); and we define $(a) as an arbitrary element of M otherwise. Clearly we succeed in

OH. VII, 8 13 SKOLNM FUNCTIONS 40 1

expanding M to an L(l)-model of T(l), so condition (B) holds. Con- dition (C) is immediate.

(2) We define, by induction on n, Ln and Tn: Lo = L, To = 0; Tn+l = T(l)(Ln), Ln+l = (Ln)(l). We let LsK = UneopLn, TsK =

Un < Tn. Clearly Ln is an increasing sequence hence any formula 'p in Lax is in Ln for some n, hence in Tn + we can find a Skolem sentence of 'p, so in TsK there is a Skolem sentence of 'p; hence condition (A) holds. As for condition (B), if M is an L-model we define by induction on n, the Ln-model Mn: Mo = M, Mn+l is an expansion of Mn to an Ln+l- model of Tn+l (by (1) this is possible). In the limit we get an LsK- model of TsE = Un Tn. condition (C) is immediate.

THEOREM 1.2: (1) Sugqme M is a model of T,, N a subnzodel of M, ( M , N are L,mOdels) L, = L(T,) and Lo E L,, and T , hae Skolem function% for Lo. Then the Lo-reduct of N ie an elementary subnzodel of the L,-reduct of M .

(2) Suppose M is a model of T,, T , hae Skolem functim, and N is a mbmodel of M . Then N is an elementary submodel of M.

Proof. (1) By the Tarski-Vaught test I, 1.2 it suffices to prove that if a E IN1 'p(s, g) E Lo and M C (3s)'p(s, a) then for some b E IN1 M C 'p[b, a]. But by assumption there is in L, a Skolem function P of 'p

(in T,), hence M C (3s)'p(s; a) implies M C 'p[P(a), a]. As N is a submodel of M , P(a) E INI, so we finish.

(2) A particular caw of (1) where Lo = L,.

CONCLUSION 1.3: Suppxe Th(M) hag Bkolem fundim, A c 1M1. Then the Skolem closure (=Bkolem Hull) of A, { ~ ( a ) : a E A, T a term} = do1 A = acl A is (the universe of) an elementary submodel N of M . So for every 6 = (bo, . . . , b,) E IN1 there are a E A, and t e r m T ~ ( E ) , . . . , T,(@

80 that bi = .;(a). we Wite it in 8hort 6 = ?(a), ? = ( T ~ , . . . , 7%). If A is (totally) ordered, we can aemme is an increaeing sequence. In any m e we can aasume there are no repetitions in a.

THEOREM 1.4: Suppoee [MI = {ai: i < A}, L = L(M), h > ILI, h ie regular. Then the set C of a < h which satisfy the following, is a closed unbounded 8ub8d Of h:

A , = {ai: i < a} is (tlte universe of) an elementccry idmodel of M .

Proof. By I, 1.3(2) C is closed. Let us prove it is unbounded. By 1.1(2) we can assume w.1.o.g. that T = Th(M) has Skolem functions. So

402 OONSTRUUTION OF MODEL8 [OH. V'II, 9 1

a E C iE A, is olosed under the functions of M. Clearly Idd A,1 < A hence for some /3 = /3(a), do1 Aal G AB. 'SO for m y a. c A, if a,,,, = /3(an), a = Un a,,, then do1 A, = A,, so we finish.

DEFINITION 1.2: (1) &(A) = { B : B c A, IBI < A}. (2) %(A) = { B : B c A, IBI 1 A}. (3) S A ( A ) = { B : B G A , IA - BI < A}. (4) #$(A) = { B : B G A, IA - BI 2 A}.

DEFINITION 1.3: ( 1 ) H is a set function if it is defined on sequenoes {ti: i < A} (ti any entities),

H({t,: i < A}) c ( 8 : s c {ti: i < A}} and

H({ti: i < A}) = {{ti: i ~ q : 8 E H(A)}.

(2) We sometimes use aa a& of indioee sets other than A, mainly orders. H is a pure set function if any permubtion of A maps H(A) onto itself. H , 5 Ha if H,(I) E Ha(I) whenever H l ( I ) is defined.

Renucrk. Clearly SA, S f , SA, S$ are pure set functions.

DEFINITION 1.4: Suppose H,, Ha a m set functions, M a model, A a cardinality. Then

(1) M is (A , H I , Ha)9aturated when for every set A s 1 M 1 of cardinality 5 A, ordered in some way in order type A, and p c S m ( A ) ; if for every B E H,(A), p 1 B is realizes in M , then for some B E H,(A) p tBisre&edinM.

(2) M is (A , H,, Ha)-compact when for every sequence r = {qt(2; a,) : i c A} (ai E 1611, q, E L ( M ) ) if every p E H,(r) is realized in M , then some p E H a ( r ) is realized in M.

LEMMA 1.6: M i s A--act iff it i s (p , SN,, S1)-wm~pact for my p < A. M is A--eneoua iff it k (p , rSn,, 8')-eaturated and No---, for every p < A.

(2) If H1 5 H1, Ha 5 H' , M k (A , H I , Ha)-saturated [-comw] then M is (A, H1, H " - s a t u r a [-compact].

(3) (A, H I , Ha)-Baturatim and mpa&ne%8 are p e r v e d by &fin;- tional expansion. (See I11 6.14.)

(4) (A, H,, H 2 ) - ~ p a & n e % 8 is preserved by adding individual constants and by taking reducts.

Proof. Immediate.

CR. VII, 5 11 SKOLEM FUNCTIONS 403

DEFINITION 1.5: Let M be a model of T, (L = L(T) = L(M)) 11M11 = A 2 K + ILI , K is regular. Then M is called (A , K)-wturated if the following holds: (R(I?), E) is a model of ZFC, consisting of the seta of hereditary power < I?. Some N E R(I?) is a (2”) + -saturated model of T and as a set N is of hereditary power 5 llNll, 94, = SA,, (a 5 K ) is an increaeing and continuous sequence of elementary submodels of 93 = (&I?), E, N , A ) (N, A serve as individual constants) such that:

(1) 93, has cardinality A. (2) If a E l93,l, and a is a set (in R(R)) of cardinality 5 A then b E a -

b E ISal. (3) (IS41: p 5 a) ~ 9 3 , + ~ (hence, e.g., there is an ordinal in 93a+l

bigger than any ordinal €BE, hence the set of ordinals €!BK has cofinality K).

(4) Let N , be N as interpreted in 93,. Then M is isomorphic to N,. (Notice N , E by (3), and that N, is an elementary chain as 93, is, and L E S,.)

Remark. It almost always suffices to msume N is A- or at most A + - saturated. We can replace “(R(I?), E) is a model of ZFC” by weaker conditions to ensure the existence of such a’s, but our theorems will not be changed.

THEOREM 1.6: (1) If A 2 K + ITI, K regular, M a nzodel of T of cardinality 5 A, then M has an elementary extension N which is (A , K ) -

eaturated. (2) If Lo c L, the saturation of DeJinition 1.5 holde, and Lo E 23,

then the reduct of M g N , to Lo is ako (A , %)-saturated. ( 3 ) Let M be (A , ~)-saturated. If 2IL1 I A, M is K,-saturated. If Ax = A,

K > x , M is X+-compact and X+-honzogeneowr. If 21LI 5 A = Ax, M is

Proof. (1) and (2) are immediate. For (3) see the proof of Theorem 1.7( 1 ) below.

THEOREM 1.7: Suppose M is a ( A , K)-saturated madel, p a cardinal s A. If A@ > A > 2, let N, = mink: f 2 A} (hence 2, < N,, cf(H,) 5 p; p < a - N$ < 8,). Let A = K,. Then:

X+-BdUV‘d&. M is No-hO?TlQgent?W.

(1) If A = A<#, K 2 p then M is p-compact and p-homogeneous. (2) If 2” < A < A”, y - a < cfp, cf K, # cfp, and K # cfp then

(3) If 2’ < A < A,, y - a < cfp, cf N, # cfp, p < K , then M is M is (p, sn,, s : ) - c o m p ~ .

(p, s,, S1)-compact.

404 O O N S T R U ~ O N O F MODEL8 [CH. VII, g 1

(4) If 2Y < A < AY; and for any 8, a < /3 5 y =+ /3 - a < Ks; p < x 5 A < K a + x y x # K then M ie (x, SHo, S;)-~rn~)a~t.

(5) (i) If 2fl < A < AN; and for every 8, a < 8 5 y =- 8 - a < NE and p < x < K 5 A < t h a M is (x, S:, S1)-compact.

(ii) If A = AMo and x < K 5 A, then M is (x, Si, r91)-com~t. (6) The dde7nentS (1)-(6) b l d with 8d~t'dion of c0rn'p&12e88;

but for (1) we s b l d aesume 2ILI s A, in addition.

Remark. In parte (4) and (6)(i) we can replace ''8 - a" by "/3".

Proof. W.1.o.g. we shall use the notation of Definition 1.5 and assume M = N,.

(1) Suppose p is a sequence of m-formulas over M of length p, ) which is finitely satisfiable in N. Aa K is regular and >pl, p is over Na for some a < K. Now L, Na E IL( 5 IINaII = A, hence the set @ of m-formulae over N, belongs to b,+,, hence SN(@) E Sa+ ,, but [ & ( @ ) I = A'" = A, so p E BY(@) == p 6 (by (2) in the definition). So p is realized in Na+, (as Sa+, is an elementary submodel of (R(K), E, N, A) and N is (2"+-saturated; we use such reasoning without saying). Hence p is realized in M = N,.

The p-homogeneity is proved similarly.

CLAIM 1.8: Suppose u E IS^,^^, u c 1SASEl, S c u, IS1 < Kt 5 IuI. (1) If IuI < then for some VES~,, , S G v, lvl 5 Kc and if

cf Kc > JSl even I v I < Kc. (2) If I u I s Kc+,, cf 181 > j, then for sow v ESA,~ IS n v( = (81,

I v I 5 Kc and if cf Kc # cf (Sl even lvl < Kc. (3) V c f K c > 181; and x < KI;=xlSl < A, then in (1 ) K p , < A+K5+,

so S E %A,B; and if cf Kc # cf 181, x < Kc == xlSl < A , then in ( 2 ) we can aesume v c 8.

Proof of 1.8. (1) Let lul = NC+,, and we prove it by induction on n. Clearly Kc+, 5 A, and there is in a one-to-one function g from Kc+, onto u. For I < Kc +, let u, = {g(i ) : i < e}, so clearly u, E SAPE and if 181 < cf Kc+,, then for some I , S E u,; as )u,J < Kc+, the conclusion follows by the induction hypothesis. If (81 2 cf Kc+,, n = 0, u = v ie sufficient.

(2) Let lul = Kc+ ,, and we prove it by induction on I.

OH. VII, 5 11. SKOLEM FUNCTIONS 405

If ( = 0 or ( is a successor the proof is as in (1). If ( is a limit ordinal cf I u I < cf l#l;hencedefiningu,,i < Nc+easin(l) ,there~((i) , i < i, < cf 181 so that u = U ,,,, u ,,,,, hence for some i, 18 n ue,,,l = 181, and of course lue,,,l < IuI, so using the induction hypotheeis on ueco, 8 n ueco we get the desired conclusion.

(3) Immediate, by Definition 1.6, condition (2).

Continuation of the proof of 1.7. (2) Let p be a set of m-formulas over N,, lpl = p , p finitely satisfiable. We should find some p' G p, Ip'l = p which is realized in N,; so we can replace p by any subset of cardinality p. As of p # K for some B < K, the set of formulas in p which am over N,, ply has cardinality p. So p1 is a subset of @ = the set of m formulas over Np, which E % ~ , @ + ~ . By Claim 1.8(2), (3) (taking p, = S , @ = u , 5 = a) and the assumptions, some p, E ply !pal = p, is a member of S,,, + ,, hence realized in N, + hence in N,.

(3) Let p be a set of m-formulas over N, of cardinality p sucih that every p' E 8Jp) is realized in N,. We should prove p is realized in N. Let p = {p,(Z, ail: i < Cl) and choose 6j EN, which realizes pj = {p,(Z, at): i < j}.

As K > p there is 8, a 5 8 < K such that a,, bj E N,. Using 1.8(2), (3) with {6f: j < tc) = 8, the set of finite sequenoes from N, = u and a = 5 ; some AS" r {Ef: j < p}, 18'1 = p, belongs to 8,. As 93 satisfies "there is a sequence 6 E N which satisfies any formula p(Z, 7i) (a E N6) which is satisfied by 6i for every big enough ~ E S " also SF+, satisfies it, so this 6 necessarily realizes p (in N,). So we have proved (3).

Now we need

CLAIM 1.9 : Let Sc be as in Definition 1.5, N, = min{N, : Nz > N,}, 2 ~ < N,< N t ; a < / 3 ~ ~ * / 3 < K B ; p < ~ 5 K r < N a + x . I f ~ ~ 1 8 t I , S E u E ISc(, lul < K,+,, IS1 = x then there is S' E S , JS'I = p, S' E lScl.

Proof of 1.9. We prove i t by induction on lul and for fixed u by induction on X. Always there is in 93, a one-to-one function g from lul onto u, and we let ue = {g(i): i < 8 for I < lul.

Case I : lul s 24,. Then, of Ha s p < x, for some < Ha lu, n 81 2 p, hence (by Definition 1.6(2)) 8' = ue n 8 E SC, so we get the conclusion.

Case 11: lul > Ha, x a limit cardinal. Then for some ~ ( 1 ) < X, p < ~ ( 1 ) and I u I < Na+xcl,, so choose 8, E 8, l8,l = ~ ( 1 ) and use the induotion hypothesis on u, 8,.

406 OONSTRUCTION OF MODELS [OH. VII, 4 1

Cme 111: IuI > Xu, x is a successor, 1.1 > x, IuI singular. As X u < 1.1 < Xu+x , x regular, necessarily IuI has confinality <x, hence for some 6, Iu, n Sl = x so use the induction hypothesis on u,, u, n S .

Cme IV: 1.1 > Xu, x is a succesaor, 1.1 > x, 1.1 regular. Then for some 6, u, z 8, so use the induction hypothesis-n u,, S.

Cme V: 1.1 > Ha, x is a successor, IuI = x. Let 1.1 = A:, then for some 8, luel = Iue n Sl = Al, so uee the induction hypothesis on uey u, n 8.

So we finish the proof of 1.9.

Continuation of theprmf of 1.7. (4) Let p be an m-type over N, of cardinality x (which is finitely satisfiable). If cf x # K then there are /I < K ,

p* G p , Ip*l = x, p* is over N,; and then we proceed as in the proof of 1.7(2) using Claim 1.9 instead of 1.8. If cf x = K , x # K, x is singular (so a limit cardinal) hence there is x(1) < x such that p < x(1) s A < Na+xcl,, x(1) is regular and # K . So replace p by p* c p , 1p*1 = x( 1) and proceed as before.

( 5 ) (i) Let p be a set of m-formulas over N of cardinality x; such that every q E S; (p ) is realized in N,. Let p = uf pr, pi has cardinality x, the pi's are pairwise disjoint. Let zf E IN,] realize p - pf, and aa K > x for some < K , 6f E lN,l and p is over NB. The rest is like the proof of 1.7(3) Using Claim 1.9 instead of 1.8.

(ii) Now the proof is trivial. (6) The proofs for eaturativity are similar.

DEFINITION 1.6: (1) Let i be an infinite sequence from N, ii = TP(i i ,N) = {tp,(in6,0,N):6E~bI((6afinitesequence)}.

(2) TPA(N) = { T P ( i , N): i is a sequence from N of length A}.

DEFINITION 1.7: For X = S, C

TPX"(a', N) = {E r is a set of formulas p(3; qCl,, . . .) such that for some 6 E IN[, N t p[6; afo,, . . .] for every p E and X = C * Irl s K , X = S == only S K

variables appear in r} TPX;C,(M) = {TPXK(iiy N): 5 a sequence of elements of N of length A}.

Remark, Clearly from TPSi(M) we can compute TPA(N) and vice voi%a. .

OH. VII, 8 11 SKOLEM FUNCTIONS 407

LEMMA 1.10: Let L(M) = L(N) = L, and add new variablm q, i < A, to L.

(1) 2A s ITPA(M)I s min[2aA+'L'y llMllA]. ( 2 ) TPX$(M) is a monotonic function in K , A ; but if K 2 A , [X = C *

K 2 11.13, it is a constant function in K ; and if K 2 ILI:ITPC;(M)I = ITP&(M)I (in fact, one is computable from the other); and ITPSi(M)I = I TPA(M)I (in fact, m e is computable from the other). A180 1 TPQ(M)I I I T P m w I *

(3) 2A s ITPfl(M)l s min[2(A+ILI)x, I I Jf 11"- (4) 2A s ITP8;(M)l s min[2(Ax+a'L' )Y 1 1 ~ 1 1 " . (6 ) If M i s K+-compact, ITPCz(M)I s 2A+ILI, and if N is elementarily

equiwalent to M and is K+-compct, TPCZ(M) = T P a ( N ) . (6) If M is K+--eMOUB (e.g., K+-sdUrded) t h ITP&(M)I S

2A+ILI.

( 7 ) If M is stable in A (i.e., m < o, A E lMl, IA1 s A implim ISm(A, M)1 s A) t h ITP,(M)l s 2A+lLI.

Proof. Immediate.

The next theorem will be used in Ch. VII I in computing the number of models for an unsuperstable theory.

THEOREM 1.11: Let 2" < A; K < A =- K" < A; Man L-model, llMll s h, ILI s p, then there are x I p, a hngmge L, = L u {Q,: i < x s p} (&,--one place predidm) and an L , - d l M, w h tirat

(1) M , b p+-lumLOgeneow,; weover if Lo c L,, &, E Lo, then the

(2) The reduct of M, to L is an elementary extenAm of M . ( 3 ) M , ?ma cardinality A.

L,-reduct O f MI i8 p+-homogeneOU8.

(4) p l l = Uf<X&f(Ml)Y m d i < j =+ QdMd = Q,(Ml).

Proof. W.1.o.g. llMll = A. If A' = A, take x = 1 and extend M to a p+-saturated model M , of cardinality A. So assume A p > A, and let x = cf A s p. Choose an increasing continuous sequence Mf, i s x so that L(Mf) = Lf = L u {&,: j < i}, Mf < (My &,),<, where &, = 1Mj1, and IM*l = 1611, /[Mi 11 < h for i < x. Expand M to an L,-model M, by defining Q,(M,) = lMf I . Let D be a good N,-incomplete ultraflter over p, (exists by VI, 3.1) M, = MY,/D, and let M , be the submodel of M , with universe U,<,&,(M,) . By VI, 2.3 M , is p+-saturated. As Q,(M,) is (the universe of) an Lf-elementary submodel of M,, &,(M3) is (the

408 OONSTRUCmON OF MODELS [m. MI, 0 1

universe of) an LWementary submodel of M,, hence M, is an L,-elementq submodel of Ma hence an L,-elementary extension of Ma. As IQr(Ma)l < h a h IQWdI = IQWdI’/D < &hence 11JfiII = A Now if A c ]Mil, IAI 5 p, p E P ( A ) , thenp is realized in M, iff it is realized in Ma in some Qi(M,) iff it is finitely satisfiable in some Q,(M,) (by the p+-saturativity of Ma); and this depends on the type of A and on p only. Hence Ml is p + -homogeneous.

Remark. If the other oonditions are satisfied but p < ILI 5 h we can get M, satisfying (2), such that for any Lo c L,, lLol 5 p, Q, E Lo the redud of M, to Lo is p + -homogeneous.

(By 1.10(2), we can wume p 5 x) .

Remark. If h 2 2 X + I L I , we oan find suihble K, or hX+ILI = X and the conclusion holds by 1.10(6) aa a (A , A)-saturated model is X+-s&turated by 1.6(3). But we cannot always satisfy “ y - a < x + ” .

Prmf. We use the notation of Definition 1.6. Note that aa 2ILl < K 5 A, M is Ko-saturated by 1.6(3). By 1.10(2), (3) it suEces to prove ]TP.8;(M)l 4 2X+ILI. Suppose not; then there me sequences it from M of length x for i < (2X+lLI)+ such that for i #j TPSfi(i , ,M) # TP.8’(iijS M ) . As ( 2 X + l L I ) + < K , and TPS”(i , ,M) ha9 cmdinality s 2x+IL1 < K, for some C < K, i, is from N , and TPS’(iif, M) = TP.8’(iii, Nc). W.1.o.g. N, = min{NB: Nj 2 A}; so we can assume of Nu 5 p (otherwise N: = N, hence h = N,, hence M is p+-saturated, so our conclusion follows from 1.10(6).), and ( 2 X + l L I ) + < Nu. Let D be a regular ultrafilter over p(1) = p + ILI (exists by VI, 1.3), which belongs to So; and its power is 2’+ILI s 2x+lLI s K s h so it is ~ 8 , . As Sc+, C “ N c haa cardinality A”, in SC+, there is an elementary embedding of Nf(l)/D (as interpreted in 9, + l ) into Nc + , = PC + 1, which is an inverse to the natural embedding of Nc into Nf( l ) /D .

Now define TPS’(S, B, M) as we defined TPSN(i, M), but restricting ourselves to 6 E B, end let TPS;(C, B, N) = {TPSM(’a, B, N ) : ii a

CH. VII, 0 11 SKOLEY BUNOTIONS 409

sequence of length x of elements from C). Let us prove by indudion

(*I on PI

IfC, BE^,, C, B E N, then there is B ' E ~ , , B c B' E N,, 123'1 < IBI+ + N, and TPB;(C, B , N,) haa adin- ality s 2X+ILI. Moreover there is a formula B E L(93), with parameters from 93,+1 such that for suitable B, C, 93, ( x q ( 2 , B, c), and 9, c e[E, B, q.

Proof of (*). For the definition of B use the construction below and a choice function (in 93,) for the family of sets of hereditary cardinality I llNll (we essumed N is in it).

Case I: I BI < K,. B will be the image of Bfi(l)/D (ultrapower computed in 93,) by an elementary embedding f of N$(l)/D (ultrapower in 93,) into N , where B u C c IN*] N , < N , N , E b,, N , is KO-saturated, '29, C " llN*II = A"; where f is an inverse to the natural embedding of N , into N$l)/D. As I BI < K,, all functions from p( 1) into B with range of cardinality s p are included, aa IBI" < N,, 2fi(l) s A. As D waa regular, clearly any m-type over A E C, IAI c p, is realized in B' iff it waa finitely satisfiable in B. It is eaay to check that B' satisfies our demands.

CaseII: 1B1 2 K,, IBIisregular.SoIBI > (2x+ILI)+.In93,thereisa well ordering of B, so let B = {bi: i < IBI}, Bj = {bf: i < j}, and let BF = (B, u Uf<j Bt)'; and B' = Ul<lBl 23: satisfies our demands; otherwise, aa cf I BI > (2X+ I L I ) +, some Bj' fail our demands. (Notioe that )B, u Uf< BTl < I BI, so B; is well defined by the induction hypothesis.) [The primes are from (*).]

Case 111: 1B1 2 Nu, IBI is singular. Define B' aa in Case 11: by wumptions cf IBI x, so let a(i) , i < 8 = cf IBI, be an increaaing unbounded sequence of ordinals < I BI, then B' = Ute B&, and B,*) c B& for i < j. Hence I TP(C, B', N,)I s I[rP(C, B&, Ne)l s (2x+ILI)x = 2x+ILI. So we have proved (*).

= f [ + 1, then, we get a contradiction by B' and the properties of the &'a.

Let us finish the proof of 1.12. Choose B = C = IN,l,

DEFINITION 1.8: H is a bi-set function if H({ul: i < A}) is a family of pairs of subsets of {al: i < A}, and H({ui: i < A}) = {(Al, A2): for some (w l , wa) E H(A), A , = {af: i E q}}. We can use index sets other than A. A set function H is used also aa the bi-set function 8,1?({af: i < A}) = { ( A , 0): A E H({a,: i < A})} .

410 CONSTRUCTION OF MODELS [CH. VII, 5 1

DEFINITION 1.9: M is (A, H,, Ha)-comp&, provided that for m y set F = {vf(z, 4): i < A} of m-formulas over M , if for every (F,, Fa) E

H,(F) some sequence in M realizes (I',, Fa) (i.e., realizes F,, and fails to satisfy any formula in Fa), then some sequence from M realizes some pair from H2(I').

DEFINITION 1.10: For an ordered set I: (A) DC(I) = {Ia: a E I), 1, = ({b E I : b < a}, {b E I : b > a}). (B) The pair (A, B) is a ( A , p)-Dedekind cut of I if A < B (h.,

a E A , b E B =- a < b) and A v B = I, A is the cofinality of A, p is the lower cohality of B (i.o., the cohality of B with inverse order).

DC,(I) = {(A, B): (A, B) is a (A, p)-Dedekind cut of I , (A, p) E W), DC*,(I) = ( (A, B): A , B E I , A < B, and for no c E I A < c )) E W } .

DIPINITION 1.11 : (1) A (A , +family is a family of sets of cardinality A such that the intersection of any two distinct members has cardinality

(2) AD(x, A, p, K) holds if there is a (p, K)-family of x subsets of A; < K.

where x > A 2 p 2 K.

Remark. Baumgartner proved the consistency of AD(2n0, N,, K,, No) -t 2*0 > N,.

EXERCISE 1.1: (1) If AD(x, X,p, K) holds, x' 5 x, A' 2 A, p' 5 p, K' 2 K , and x' > A' 2 p' 2 K' then AD(x', A', p', K') holds.

(2) If AD(x, A, p, K ) holds then A" 2 x; if x = 2A then A" = 2A. (3) Prove that A D ( A + , A, p, p) holds when of A = p. (4) Use (3) to show 1.8 cannot be improved in this case.

CONJECTURE 1.2: Claims 1.8 and 1.9 cannot be improved (at least in ZFC).

EXERCISE 1.3: Find the order between the set functions S,,, S:, S', 8: (dso Werent p's).

QUESTION 1.4: Investigate the logical connection between the various notions of saturativity (implications, and independence).

QUESTION 1.5: Investigate the various kinds of compactness for dense linear order.

OH. VII, 8 21 GENERALIZED EHRBNFEUCFI“-MOSTOWSKI MODELS 411

QUESTION 1.6: Let I = z,,, I,. Investigate the connection between the compactness (and saturativity) (with various notions) of I ; J, I,.

EXERCISE 1.7: For every order J, IJI I; K there is an order I 2 J, IIl = K such that DC((A,fl)l(I) = 0 if A, p + KO, A # p.

QUESTION 1.8: If we use the construction of Definition 1.6 but without assuming that N is saturated, does the big model still bequeath its properties (such as (p, Sh, S,*)-compactness, -saturation, etc.) to its submodela.

EXERCISE 1.9: Show that in 1.6(3), 1.7(1) and 1.7(6) we can replace (6 X-homogeneous” by “strongly X-homogeneous”, if K s x or N is strongly X-saturated.

VII.2. Generalized Ehrenfeucht-Mostowski models

Here we shall deal with the pseudo-elementary class PC(T, , T) where

DEFINITION 2.1: (1) PC(T,, T ) is the class of L-reducts of models of T,, of cardinality 2 IT,!, L = L(T), L, = L(T,), T is aa usual complete, T G T,, T , has infinite models.

(2) T , is a conservative extension of T , if any L,-model of T1 of cardinality 2 I T , I can be expanded to a model of Ta.

Clearly if T , is a conservative extension of T , and I Tal = I T , I , then PC(T2, T ) = PC(T,, T ) ; hence we can replace T , by T , without loss of generality. Also if we want to prove the existence of some models in PC( T,, T ) we can replace T , by any extension of the same cardinality. Hence if T , has an extension of power I T , I which hae some additional properties we can assume T , has them. We list below some such assumptions on T, . Note that if Ti + is a conservative extension of T,, T , = U, <, T,, I Ti I = I T,I, then T , is a conservative extension of To.

Assumption I: T , has Skolem functions. (See Theorem 1.1(2).)

Assumpth 11: T , is complete.

Assumption 111: There is a countable LC, E L, such that for every formula ~(3) and term 7(2) in L,, there are a formula R(3,y) and

412 OONSTRUOTION OF MODEL8 [CH. VII, 8 2

function symbol P(Z, z) &om LE and individual constants c,, c1 in L, such that

(1) (W[Cp(Z) = m, c1)3 E T1, (2) (W)[.(Z) = P(Z, C')] E Ti. For this it suffices to prove:

CLAIM 2.1: 80me coneervative ezteneiOn T , Of TI, = 8&b$eS

Aem?n#on m. Proof. It suffices to define by induction on n theories T", and countable languages L* s L,, = L(Tn) such that To = T,, T"+l is a conservative extension of T", IT"I = lLnl = lTll and for every p(Z), ~ ( 2 ) in L, there are R(Z, y), P(Z, y) in L,*+, and individual constants cl, c1 in

such that (1) and (2) in assumption I11 holds for Tn+l. (Then T , = P, L; = U b* satisfies the claim.) If T" is defined let &*+ , consist of (m + l)-place predicates RE(Z, y) and function symbols P:@, y) for m < w , and

Tn+' = T" U {(W)[q(Z) e ( Z , c:)]: (p(Z) EL,, Z(Z) = m}

u {(vZ)[T(Z) = 4)]: T(Z) E L,, d(5) = m}.

Clea,rly d the demands a m satisfied, so we have proved the olaim.

I denotes an index-model, i.e., a model whose elements serve as indexes; usually I is just an order.

DEFINITION 2.2: If J c I , 8 E I , then the atomic type of 3 over J (in I ) is

atp(Z, J, I ) = {tp(Z; i): Z(Z) = Z@), Z = ( ~ 0 , . . .), i E J , I C ~ [ 3 ; Z] and cp(Z;jj) is an atomic, or negation of atomic, formula in L(I) }

We omit I when its identity is clear. Clearly atp(3, J, I ) = tpd(3, J , I ) where A = the set of quantifier free formulas.

DEFINITION 2.3: If 8, t E I , J c I , 3 - t(mod J) (in I) if atp(3, J) =

atp(t, J ) . In both definitions, if J is empty we omit it.

EXERCIBE 2.1: Check the meanings for I an order.

NoWh. If {&: 8 E I} is an indexed set, 3 E I, 3 = (8(O), ~ ( l ) , . . .) then sj = ti#(o)-6r(l)n....

CH. 5 21 QENERALIZED EHRENFEUCHT-MOSTOWSIU MODELS 413

DEFINITION 2.4: (1) The indexed set {&: 8 E I) (& E 6) is called A-n- indiscernible over A :

(A) If 8 - t then I ( & ) = l (6J. (B) If 3 - t , Z(8) = l ( t ) = n, ~(3; g) ~ d , a E A then Q C &; a] =

Q[k a]. (2) If 8 # t implies 6‘ # & we d our indexed set non-trivial (we

use (2) rarely-mainly in estimations of cardinalities). (3) We adopt the same conventions for shortening aa in Definitions

I, 2.3 and I, 2.4. (Note that this definition and Dehitions I, 2.3 and I, 2.4 am Consistent.)

Assumption IV: When I is an index-model we wume for every 3 E I there is a quantifier-free formula d(Z) in L(I) such that for 8 E I, B - 8 iff I b 0[8].

DEFINITION 2.5: K ( I ) is the c l w of L(I)-index models I, such that for any I E I, there is 8 E I for which atp(3,0, I , ) = atp(t, 0, I). So for every I there is I, E K(I ) , 1111 s IL(I)I, such that &I,) = K ( I ) (without assumption IV, we would have only /Ill s 21uOl). The c l w of orders is Kord = K((w, c)) = K(w).

LEMMA 2.2 : If I , E K ( I ) , {a, : s ~ l ) is an indiscernible indexed set in a ntodel M of T, , then there is an indhcernible indexed 8et {a8: 8 E I,} in a d M‘ of T, , ~h tirat if I E I , 8 E I,, atp(8,0, I) = atp(t, 0, I , ) t h tp(&, 0, bl) = tp(Zj, 0, MI). A8 T , h Bk&m f u n d h a we mn a88um lM’1 = c1{a8: B E I , } and then M’ is uniquely determined (wp to ieo- w p h h pmerving the 4).

Proof. Immediate, by the compactness theorem and the dehition of W).

DEFINITION 2.6: If {68: 8 E I) is non-trivial and indiscernible in M, then the # of {g8: 8 E I) which we define as {&): M k &I} and the index- model I, E K ( I ) determine M’ so we denote M’ by N1 = EM1(Il, #), and its reduct to L by N = EM(I, , a). Such # is called proper for (I, T,). We call {a8: 8 ~ 1 , ) the skeleton of N1 (and N) (of course, really we cannot reconstruct the skeleton from the model, but we behave a8 if we can).

LEMMA 2.3: T h e is #proper for (w, T,) .

414 OONSTRUOTION OF MODEL8 [=. 8 2

Pruof. It suffices to prove in some model of T, there is a non-trivial indiscernible sequence {a,: i < w}. Let M, be a model of T,, & distinct sequenoee &om M,. It suffioee to prove

I' = Ti u {v(%) ~ ( e ) : v(E) E L,, I N 8; I , f E W }

U {zs # Zt:8 # t < w }

is consistent, for this it suffices to prove the consistency of a finite

where I(Z) - f(Z). Let A = {pl: Z < n}, m = maxk<,, Z(b(k)), a d let {& k < w } be a A- S m-indiscernible sequence (exists by I, 2.4 and I, 2.3(3)) and interpret Ek by 6Nk); 80 in M,, r' is satisfied.

Asmmpth V: If T is unstable, for some formula 3 < g in L and some 6f, i < w , (I(!$ = I ( @ = Z(q)),6" < 6k iff n < k. Then (Z < ~ ) E L P . If T has the f.0.p. then T n L; has the f.0.p.

subset I" c T1 u {vl(Zal)) ql(Zi(1)): Z < n} u {iE, # 3: I # t < w}

LEMMA 2.4: If T W unstable, thre W 9 proper for (0, T,) whkh W ordered by < , i.e., (3,, < ?Em) E Q iff n < m.

Pruof. AS of 2.3, only to I' we tldd {(Z,, < Em)u(n<m): 7~ < m < w}, and use the &,'a from assumption V.

Similarly we can prove

LEMMA 2.6: If 9, i8 proper for (w, T,) and T , is complete, T, 2 T,, t h there W 91 2 whkh is proper for (w, T,).

For Q, proper for (w, T,), we can investigate properties of EN(1, Q,) (I E R(w)) , but for casdinals s I T,I we cannot get much. Hence we first introduce a generalization.

DE-ON 2.7: If 9 is proper for (I, T,) let N1 = EM1(I , @), N' be the submodel of N1 with universe the set of interpretations of terms (with no free variables). (As T , has Skolem functions, N' is an element- rtrg submodel of N1.) Let N* be N' expanded by adding the relations:

R ~ , ~ , ~ = {a E N': ~1 c 'p[~(q, a)], when I E I, I c em} for every atomic formula 'p(Z) E Lf, term c(3, 9) E LE(l(Z) = l ( ~ ) ) and quantifier free formula e(3) E L(I) satisfying assumption I V (we cthoose one e(Z) for each atp(B, 0, I), B E I). Let T* = Th(N,), = L(N,),

= Lz u {RB,o,s: 8, 'p, c as mentioned above}, so IQI s No + 1L(I)l and T* satisfies assumption 111 when L(I) is countable.

OH. WI, f 21 GENERALIZED EHRENFEUCE’FMOSTOWSKI MODELS 415

We behave as if T, determines 9, and say, T,, or a model of T, has a property, if 9 has it.

LEMMA 2.6: 8wppoee N i s a maEel of T, (heme an elementary extenaim of N,) where 9 is proper for (Io, T,), and I E K(Io).

Then there is a unique L,-modeZ W , w1 = cl(m U {a#: t?~I) ) m h that

( 1 ) F o r e v e y R , , , , ; ~ L , - L , , ~ ~ ~ E E I N I , I E I

(*I when I C ep], N C R,,q,t[E] o M1 C p[z(h, a)].

We denote this d by EM1(I, I?), and it8 L-red& by M = Ehf(I , N ) . We call {a8: s E I } the skeleton and N the ba& of M1 (and of M ) .

( 2 ) M’ is a &el of T,, and an elementary extension of the L,-reduct of N . The indaed set {Z8: 8 E I } is indi8cernible over IN1 in AP.

(3) For every E E IN1 I there are I E I I I, 6 E IN I and z E Lf (we k d e r them aa determined by E ) m h that i5 = T(%, 6). Also (*) holds f o r any T ,

p E L, for suitable f m d m (wring mitable individual mtant8). The elementay type of Z is determined by atp(I, 0, I ) , tp(6,0, N). Also

(4) If in I only finitely many atomic types of sequences of length m are realized f o r each m: then for every fornula V(T(%, g)) ( p , z E L,, I varied over I , $i over N ) there is a Boolean cornbination #(S, g) of formdm O,(a) E L(I), e,(g) E L, m h that M1 C p(z(q, E ) ) ifs #(& a). [Being formalistic # is not a well defined formula, but its meaning is clear.]

lP1ll = IlNll + 11I11*

Proof. There is no problem in the proof.

Remark. If M , satisfies (1) it automatically satisfies (2), (3) -and (4).

DEFINITION 2.8: I is A-atomically-stable if J E I , IJI 5 A implies 18T(J, I ) ! 5 A where A = &(L(I)) is the set of quantifier free formulas of L(I). (Note that if IlIll 5 A, I is h-atomically stable.)

LEMMA 2.7: ( 1 ) If N is proper f o r ( I , T,), N stable in A, and I h- atomically-stable then M1 = EM1(I, N ) is stable in A.

(2 ) If N is proper for (0, T,), IlNll 5 A and I is a well ordering then EM1(I, N ) is stable in A.

416 tYONSTRUCTION OF MODELS [CH. VII, 0 2

Proof. (1) Let A E IWl, IAI 5 A, then there me J E I , IJI 5 A and B E N, IBI 5 A, such that for every a E A there are %(a)EJ, ~ , E B and 7, E Lt such that a = 7,(7ii(,), 6,). Noticing that if at = ~(79,,,, h1) %(l) E I , 6I E IN1 for Z = 1, 2, and %(1) - S(2) modJ, tp(6,, By N) = tp(Eay By N), then tp(@, A, iK1) = tp(aa, A, Ml); the conclusion is clew.

(2) Immediate from (1) and part ( 5 ) of the following claim.

CLAIM 2.8: Let I E K(w), J s I .

and t > 81 0 t > 81.

(1) 81 - 81 mod J if for any t EJ, t < 81 o t < 81, t = 81 o t = 81

(2) (80, . . ., 8,) - (to, . . . , tm) modJ ifst - ti mod J for i = 0, . . ., m

(3) The number of atom&? typea of 8eqwnces of length m from I i8$nite andat < 8 , 0 t t < t, for 0 5 i , j 5 m.

(even <(2m)!).

where d = &(L(I)). (4) I i8 h-&?&dy 8tabk if 1&(J,I)I 5 IJI f or J E I , IJI 5 A,

(6 ) If I k well ordered, I k A4omictally 8tabk for euch A 2 No. (6) If A 2 2b, I k A-Ut~WnMy-8tabk iff I k A-etabk. (7) If I = zae, In; J, I , are A - a t ~ W l y - s t a b k Orders th I i8 A-

atoncieauy ekrbk.

Proof. (l), ( 2 ) and (3) are immediate. (6) will not be used hence we leave it as an exeroiae fo the reader. (Hint: use the Feferman-Vaught theorem, see, e.g., [CK 731.)

(4) Suppose I , J are a counterexample. So IJI = A, J E I , p i = atp(%{, J , I ) l (SJ = my for i < A + , pi # p ,

for i # j. As we can replace {p,: i < A+} by any subfamily of same ccudinelity we can aasume w.1.o.g.

(A) atp(Siy 0 , I ) is constant. (B) Letting Si = (a!, . . . ,Y-l>, then for each k, 0 I k < m,

atp(6, J, I ) dependa only on k (for this use m times the aasumption I(atp(8, J, I ) : 8 E r)( 5 A).

So we get a contradiction by (2). (6) Using (4) we prove lRj(J, I)I 5 A when I JI 5 A. Define a function h on I : h(8) = min{t: t E J , 8 < t } and if for no

t E J t > 8, let h(8) = a0 (h is well defined aa I is well ordered). So the range of h has cardinality I JI + 1, and if 8, t 4 J, h(8) 3: h(t) then by (1) atp(8, J, I ) = atp(t, J, I ) . Hence we finish.

(7) by (4) it is immediate.

CIE. m, 5 21 O E N E U E D E~ENFEUCET-MOSTOWSKI MOD- 417

CONCLUSION 2.9: (1) If A 2 ITl] then there ie M E P C ( T ~ , T ) of cardinality A, which W 8tubk in evey p 2 IT1]; 80 i f T W unatabk in lT1l, Af W not IT,I+-univereal.

(2) I f p 2 2”, h 2 lTll then there is Af E PC(T1, T ) of cardinality p, whhh ie A + - ~ n i ~ s a l , x-etabk for x 2 2”, h when T W tlnekrble Af is ?lot (2”) + -universal.

Remark. In ( 2 ) if 2” = A+ the mult on universality is sharp, but in general not. If pA = p we can assume M is A+-satmted x-stable for x = XA (using ultrapowers). Sharp results, using finer methods, appear in Section 4.

Pvoof. (1) Let N be proper for (w, Tl ) , and be of mrdinality I TII. Then EM@, N ) is of oardinality A by 2.6(3); it is stable in p 2 lTll = llNll by 2.7(2).

If T is unstable in I T , I, there is a model M of T which is unstable in I T l I, of mrdinality I Tl I + , hence Af cannot be elementarily embedded in EM@, N ) .

(2) Use a A+ -saturated model N of cardinality 2” whiah is proper for ( w , TI). Take EM(p,N) as the required model and use Theorem 11, 2.13(2).

LEMMA 2.10: If M1 = EM1(I , N ) , N p o p ~ f ~ ( I , T l ) ; t h ITPX;(Afl)l s ITPX;(N)I + lATPXf(I)l where X = 8, C and where ATP metam3 that in &$nition 1.6 we replace t p by atp.

Proof. Similar to that of 2.7.

LEMMA 2.11: S-e I is a h e and h+-saturated or& and N W h+-cumpact a n d p ~ for (0, Tl ) . Then Afl = EAfl(I, N) W (A, SN,, S1)- compact.

Proof. Let p be an m-type over M1 of cardinality A, and every q E

SNl(p) is realized. W.1.o.g. h 2 lLll.

(*) There is F = ~ ( g , Z) E Lt such that any q ~ & ( p ) is realized by some F(&, 6), 3 E I , 6 E N .

Otherwise for each F EL: there is a counterexample qo so q = Us qr is a countable subtype of p , and is not realized, contradiction. So for every q E SNl(p) there are S(q) E I , 6(q) E N such that T(%(~, , b(q)) realizes q. Let D be an ultrafilter over Sh(p), such that { q ~ S ~ ( p ) : ~ E ~ } E D for any v ~ p . Let J E I, IJI s A, B E INI, IBI s h be

418 00NS"RUUTION OF MODELS [OH. WI, 8 2

such that p is a type over dcl(B u {a8: 8 E J}). We can find, by the A+-sa.turation of I and N , f E I and E E IN1 such that:

(1) For every 'p(iZ, E), 5 E B, 'p E L ( N ) ,

N C v[E, 51 * {q ~ S b ( p ) : N b &q), 51) E D-

(2) For every # E L(I ) , 3 E J ,

I #[t, 31 0 {q E SNo(P): I f= #P(q), 311 E D.

It is easy to check ~(s, E) realizes p.

QUESTION 2.2: Investigate the connection between (A , H,, Ha)- compactness (or saturativity) of N , I and of EM(I , N ) ; in particular when the H a am S,,, S:, Su and S$.

QUESTION 2.3: (A) Investigate the connection between non-(A, H,, Ha)-compactnees (or saturativity) of N , I and of EM(I , N ) when N C T,, @ is ordered (see Lemma 2.4).

(B) Show, for unstable T , the existence of models which satisfy simultaneously various compactness and non-compactness conditions (also for mturativity; we mean of course for the various (A, H,, Ha)- compactness notions).

EXERCISE 2.4: Suppose A > p > lTll am regular, bl = EM(I , @) I a dense order, @ proper for (w, T,) . Show N is ( A + p*, DC, DCo,,,)- compact, when I has no (i) (A, p) or (p, A) Dedekind cut, (ii) (1, A) or (A , 1) or (0, A) or (Ay 0 ) and (1, p ) or (py 1) or (0, p) or (p, 0 ) Dedekind cut.

EXERCISE 2.5: Show that if T1 i < 8 is an increasing sequence of complete theories, @: proper for (I, TI) 4 5 @, for i < j < 8, then Ul<d @: is proper for ( I , U l < d T1).

DEBTNITION 2.9: I hw the extension property i f When T, is complete, Q1 proper for ( I , T,) , Ta 2 T I , then there is @a z proper for ( I , TI) .

EXERCISE 2.6: Show that if is proper for (I, T,), I has the extension property, then there are T , 2 T,, I TaI = I T,Iy d& z a, proper for I, T,, such thrtt for any J E K ( I ) , EM(J, 0,) is KO-homogeneous.

OH. VII, 8 31 ON THE f.c.p., AND UNIFORM TF~EES 419

EXERCI8E 2.7: Suppose T1 = Th(M,), M , = (w, < , R,, . . . ), T, has Skolem functions, and that we get @, proper for (w, T,) &B in 2.3 for b, = n. Then for any I E K(w), N = EM(I , @), 8 E I

I{t €1: t < 8}1 5 l{b E IN]: N b b < aa}l

5 I{t E I : t < 8}1 + min {2No, 1T11}.

Moreover for each K we can find such a T,, lTll = K , such that the second inequality becomes an equality. (Hint: See the proof of 3.1.)

EXERCI8E 2.8: Let w denote here a finite set, and say M is (A, H,, H,)*-compctCt if for any indexed set {y,,,: w c A} of formulas over jK if for every 8 E H,(A), ps = {yW: w E 8) is realized in M then for some 8 E Ha(A), ps is realized.

(1) If H,(A) is closed under countable unions, N is A+-compact, I is (A, If1, El,)*-compact (or vice versa) and N is proper for (0, T,), then EM(I , N ) is ( A , a,, Ha)*-compact.

(2) Generalize from I E K ( o ) to I E K(Io). (3) Generalize to saturativity and Definition 1.8. (4) Check the implications among these notions.

VII.8. On the f.o.p., uniform trees, and ID(T)I > IT1 = KO

We continue the conventions of Section 2, with various additional assumptions on T.

THEOREM 3.1: 8wpp8e T 7uza the f.c.p. (finite cover property). If @, i8

proper for (w, T,) then there are T , 2 T,, lTal = ITl] and @a and y(z; g) E L, O(g; Z) E La m h t7&:

(1) @2 2 0, is proper for (w, Ta). (2) If N 1 = EM1(I , @a), 8 E I , then O(N1; 4) 7uza cardinality

I{t E I : t < 8}1 + min{lTll, 2&}.

(3) {rp(z, a): a E N1, N 1 C O[E, an]} is not realized in N 1 but any proper &set of it is realized in N 1 .

Proof. Let M' = EM1(w, 0,). By assumption for some y(z, 8) E L and infinite set W E w for any n E W there are (i < n) such that

M' C A (W[ A d z , 5,,.,)] A -(W[ A t<n d z , 5,J] - f e n j-8

jrrt

420 O O N S T B U ~ O N OB MODELS [OH. VII, 9 3

We expand M1 by: P = {ti,,: n E v, sequence of functions F, P(7i,,~i0 = 6n,1andQ = {6n,,:i < n , n ~ W}andR = {6n,1n~n:i < n, n E W), and by adding Skolem and other funations so that we get a model M" such that T a = Th(Ma) satisfies all the assumptions from Seation 2. Now for any finite A E La = L ( T 7 , by Ramsey'e theorem (eee I, 2.4) we can find an infinite J A c {Zi,: n E v which is A-indiscernible. So we culn find a maximal consistent set of formulas 0, in La in the variables itn, n < o, such that for any formula $(Zo, . . . $4) E

La if for some finite A, E La for all finite A, A, E A c La, and all Z,,(o,, . . . , E Jb(n(0) < n(1) < . .), Ma C $[aM0), . . .] then $(%, . . .) E 0,. So clearly 0, 2 0,, and is proper for (o, T,).

Let B(E; g ) = R(g; E) , then (3) is easy to cheok so only ( 2 ) remains. suppose 8 ie a counterexample. Using P we see 1 e(lN1 1 ; a,)! r I{F(q, q): t < 811 = I{t: t < 811.

By d e w g Ma properly we am ensure that 1 B(lN1 I ; G,) 1 r m i 4 1 T 1 I,%). So we prove one inequality. Now suppose I B(N1 ; 4) 1 > A,, = I{t : t < 811 + rnin{l T I 1 , 2Ho). So there are distinct ri(b(,)) E B(IN1l; 4) for i < &+. As we can replace them by any subset of &ality A,,+, we can assume all the J ( i ) are similar over {t: t < 8).

Also, if A, r ITII we can assume 7, = r for any i. If A, < ITII n e d y A,, r %. Define an equivalence relation among the 7,: 7, N

r j if for any f E W, rl(&) = ~ ~ ( i 3 ~ ) , or rl(i&), r j (B f ) Q in the model Ma. Clearly the number of equivalence classes is s 2n0, so we can assume all the riys am equivalent. As q(%(,,! E Q(N1). q(7i8(,,) = r0(8,,,); we can assume a ( i ) = tn(8)"€(i), f[l] c 8, t ( i ) [ j ] > 8. Now if R@, rO(Z, 8, Z l ) ] + [rO(it, g, z l ) = r0(itY g , ~ , ) ] E A and E, m, By i E {i: a, E JA), (I(@ = I ( € ) , z ( E ) = ~ ( i ) = Z(t(i)), and ~ [ i ] < m < E l i ] , iu] then necemady

(otherwise n 1 I{Z: R(amy a))] r l{rO(a;, am, aE): suitable B}I r No). So all the rO(ZZHi)) are equal, contradiction.

THEOREM 3.2: 8 u ~ e T is stable, M1 = EM1(I, N ) , N proper for T I ) .

( 1 ) Let v(E, 8) E L, E, T E El. If tl, f a , J1, ga are i n ~ r e ~ n g sequences from I , = l ( 6 ) = m,,l(s,) = l ( 4 ) =.m2 and g1[i] = g 2 [ j ] o t ; [ i ] = fa[ j ] , ccnd 6, E N then M1 t T[T(%,, a), 6(qa, 611 = v[.r(aZl, a) , d ( ~ i ~ , , 6)]

(Remark: I7cstead of if,, aa, we could have taken if1, . . . , &.)

CH. VII, 8 31 ON TILE f.C.p., AND UNIFORM TREES 42 1

(2) #uppose A > 11 N 11 is regular, and every interval of I hm cardinality # A (including the intervds with an endpoint &a). Then M , the L-reduct of M', is ( A , # ~ , # ' ) - c r n n ~ ; and moreover (A,#A,#l)-mpact. (The aame h0ld.a for saturation.)

(3) In (2), instead of A regular, it aumes to msume I hQs ru) interval of cardinality cf A; or for some A, < A, any interval of cardinality 2 A, has cardinality > A. We can replace " A > 11 NII " by " N h+-compact ".

Prmf. (1) Here we can wume I is dense. Suppose we have a counterexample. Then, by assumptions we can find go, . . . , Ik such that:

(i) So = and glniik - 81-82. (ii) a2[i] = 8,[j] -Qi] = <[j] (0 < 1 < k, 0 < i < m,, 0 < j < m2). (iii) For each I there am i (Z) , j (I) such that for n # i(1) S"[n] = g'+'[n]

and gl[j(Z)] < a'[i(l)] if€ not gl[j(l)] < i?+'[i(I)], but for j # j ( l ) , S,[j] < d[i(I)] iff gl[j] < #+'[i(l)].

BY (i)

M' c 'p[t(%,, a), 6(ha, 6)] 0 M' c 'p[t(%,, a), 6(Q, i)],

M' c 'p[t(a~,, a), 6@&, 6)] 0 M' c 'p[t(%,, a), 6(%&, 6)] .

But as we are dealing with a counter example

M' c 'p(C(%,, a), qqa, 6)) 0 M' c 7'p(e(afl, a), 6(afp, 6)).

M' c 'p(t(%,, a), 6(%, 6)) = l'p(?(%,, a), 6(7ij+l, 6)).

So for some 1 s k

Let U, E I, U, < U,+', and U, is between #[i(l)] and $+l[i(Z)]. Let v:, vt E I for n < o be defined such that: they are of length l(gl), I(&J if j # j ( l ) , V:[j] = S,[j], if j # i(Z), v 3 j ] = 8 [ j ] and "[j(4,] = U,, vi[i(Z)] = U,. It is eaay to check that for n # k M1 I= 'p[d;, a;] = +a:, a:] where a: = ~(9"~ a), at = 6(az, 6). Hence for # = 'p or # = 7'p M1 C #[a:, a:] A d: # d; iff n < k, ao T has the order property contradiction by 11, 2.2 and 11, 2.13.

(2) Suppose every q E &(r), I' = {'pf(3, &): i < A} is realized in M1. So let ~ E W I , M ' + r p t [ ~ , b , ] for i <j. So let ci = ~ ~ ( a ~ ( , $ ) ($EM, Tt E LE). Of course, each ct is increasing. As A > llNll by renaming we can assume that for i < A cf = c,

8 = 8, (as we can replace Ef by a, when j > i). In the same way, we can assume that for each I, either all Zif[Z], i < A are equal, or for every i, j S,[l] = q z ] =$ i = j.

422 CONSTRUCTION OF MODELS [CH. WI, 5 3

Let Z(0) < - - - < Z(k) be those for which the firat w e occurs, and again we can wume that for n, m # Z(O), . . . , Z(k) i,[n] = 8,[m] iff i = j , n = m. So Z(n) + 1 < Z(n + 1) implies {e E I: eo[Z(n)] < s < 8o[Z(n + 11) hae cardinality T A hence > A (for - 1 s n s k + 1, where we stipulate I ( - 1) = - 1, Z(k + 1) = the length of a,,). Hence we can define i E I, @) = Z(io), %[Z(n)] = i#(n)], and if n # Z(O), . . . , Z(k), then a[n] does not appear in ti, where b, = ah, 6,). Then by part (1) ?(ai, a) realizes p.

(3) Proof similar to (2).

CONCLUSION 3.3: 8 p e T i8 S a k , with the f.0.p. A 2 lZ'11, and 8 a set of c a r d i d 8 A E 8. T k n there is M E PC( T, , T) such that:

p 4 A, ~ € 8 , then for s m v(z,a) E L , bl omits a T-type p of cardinality p, m h that any proper &type of p b realized in M .

(2) If m i n { * o , lT1t} s p < h,p$8,pier@arorp = min{*o, lTll}, then M ie ( A , 81) -compt .

(1) If min{lT1l, 2no}

Proof. When lTll < 2Mo this is immediate by 3.1 and 3.2, aa there is an order I, IIl = A such that for every p, lTll 4 p 4 A: p E 8 * for some s E I I{t E I: t < 8)) = p, and p $8 == I haa no interval of cardinality p. When lTll 2 2% we can use Exercise VI, 6.4, 6, Exercise 2.9 and take an elementary submodel, or 3.2(3), laet phrase.

CONCLUSION 3.4: 8uppo8e T i s sa le with the f.c.p. If Nu 2 N, + lT1l, t48 = min{lT1l, P o } , then there are in PC(T,, T)

at Zeaat 2Ia-B1 n o n - i e ~ p h i c models of cardinality Nu.

Proof. If a - 8 2 w, this is immediate by 3.3. When a - 8 < w we look at &: for some finite A c L, n < w, there is a finite A-n-indiscernible set I c N such that dim(1, A, n, p ) = p}. For suitable T, this is not difficult, and we leave it to the reader. (Compare with VI, 3.9.)

DEFINITION 3.1: For an order J, I is called a (J, /3)-tree if I is a model with universe G JsB = (7: 7 a sequence of elements of J of length sp}, such that 7 E 111 =- q a E 111; and with the relations Pa = (7 E 111: Z(7) = a} for a 4 8, the lexicographic order < , and order 4 of being an initial eegment, and the function h, h(7, v) = the longest common initial segment of 7, v. I is the full (J, /3)-tree when 111 = JsB. We denote I by JsB. Note that up to isomorphism the claas of (J, /3)-trees for a11 J's is K(w'8). If for some J, I is a (J, /3)-tree, then I is a 8-tree.

OH. VII, f 31 ON THE f.c.p., AND UNIFORM TREES 423

Remark. If I , f E I ; I ,., f iff Z(I[i]) = l(&i]) and for any i , j ; ai,j =def

max{a: B[i] 1 a = B[j] 1 a} = max{a: t[i] 1 a = f [ j ] a}. (In another notation, l ( h ( g [ i ] , ~[ j ] ) ) = Z(h(E[[i], E[j])) a n d ~ [ i l < g[j]e-E[[i] < E[j]).

DEFINITION 3.2: (A) T has a uniform ,%tree of the form ((cpa,ma): a < 8, a successor) (cpa E L, ma s w ) if there are a,, for q E ASS such that :

(1) a,, realizes p,, = { c p a ( f , anla): a < 8, a successor} for q E B A .

(2) For any a, r ] E Aa,v E AB,7iv realizes <ma + 1 of the formulas ~ ~ + ~ ( f , Sin-<r)), i < A. (We can aasume that if 6 < /3 is not a successor q E Ad, then a,, is empty.)

(B) The tree (and the form) are called strong if for any set w s A,

LEMMA 3.6: (1) Definition 3.2 holdejor s m e A iff for every A. ( 2 ) If 8 < K ( T ) then T h a uniform ,%tree, ma = 1 for all a. (3) If T is stable, /3 < K ( T) then T h a strong uniform 8-tree, ma = 1,

(4) If T hua the strict order property then T h a strong uniform 8-tree,

( 5 ) If D ( s = 2, L, GO) = GO then T h a strong uniform Ho-tree,

(6) If K < KrOdt(T), K is regular, then T h a strong uniform K-tree,

for all a.

with ma = 1 for all a.

ma = 1, for all a.

ma = 1.

Proof. Immediate, see 111, 3.1 for (3), 11, 3.9 for (6) and 111, 7.10 for (6). Clearly, (1) and (4) are immediate.

For (2), note that if on J = As* x {0,1} we define an ordering < : (7 , i ) < <v, j> i fq 2 v, i = 0 or q = v, i = 0, j = I , 01 p Q q , v where p = i ( q , v ) # q, v and q < v (in the lexicographic order). Then assuming T is unstable (as (3) has been proved) by the order property, there are cp E L, a, (8 E J) such that tcp[Z8, at] iff 8 < t. Then let

a = ~ < , o > - ~ < , * l >

cp(J ln~ , , a,) = [cp(fl, B<,,,OJ = -lcp(fl, ~ < W , I > ) l .

and

THEOREM 3.6: (1) If T h a uniform @-tree of the form ((cp,, ma): a < 8, a a 8ucce88or), then it h such a tree which i s indisernible ( w h

424 CONSTRUCTION or MODELS [OH. VII, 8 3

we ukw the 8-tree acr a n index set). In fact, JsB h a the extendm property (see Definition 2.9).

(2) B y the assumption of ( I ) , there is a @ proper for 8-trees and T I , euch that V M = EM(hs4, @), then {a,,: q E hsB} is a uniiform 8-tree of the form ((v,, ma): a < 8: a memasor).

(3) I f we dart in ( 1 ) and (2 ) with a strong tree, we get one. (4) BY rep hi^ cp , ( f ; g ) by & ( f ; 81, gal = v a ( f ; a11 A i , ( f ; Y a ) ,

we can wmme in (2) t7& if M = E M ( I , @), q E I n ha, no seq- in M satis+ infinitely many formulae rpa+ a,,-<,>), qh(i ) E I . That is, we get a strong uniform tree of the form ((cp,, No): a < 8, a a memasor).

Proof. (1 ) Suppose {Si,: q E hsB} is a uniform 8-tree; and choose h 2 sa+,, a, r IT(. By adding dummy variables to the qaYs we can assume the a,'s ctre distinct.

X t sufXces to prove the consistency of

u {7va(gn, g v ) : a = y + 1,l(v) = a, l(q) = 8, v r Y = q r Y Y V ( Y ) Z r](Y)l

u {!I,, # !I" : r ) st v7 l(!I,,) = l(!J")). So it suffices to prove the consistency of all finite I" s r. In I" the set of a which me l (q) where appeam in I" is finite. So by renaming we can aasume all such a's me <no. (We rename in a corresponding way the anye.) By the combinatorial theorem 2.6 of the Appendix we can fhd an aeeignment showing the consistency of I".

(2) , ( 3 ) Follow easily. For (4) notice that for the @ from 2, for no 6 E &I 80 S I{i: cpa(6, a,,-<l>) 7d6; a , , - < ~ + ~ > ) } l .

THEOREM 3.7: 8 w p p ~ e IT1 = KO and let v,,(P) E L ( T ) for q E 2<@ be euch thud

(a) q 6 v =- w ' a v v ( f ) + v,(f)l, (8) W 3 ~ ) [ ( ~ n - < 0 > ( 5 3 A ~ , , - ( f ) I ,

(7) b(3x)vn(f). ThenTha;sanzodelMand7i,~Mforq~2",andafunctionh:2<"+

2<" such that: ( 1 ) q 4 p impl iu h(q) 6 h(p), and h(qh(0)), h(qn<l)) are 6-

incomprable, so for q E 2" we can define h(q) E 2" w the unique v E 2", euchthuth(q 1 n ) Q v f o r n < w.

OH. m, 5 31 ON THE f.C.p., AND UNIPORlK TREES 425

(2) a,, realizes p: = pi(,,), p t = {‘p,+,(i~): n < 0).

(3) Call i j , i E 2’ 8imiZar over my i j - 6(mod m) if (i) ij[ll r m z ij[kI t .zfor k # 4 (ii) $1 m = $3 1 m for any 1. Then for p(0) E L there i 8 m, euch t7td if i j , i are 8imilar over m 2 m,,

(4) If T E L, 7(aq) realizes p: then v = q[Z] for some 1. Moreover there ie nz, such tW if i j - +j(mod m), m > mi and v E 2m,

(6) If r, are j h i t e set8 of formulas in L, r,, E r,,,,, UneP, r, i8 the

then M C cp[a;] = cp@~].

v # go] r m, . . ., then MI= -tpncV,[~(aq)].

set of formdm of L, then in (3), if ‘p ( j j ) E r,, we can chooee m, = n.

Proof. Choose a model M , of T , and Z,, E lMol , M , C cp-,[Z,,] for r) E 2. We use 2.4 ofthe Appendix for the reasonable fm’s (for each ‘p(Zl, . . . , Z,) there is m m such that f ( i j ) = 0 if€ f ( i j ) # 1 iff M C ‘p[s]). Now letting M = Mb/D, a,, = (. . . , E,,i~,,,nly . . . ) /D , D be any non-principal ultrafilter over w, clearly (l), (2) and (3) are satisfied. Now it is trivial to get (6) by renaming, and (4) is quite easy.

EXERCI8E 3.1: We can assume in 3.7, that there are distinct b, E M such that

(i) {bn: n < w} is indiscernible over U {a,,; r) E ‘2}, (ii) (My b,, b,, . . .) satisfies (1)-(6) of 3.7. [Hint: Let N < M y b, E IMI, {b,: n c w} indiscernible over N. Apply

3.7 to T’ = Th(M, P, boy . . .) (for P = N) and cp,,(z) such that CP[a,,] (more exactly, we have to repeat the proof).]

EXERCISE 3.2: Show that in 3.2(2), we can replace “llNll < A” by “ N is (A , SAY S1)-oompact.”

EXERCIBE 3.3: Suppose #,, E L(T), T countable, p an m-type over 0, N a, model of T , omitting p , and {{I,$,(??): N C #,[6]}: 6 E INI} has cardinality 2Mo and 2Ho is real valued measurable (see [So 711).

Show that T has a model M omittingp, and there me ‘p,, E L(q E @’2) which are Boolean coqbinations of the #,’a, and a,, E 1M1 aa in 3.7. [Hint: Let D be a normal measure over 2n0, and 5, E INI, (i < 2Wo) such that p , = {#,(if): N C1,4,,[5~]} are distinct; and w.1.o.g. T has Skolem functions. Now we define inductively Ela E D (a < wl) such that S6 = nf<a8f, S,,, c 8,, 8, = 2n0, and: if i (O) , . , ., i(n - 1) are distinct, and ‘p E L, N C p[b,(,,, . . . , 6,(,- ,)I, then for some j E Su - Su+

426 CONS!l'RUCTION OF MODEL3 [m. VII, 8 4

N C cp[g,, &l), . . . , &,-l)]. NOW we shall define indu~tively tp,,(. . . , z,,, . . . such that

(i) for every a for some i(9) E Is,, N C t p ~ . . . , & ,,), . . .I; (ii) If v(7) E {f(O), ~"(1)) for 7 E 2', then

tpn+i( - . ., zD, - - . )pea-+l I- tpJ. - - 9 ~ v ( n ) , - - )"ep;

(iii) When 7 4 v(r)) E 2" for r) E P, tp,,(. . . ,z", . . . ) is an approxima- tion to the desired properties of ~n7iv(,,)n--- (i.e., we allow only finitely many terms, and note here we should decide why a term does not realize p). For further help SIX [Sh 761.1

M . 4 . Semi-definabilify

DEFINITION 4.1: (1) If D is an ultrafilter over I (I E 161") then

Av(D, A) = {tp(Z; a): si E A , (6 E I: ktp[6; a]} E 0)

(Clearly it E Ism(A), it is consistent as D is closed under intersection). (2) An m-type p is semi-definable over I if there is an ultrafilter D

over I such that p c Av(D, Dom p). (3) An m-type p is semi-definable over A if it is semi-definable over

Am. (4) tp*(C, A) is semi-definable over B c A, if for any E E C, tp(E, A)

is semi-definable over B E A. (This naturally extends to not-necessarily complete t y p e s with infinitely many variables.)

( 6 ) p E Ism(A) (B c A) is stationary over B, if p is semi-definable over B, and it has no two contradictory extensions which are semi- definable over B. (This is not necessarily consistent with the previous definition of stationary Definition 111, 1.7, but our meaning will always be clear.)

Remark. Notice that if T is stable I an indiscernible set, D a non- principal ultrafilter over I then Av(D, A) = Av(1, A).

LEMMA 4.1: (1) A type p ( p s i b l y with injnitely many variables) ia semi-dejnable over I(A) iff everyjnite subtype of p is realized in I(A).

( 2 ) Every type over a model M , i8 semi-dejnable over ]MI. ( 3 ) If p is a type over A , and is semi-dejnuble over B, then there is a

complete type over A extending p which is semidejnable over B (the 8ame hold8 for I instead of B).

427 CH. VII, 5 41 SEMI-DEFINABILITY

Remurk. Part ( l ) , implies, by 111, 4.10(1) that if T is stable, B = lUl, then p is semi-definable over B i E p does not fork over B.

Proof. ( 1) Let p be an m-type semi-definable over I. If p c Av( D, Dom p) for some ultraflter D, and q c p is finite, then

{C I: c realizes q} = {C I: b Q [ ~ ; a]} E D ,(Z;;a)sq

(as D is closed under intersection). So {C E I: 5 realizes q} # 0 80 q is realized in I.

Suppose every finite q c p is realized in I. For each Q EP let J, = {EEI: C satisfies Q}. By clssumption the intersection of any finitely many J, is non-empty, hence there is an ultrafilter D over I such that Q ~p J@ E D, by 1.1(2) and (3) of the Appendix. Clearly Q ~p * Q E Av( D, Dom p).

For types with infinitely many variables the proof is similar. (2) Every type over a model is finitely satisfiable in iK, hence the

(3) It suffices to note the following; and use (1): (i) If p,, i < 8, is an increasing sequence of types, which are finitely

satisfiable in B, then Uled p , is finitely satisfiable in B. (The proof is immediate.)

(ii) If p is finitely satisfiable in B, ~(3; 3) a formula, then p1 = p U {v(Z;a)} or p , = p U {-q(z;a)} is finitely satisfiable in B (or both are).

Because otherwise there are finite q1 G pl, qa G pa which me not satisfiable in B. So q = (ql u qa) n p 5 p and is finite, hence satisfied by some 6 E B. If b ~ [ 6 ; a] then 6 realizes ql , and otherwise it realizes q2, contradiction.

result follows from (1).

LEMMA 4.2: (1) If C is a subsequence of 6, tp(6, A ) is semi-&$nable over B thn tp(C, A) is eemi-de$nuble over B.

(2) If p k q (e.g., q c p ) and p is semi-dejnable over B, t h q is semi- dejinuble over B.

( 3 ) If B' c B, p is semi-&$nuble over B , t h p is semi-&$nuble over B.

Proof. Immediate. Note this lemma shows the consistency of the definitions of semi-definability.

428 OONSTRUCTlON OF MODELS [OH. VII, 6 4

LEMMA 4.3: (1) If p is semi-de$nable over By then it doecr not &it over B. (2) If p is a complete type over A , eemi-&$dle over By B c A, and

every qESm(B), m < w , is realized by some F E A , then p is stationary over B.

Proof. (1) Supposep c Av(D, A), and tp(6, B) = tp(E, B), 6, E E A . If v(Z; 6) E p , then 8 = {8 E B: hpp; 51) E D. But for any 8 E B bp; 61 = g(G; Z], hence 8 = {8 E B: b[8, a}, hence v(E; Z) E Av(D, A) hence +P; 5) $ p , so p does not split over B.

(2) Suppose ply pa are contradictory extensions of p which are semi- definable over B. By 4.1 (3) we can mume they are complete types over some C 2 A, SO them is iz E C q ( E , 8) €ply -Q(Z, 8) Choose 8, E A, tp(8,, B) = tp(iz, B) and w.1.o.g. p(2; a,) E P . So p2 splits over B, contradiction.

DEFINITION 4.2: q = { ( p , B): p E ~ ~ ( A ) , B E A , IBI < A, p is semi- definable over dcl B}. LEMMA 4.4: (1) satiejk9 the fobwhg a&ma (eee IVY 8ectbn 1):

and (XI, 2). Ax (I), (11, 11, (IIY2)Y (III,l), (111, 2), I V Y (VII), (VIII), (IX), (XI, 1)

(2) A(Fi) = A; p(Fi) = a. (3) If T hm 8kolem functions, then also Ax(X.1) and (X.2) hold.

Moreover if IDompI < A, p i s over B E A, IBI < A, t h there ie a complete type q over A &ending p mch tluct (q, B) E Fi. Also Ax(II.3) and (11.4) hold.

Proof. (1) For Ax(II.1) notice that if 8 E B c A , l(8) = m; then D = (J c B"': ii E a) is an ultrafilter over B"' and Av(D, A) = tp(iz, A). Ax(1V) is Lemma 4.2( 1). For Ax(VII), it suffices to prove the following: suppose dcl B = By dcl C = C, B c C; and B s A, tp(Z, A u C) is semi-definable over B u C, tp*(C, A) is semi-definable over By a E A and v(Z; a) is realized by some Z E iz u C. We should prove q(Z; a) is realized by some i? E B. We can assume Z = 8-Zl, El E C, so Cq[Z, E l ; a], hence for some izl E C C @,, El; a]. As Z,, El EC, for some a;, Z; E B, C &, 5;; a] and we finish.

The other axioms are trivial. (2) Immediate. (3) If T has Skolem functions, for any By cl B is the universe of a

model. So by 4.1 (2) every type p over cl B is semi-definable over cl B. So 4.1(3) implies our conclusion. The proof of Ax(II.3) and (11.4) is easy.

CH. VII, Q 41 SEMI-DIEFINABILITY 429

LEMMA 4.6: Let P = IN[ , and (N*, P ) be an elementary submodel of (Q, P ) m h tlrat IN1 E IN*I. Then for any sets A , B there are elementary mappings f, g which are the klentity over IN!, IN*! r q . m h that if A' = f ( A ) , B' = g(B) then:

(1) tp,(B', IN*I u A') is eemi&$nuble over IN*I, (2) tp*(A', IN*I u 8) i.9 smi-de$nuble over INI.

Prmf. By 4.1(2) and (3) we can extend tp*(A, INI) to a complete type over IN*I which is semi-definable over INI. Hence we can define an elementary mapping f, which is the identity over INI, such that tp*(A', IN*I) is semi-definable over IN], where A' = f ( A ) . Now let

I' = tp,(B, IN*[) u 0 u Y, where

@ = {-,cp(Zs; a, 5): 6 E B, E A', E E IN*I (Q E L) and

Y = {l$(Z6; B, 5): 6 E B, a E A', E E IN*I (+ EL) and Q@E; a, 2) is not realized in N*},

#(6, g, a) is not realized in N).

Clearly it suffices to prove I' is consistent (if x,, I+ b' is an assignment satisfying I', let g map c to c for c E IN*I and b to b' for b E B. Now g is elementary aa tp,(B, IN*I) E it satisfies (1) aa @ E r m d (2) &B

Let r' be a finite subset of I'. As tp,(B, IN*[), 0, Yare closed under !PG r).

conjunctions, we can mume, by adding dummy variables,

r' = {6(36, q, 7q(5, a, q, -@(s, a, q> where a E A', 6 E B, E E IN*I, CO[6, E l , +(6, gy 5 ) is not realized in N. So, as P = "1,

a, a) is not realized in N*,

(Q, P) C (W[ A P(#[iI) -+ +@, a, El] A e(6, el. t

As (A'*, P) is an elementary submodel of (a, P) and 6' E IN*I

E IN*I, there is

By the second conjunct, 6' satisfies O ( q , a); by the first one for every a' E INI, C7+[6', a', 131, but tp@, "*I) is semi-definable over IN1 hence by 4.1(1) finitely satisfiable in N, hence Cl+[6', a, a. So in order to realize r', 6' has to satisfy l Q ( f 6 , a, a), but as 6' E IN*I this follows from W s definition.

430 <IONSTRUO!l'ION OF MODELS [m. VII, 8 4

THEOREM 4.6: Por every p 2 2A1, A, 2 2IT1I, PC(T,, T) has a A,- universal model M of cardinality p which is stable in A,. (Hence if T is A,-unstable, M is A,-universal but not A:-universal.)

Remark. The construction here wi l l serve additional theorems.

Proof. We shad describe here a construction for a cmdinal A > I T,], 2<" s p such th t for A = A t we get the desired model. Of course, we can assume T, has Skolem functions.

Let N be a model of T, of cardinality lTll, P = INI, and (N*, P) an elementary submodel of (El, P) of cardinality ITII. Let M1, Ma be p-saturated models of T, extending N. By Lemma 4.5 (and 4.2) we can aasume tp*(lM1l, IM'I), tp*(1HaJ, lM1l) are semi-definable over IN*I, IN1 reap.; hence do not split over them (by 4.3(1)). By 4.3(2) tp,(Wl, PI) is stationary over N. Note that if a, 6~ wl, t p (q W*l) = tp(6, p*l) then t p ( a , w l ) = t p (6 ,wl ) (as for any CEWI and tp(z,g), t=tp(a,c) EE ~ ( 6 , c ) as tp(q PI) does not split over p*l); hence the number of ~ES"(W) , m < w , realized in 1M2 is < IUmSm (N*)l Q 2ITll. We can interchange the roles of JP and M2 (with N instead of N*) .

Let {M,: i < p} be a list of models of T, of cardinality < A, such that up to isomorphism, each model of T, of cardinality < A appears in it. As Ma is A-saturated, we can assume all the M i are elementary submodels of Ma. We define by induction on i models Mf such that:

(i) tp,(lM:l, lM1l) = tp,(lM,I, IMlI); more exactly there is an elementary mapping f,, f, is the identity over M1, fi maps M , onto M?.

(ii) The type of iKf over M' u UjCi M; is =mi-definable over IN!. Let M be the Skolem closure of ufCg M:. Clearly M has cardinality s p (as 2CA s p) and M is x-universal for every x < A. Suppose x+ 2 A so llMfll s x for each i, and we shall show M is X-stable. Let A E 1x1, IAI s x, but @(A, M) has cardinality >x. W.1.o.g. A is the Skolem closure of {M:: i E w} where IwI s x. Suppose Zi E 1611, i < x+ realize Werent types over A, so by the definition of M a, = T,&, . . . , 6,,J where n = n,, 6,,, E 1M&,,l. As we can replace {a,: i < x+} by any subset of the same cardinality we can assume n, = n, fi = T . Let 6,,, = f,c,,,,(E,,.l) E,,, E M,,,,, hence Z,,, E Ma. As the number of complete types sequences from Ma realize over M1 is s 2IT1I, we can assume, if x 2 21T11, that tp(E,,,, lM1l) = g,. As the order p is atomically stable in x (see 2.8(6)) we can aasume the atomic type of @(j, I), . . . , p(j, n)) over w is constant, and p(j , 1) E w * b,,, = b,. So it suffices to show that under

CH. VII, 8 41 SEMI-DEFINABXUTY 43 1

a11 those conditions all the B, realize the same type over A. For this it suffices to show:

(*) If i(1) < * * * < i(m) < p,j(l) < * ' * < j ( m ) < p, 81 E 1M,(1)lY

Proof of (*). We prove it by induction on m. Form = 1, tp(si;, lM1l) = tp(7i1, lM1l) = tp(6,, lM1l) = tp (g , lM'1). Suppose we have proven for m, and we shall prove for m + 1. By the symmetry we can wume i(m + 1) 5 j ( m + 1). Now tp(7ik+l, 161'1) = tp(pm+l, 1M:l) is station-

U {IM,l: j < j ( m + 1))) are semi-definable over INI, hence the former is a subtype of the latter, and the latter does not split over INI, so by the induction hypothesis we get (*) and finish the proof.

(By checking that when h = A,+, A, 2 2ITil, we get the result easily.) Note that w.1.o.g. Mg $ N , hence llMll = p.

ary and tP(Za+lY p ' l u u {pq: j < i(m + 1))) tp(%+,, pfll u

EXERCISE 4.1 :Let Tbe the theory ofthe rational order, IMI = (0, l ) , a = 1, b = 2. Show there is no a', such that: tp (a ' ,Mub) extends tp(a,M) and is semi-definable over M and tp(b,M U a') is semi- definable over M.

EXERCISE 4.2: Show that Fg cannot (in general) satisfy more axioms than stated in Lemma 4.4. (Hint: Ax(II.3, 4): Choose A = B = 0, aEac1A - dclA.)

Ax(V): Let M = EM1(wly 0) where we get the theory T1 and @ as in Exercise 2.7. So M Ca < a,, implies: M Ca < a, for some i < 6, and a E acl{a,:j < 6) ({a,:j < w,} is the skeleton). Let B = acl{a,: i < w

or i > w + a}, A = acl{a,:i s w or i > w + w}, 7i = (a,,,), 6 =

(a*+7)- Ax(1V) : By the previous example, for A = B = C = acl{a,: i < w }

6 = (a,), z = (a,,,). Ax(X.l), (X.2): Choose A = dcl A = 0 (Ao = dcl Ao), g, = 5 = 5.

Ax(XI1): C1 = acl{a,: i < w } C, = acl{a,: w .2 < i < w.3) B = C, U

Cay A = acl(B u {ad), p = tp(a@+,, A) (in the notation above).]

432 CONSTRUCTION OR’ MODELS [a. VII, Q 5

VII.5. Hanf numbers of omitting types

K need not be infinite

DEFINITION 6.1: (1) We define p(A, K ) as the first cardinal p such that: if lLol 5 A, To a theory in Lo, r a set of (< No)-types in Lo, ’5 K ;

and for every x < p there is a model in EC( To, F) of cardinality 2 x, where EC( To, r) is the class of Lo-models of To omitting each p E r, then there are in EC( To, r) models of arbitrarily large cardinality.

(2) We define &A, K ) as the &at ordinal a such that: if lLol 5 A, < €Lo, To a theory in Lo, f a set of ( < N,)-types in Lo, If I < K and there is a model M in EC(To, r) such that (1611, <M) has order type 2 a, then there is M E EC(To, r ) which is not well-ordered by

Remark. The difFerence between the two definitions (“for every x < p” but not “for every S < S(h, K ) ” ) is inessential, sea 6.2.

LEMMA 6.1: (1) If in EC(To, r ) there i.3 a nzodel of cardinality A, I ToI 5 p s A, t h n in EC( To, r) thre is a model of cardinality p.

(2) If s A l , K s ~ 1 , thenp(h, 4 s Ah, 4, a d W, 4 5 W , ~ 1 ) .

(3 ) S(A, 1) 2 A+. (4) S(A, 1) = S(A, A). (6) &A, 0) = w.

Proof. (1) Because N < M , M e EC(To, r ) implies N E EC(To, r) using the downward Liiwenheim-Skolem theorem I, 1.4.

(2) Immediate. (3) For a < A+, let M , be (a, <, . . ., i,. . .),<,, T, = Th(M,),

p = {z # i: i < a}. Then every model in EC(Ta; {pa}) is well ordered and has order type a aa it is isomorphic to Ma. So S(A, 1) > a; and as this holds for every a < A+, 8(h, 1) 2 A+.

(4) By (2) S(A, 1) 5 S(A, A). To prove the converse, assume ]Lo! s A, To a theory in Lo, r a set of ( < #,)-types in Lo, I rl 5 A, M E EC( To, r ) has order type S(A, 1) and we shall prove that there is a non-well- ordered model in EC( To, r).

Let r = {pa: a < a, 5 A}, pa = {tpf(~,): i < i, 5 A}, pa an ma- type. Let $(a < ao, i < i,) be new individual constants, &, P, (a < ao)

CH. VII, 5 61 HANF NUMBER8 OF OrdITTINQ TYPES 433

new one place predicates, and for a < %, Fa a new ma-place function- symbol. Let L, = Lo U {Q} u {Pa, Fa: a < ao} u {q: i < i,, a < ao},

T I = To U {Q(cf), P,(cf): i < i,, a < ao} u {(VZ,)P,(F,(Z,)): a < ao}

u {(VZa)[F,(Z,) = cf + -,Q~Z,)]: i < i,, a < ao}

" {(W(Pa(4 -+ Q(4): a < a01 u {(wrp,(4 + 1p&41: a # B < ao},

p = {Q(z) A z # 4: i < i,, a < ao}.

Clearly every model in EC( To, r ) of cardinality 2 h can be expanded to a model of EC(T,,{p}), and the L,-reduct of every model of EC(T, ,b } ) belongs to EC(T, , r ) . So we can expand bl to W E EC(T,, { p } ) ; and then aa T,, p we in L,, lLll s A, and bl' haa order type S(h, l), there is N' = W, N'EEC(T, , { p } ) which is not well- ordered. The Lo-reduct of N', N, is the desired model.

( 5 ) If in (3) we take a < w, then every model in EC(T,, 0 ) is isomorphic to Ma, hence &(A, 0 ) > a; so S(h, 0 ) 2 w. By the compactness theorem if for every n < w To haa a model bl, of order tyoe 2 n, then To u {c ,+ , < c,: n < w} has a model, so To hae a non-well-ordered model.

LEMMA 6.2: 8wppse P, < E Lo(P--a one plccce predicate, < a two place predicate) lLol 5 A, To a theory in Lo, r a set of s K ( < N,)-typee. If for every a < 6 = S(h, K ) there is in EC(To, r) a model bl, such t?@ (P*u, 4%) laas order-type l a tkn in EC(To, r) there i s a model bl in which cM doecr not well-order PM.

Procf. For K = 0 this follows by the proof of 5.1(5), so let K > 0. Let P be the function such that F ( a ) = M a for a < &(A,K), w.1.o.g. the

are pairwise disjoint and F , be the function such that for each a < & ( A , K ) F,(a,z) is an isomorphism from (a, <) into (PMu, Let 93 be an elementary submodel of (R(E), E,

F, P,, 8, To, . . . , Q, . . . of cardinality 181, {i: i < S } = S E ] % I , and let # be a one-to-one function from 6 onto 193 I, and <* an order on 1931 such that # is an isomorphism from ( 8 , <) onto (1931, <*>. Let 93, = (99, <*, #) and T, = Th(99,). For Q = ~ ( f ) E Lo let Q, = 'pl(y, Z) be an L(S,)-formula saying that y is Ma = F(a) for some a < S and f is from lM,l and Q(Z) is satisfied in Ma. For p E I', p = p ( f ) let pl = {Q,: Q ~ p } , and r1 = { p l : p E r} u {{z E Lo A z # Q: Q E Lo}}. SO in EC(T,, rl) there is a model (99,) such that <* orders it in order-type

434 CONSTRUCTION OF MODELS [CH. VII, 8 6

6 = &A, K ) , lLll s A, lF1l s K (or if K < A, IF1( s A, so by 6.1(4) 8(A, IFll) = E ( A , K ) ) hence there is 9’ E EC(Tl, Fl) which is not well- ordered by <*. Hence ord(23*) = {a E IS*]: $* C “ a is an ordinal < 6”) is not well ordered by < (use a). Choose a E ord B* so that {b E ord 23*: b < a} is not well ordered. Then P(a) is, in fact, the required model.

THEOREM 6.3: Suppoee lLol s A, To a theory in Lo, r a get of I K

( < Ko)-typea in Lo and P a one-place predicate in Lo. If for every a < 6(A, K ) there irr H a € EC(To, r), IIM,II 2 aa(IPM“I) then for every p 2 ILol, there is H E E C ( T ~ , F), 11H11 = p , IP”I = ILol.

Remark. Notice that we can replace aa( lPM~l) by aa(lPMu1 + A) for K > 0, because ~ ~ ( 0 ) 2 A, SO for a 2 A + w they are equal.

Proof. We proceed as in the proof of 6.2. Let P be a function from 6 = 6(A, x), P(a) = Maandlet23, = (R(;), E, P, 8, To,. . . , tp,. . .)(PEL,,.

So again we can find 23 elementarily equivalent to 23, such that in 23 there are “ordinals” a,, < 6 , a,,+, < a,, and for each a < 8, P ( a ) is a model of To omitting each p E F. Now w.1.o.g. we can assume that To has Skolem functions and 93 k “a,,+, + n + 1 < a,,”. Let < be, in 23, a well ordering of P(ao). Now define by induction on n, elements X, of 23 such that 23 C “X, is an increasing sequence of elements from the universe ofF(a,), it has cardinality 3aJP(F(ao))l) , and is n-indiscernible over P(P(ao))” and 8 C “Xn+l is a subset of X,,”.

This is possible by the Erdos-Rado Theorem (2.6 of the Appendix) (more exactly-as 23 Hatisfies a first-order sentence saying it).

Now we define @ proper for (w, To) so that if p(zl, . . . , z,,) E @, then if 49 C “A:* , a, E X, A A::? a, < at+,” then 23 C “tp(al, . . . , a,,) is satisfied by P(ao)”. Clearly there is such @, and EM(& @) is the required model. (It does not realize types omitted by F(ao) because of Lemma 2.2; hence, it omits the set of types r.) THEOREM 6.4: p(A, K ) = ad(h,rc) for K > 0 ; and p(A, 0) = No, 6(h, 0 ) = 0.

Proof. For K = 0 this is by 6.1(6) and the compactness theorem, So let K > 0. By the last theorem it is easy to see that p(A, K ) 5 a6(A,rc). For the other direction assume that every model in EC(To, F) is well ordered by < ; To, Fare in Lo, lLol = A, IF( 5 K. Now let Q be a new one place predicate, for every # let $0 be # relativized to Q, TOO =

{p: + E ~ ~ 1 , = {p: 4 Ep), ro = {pa: p E r}.

CH. MI, 5 61 EANF NUMBERS OF OMITTIWO TYPES 435

Let E be a new two place relation, P a new one place relation, and P a new one place funotion symbol, and ct, i < KO new constant symbols. Let

(VZ, y ) [ - ~ p ( Z ) A - IP (Y) A ( v Z ) ( Z E 2 Z E y ) + X = 3 / 1 9

(WQ(m)), (VZY Y ) ( Z w + m ) < P(Y)))U Wet): i < So},

F, = YOU {{P(X) A 2 # C,: i < KO}}.

Clearly if iK E EC(Tl, F,) then its submodel with universe QM belongs to EC(To, r ) and if the order type of the latter is a, then lldlll 5 D ~ . Conversely, if some N E EC(To, F) has order type 2 a + 1, then some bl E EC(Tl, r,) has cardinality aa + I TII.

THEOREM 6.6: Let K > 0. (1) 8(hy K ) is a limit ordinal of cojdity > A.

( 3 ) If cf h > KO then 8(hy 1) > A+. (4) If of h = KO, p = ZX<’ 21 then S(h, 1) s p+.

(6 ) If h = No

(6) &(A, 2’) = (2’)+. (7) &(A, 1) < (29+ (and &(A, K ) < (2’)+ when 2“ < 2’)). (8) For a strong limit cardinal of cojinality > KO, 2’ < & ( A ) ; and

(2) h+ 8 ( h , K ) 5 (2’)’.

h i8 a &Ong limit cardinal O f W f i d i t y KO, t h 6(h, 1) = A + .

generally cf h > KO, (vp < h)(p”‘A < A) implies &(A, 1) > A”‘”.

Proof. (1) Easy, left as an exercise fo the reader. (2) We have already proved that 8 ( h , ~ ) 2 A+ (in 6.1(3)). For the

other inequality let lLol 5 A, To be in Lo, F a set of types in Lo, s K and suppose iK E EC(To, F) has order type 2 (2’)+ (ordered

by < ). We can assume Th(M) has Skolem functions, and let a, E IiKI, i < (2’)+ , be such that for i < j, dl t at < a,. Now we d e h e by induction on n < o, sets & E (2’)+, = (2”+, and for each i E&,

ordinals ai(n, 0) , . . . , a,(n, n - 1) for which i < al(n, n - 1) < - - - < al(n, 0 ) such that: if i E 8,, j eSm, n 5 m then tp((aa,,,,,,, . . . ,

For n = 0 let So = (2”)+, (we have no at(O, 1) to define). If we have defined for n; for each j < (2’)+ we choose an i, j < i €8, and let a,(n + 1 , l ) = at(n, 1) for 0 s 1 < n and a,(n + 1, n) = i. The number

aa,(n,n- I)), 0, = t~((aaj(m.o), * * 9 aa,(m,n- 1))s 0, Jf).

436 OON8TRU(rmON OB MODELS [a. VII, 5 6

ofpo=ibletypeep;+’ = tp(<au~n+l,o), * * * Y au,(”+l.n)), 0, M)is s 2lL0l s 2” c (2A) + w h e w the number of j ’ s is ( 2”) + . So there is a type pn +

such that Is,+, = {j c (2’)+: p;+’ = pn+l} has cardinality (2”)+. Now by the compactnw theorem and our construction there is a

model N , of Th(M), and in it elements any n c o, such that tp((ao, . . . , a”), 0, N , ) = p”+’. As Th(M) haa Skolem functions, we can wume that INl! is the closure of {an: n c u}. So for every 6 E lNll there are nmd7so tha tN1 C6 = ?(ao, ..., a ” ) s o i f i ~ & + , , 6’ = ?(aul(n,o)y...) (in M) then tp(& 0, N,) = tp(6’, 0, M) so N , omits every p E F 80

As for each i E&, q(n, 0) > q(n, 1) > - - - clearly N, C a,, > a,,,,, so N, is not well ordered. All this proves that 8(A, K ) s (2”)+.

(3) Let 49, = ( R ( i ) , E, c , A+ , A, i)‘< A ( c -the order between the ordinals s A + ) and let p = {z c A A z # i: i c A}, and F = {p} and T = Th(99,). Suppose 99 E EC(T, 0, and we shall prove that {z: 99 C c A + } is well ordered, thus by 6.2 proving that 8(A, 1) > A+. For

suppose 99 C z,, + , < x,, c A+. Clearly in 93 there is an element f such that 8 C“f is a one-to-one mapping from {z: 2 < z,,} onto {z: z < A}”.

Let 9 C “f(z,,) = a,,”. As of A > No, 49 omits p , and {a,,: n c W} is countable it is bounded by some a* c A. As b C “the set of x c A+ such that f(z) c a* has cardinality c A” there is g in 49 such that 53 C “g is an order-preserving function from {z c zo:f(s) c a*} into A” so 99 C “g(z,,+,) c g(z,,) c A” contradiction to the omitting of p by 9 (aa A is well-ordered).

(4) Let M €EC(To, r ) have order type zp+, and F = {p}, p = {v‘(Z): i c i, s A}, Z(Z) = m. Let ?;(z0, . . ., $,,-,), a c A be a list of all terms from Lo where Z(f:) = m. Let for a E “1611, G(si) = min{i: M C 7cpi(i i)}. For A > No aa of A = No let A = z,, A,,, A,, c A,,+, c A, and let H(7i) = min{n: G(a) < A,,}.

The rest of the proof is just like that of (2), only p+ replaces (2”)+, and instead of tp((aul(,,,o), . . . , aul(n,n-l)), 0, M) we use when A = No

N, E EC(T,, r).

((1, FLY Q(~Lfr(aul(n.O)Y * - * Y ~u, (n ,I - l ) ) ) ) : 2 5 nY a 5 n}

and when A > KO,

OH. VII, 5 61 HANF NUMBEBB OB OMITTINO TYPES 437

At the end we choose Nl as a model of

Th(H) u {&+l < am: n < w}

u { l p p ( ~ ( U o l ..., an-’)) : p = G(q%*(t8,0p ... 1) for every i E&, n big enough}.

(6) It is immediate by parts (2) and (4). (0) By (2) it suffices to prove S(A, 2”) 2 (2”)+, or that for every

a < (2”)+, 6(A, 2”) 2 a. So let a < (2*)+ and choose for each /3 < a a subset SO that /3 # y * 86 # 8,. b t for every /3 < y < a

P6.7 = {+ c y)} u {P,(2)1f(fES0): i < A} u {P,(y)u(W: i < A} and for

To just “says” < is a linear order,

Of

every s c A p s = {P,(P)U(f€? i < A}. Let Lo = {Pf: i < A} u { < y =},

r = { & 7 : /3 < y < a} u bS: S c A, 8 # 86 for each /3 < a}

U {P # y A Pi(%) Pi($/): i < A}.

Clearly there is M, E EC( To, r ) of order type a (lHl = a, /3 E Pf 8 i E 86, < is the usual order) and every model in EC( To, r ) is a submodel of M , (up to isomorphiem) hence is well ordered.

(7) Immediate by cardinality considerations (there are essentially only 2* pairs (Tol {p}) and similarly for S ( ~ , K ) ) .

(8) Let K = cf A, and D be any H,-complete filter over K. For any function f from K to A (in fact, to the ordinals) we define its D-rank RD(f) : R D ( f ) 2 0 always, R D ( f ) 2 6 if for every a < 8, R D ( f ) 2 a,

and RD(f) 2 a + 1 if there is g : K 4 A. g/D < f/D (i.e., {a < K :

g (a) < f ( a ) } E D ) and R D ( g ) 2 a. NOW RD(f) = a if RD(f) 2 a but not R,(f) 2 a + 1; as D is 24,-complete, there is no descending sequence fn /D (n < w) SO RD(f) is always an ordinal. Let

a(A) = sup{RD(f):f: K + A}.

Let us define now a model H’: its universe is the disjoint union PM u QM u &f where ignoring trivialities:

PM = a(A), &M = {f: f a function. from K to A},

c M = the order on ordinals, F = a partial function such that F(f , i) = f(i) for i < K, f~ ah, i (i 5 A) = an individual constant, A (A c K ) = an individual constant,

&f = {A: A = K},

438 UONSTRUUTIOW OB MODEL8 [a. VII, $ 6

€M = {(i, A): i < K , A E K , i E A}, Qi = D, B = a partial function such that U ( f ) = BD(f) E P’.

Let T = Th(M) and p = {x < A A x # i: i < A} and let N be a model of T omittingp; so by renaming we can assume the interpretation of i (i 5 A) is i ifself and any member of QN is a function from K to A, and PV, i) = f(i), m d

QF = Qy, E N = { ( i , A ) : i c K,A E K , A E Q F , i , E A ) ,

so Qf = D. Now we prove PM is well ordered. If cn > c , + ~ (n < w) is a counterexample, then there is fo B ( f ) = co. (As (Vx E P) (3y cQ)(B(y) = 2) E T, as PM = a(A).) Similarly, by the rank’s definition, we can define fn E Q ~ , such that U(f, ,) = c,,, {i < K : f,,(i) > f,,+ l(i)} E Qf = D; and we get a contradiction. So in every model N E

EC(T, {p}), PN is well-ordered by <N. By 6.2 and T’s definition this implies a(A) < &(A).

Up to now we have not used any assumption on A, Kexcept K 5 A, cf K

> No, 2’5 < A; and if D is easily defined (e.g., the filter of closed unbounded subsets of K , or DEb the filter of co-bounded subsets of K ) the last restriction is not necessary.

Now we prove a(A) 2 Aofh, e.g., for D = DEb thus finishing. First note that there are Aof A functions f, : K --+ A such that (V i # j ) ( 3 a < K )

i < K ha be a one-to-one function from n,ca A, into A (exists as (Vp < A)(Vx < cf A)(px < A) ) . Let df:; i < AofA} be a list of all functions from K to A, and define fl(a) = ha((ft(y): y < a)). Thefl’s are as necessary. If a(A) < Ax, there are (29+ fl’s with the same D-rank, say cfl: i < (29+}. For any i < (29+, cf i = K, and a < K + , we define by induction on n < w, &(i, a) such that

(VB)(a f ib) ,

(iii) for any m < n, letting f = &(i, a), = &(i, a)

f d a ) > f , (a) +fc(a) ’ fM. By (iii) and (ii), fcncl,6(a)(n < w) is strictly decreasing hence for some n = n(i, a) &(i, a) is not defined. Let Sk = {&,(i, a): n < n(i, a)}, Si = Uac+ St, 80 lS1l 5 K, 9’ c i. As cf i = K + some h(i) < i bounds it, so by 1.3 of the Appendix, for some y {i < (29+ : h(i) = y } is stationary

OH. VII, f 61 HANF NUMBERS OF OMITTING TYPES 439

and also for some S : W = {i < (2")+ : h(i) = y , S6 = S} is stationary. We can find j < i, j, i E W such that for every 6 ES, t9 < K ,

fc(B) ' fdB) +fc(B) > f r ( B ) If for some a, f,(a) > h(a), j will satisfy the conditions on fna,a,(i, a), contradiction. So for every B f j ( j3) I f,(/3); but l{P: f j ( B ) = ff(/3)}I < K

hence {/3 c K : f j ( P ) c fi(/3)} E D( = D;Ib). But this implies f , /D < ff/D hence R,(f j ) < R,(f,), contradiction.

Remark. Barwise and Kunen [BK 711 have shown that it is consistent with ZFC that:

(1) A + < 2 A y 6(h, 1) < A + + , (2) A + < 2 A < 6(h, 1).

EXERCISE 5.1: Suppose lLol I h, To a theory in Lo, r a set of ( < No)- types in Lo, s K . Show that if for all a c 6(h, K ) there is a model Ma in EC(To, r) such that IP"aI 2 >a and llMall 2 > ( ~ P " U [ , a) then for all h 2 p 2 ITo[ there is a model M in EC(To, r) such that llMll = h, IP"I = p. If we omit " IPMuI 2 say' we can still get IP"1 I

PI. EXERCISE 5.2: Show that if in the definition of p(No, 1) we require To to be a complete theory and p ( F = {p}) to be a complete type the value of p( No, 1) is unaltered.

CHAPTER VIII

THE NUMBER OF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CLASSES

VIII.0. Introduction

The essence of this chapter is the construction of many non-isomorphic models in a pseudo-elementary class PC(T,, 2’) and we shall use methods developed in VII; the proofs have a combinatorial flavor.

At the beginning of Section 1 we make two important observations (Lemmes 1.3 and 1.4).

(1) If after adding K individual constants to T, there are p many ( > As) non-isomorphic models of cardinality A in the pseudo-elementary class, then the adding does not change this number.

(2) If we have h 2 lTll types in D(T) which are independent (i.e., for any subset of them, there is a model reelizing them but not the others) then I(h, TI, T) = 2” (aee Definition 1.1).

Now we can show that some properties of T imply the existence of large families of independent types, sometimes over a set A (1.6) and then by (1) and (2) we prove that I@, T,, T) is big (1.7). A more difficult theorem is 1.8 (the main crme is 2” = 2n0, ID(T)I = No).

THEOREM 0.1: If T ia countable, not Ko-8table, lTll = No, KO < A 5 2n0, then I(h, T, , T ) = 2”. If instea& of I TII = No we demand I TII < 2% we atill get a r d t when

lTll s h < P o : (1) If A satisjies the combinatorial condition (*) (“not AD(2No, A , A ,

No)”), then I (A , TI, T ) 2 2 H o (see 1.9). (2) If Martin Axiom (see [MS 701) holds, I(A, T, ,T) 2 2A (see

Exercise 2.6).

More interesting is the fact that we can generalize the theorem to higher cardinals, but the hypothesis is quite strong:

440

OH. m, 8 01 INTEODUOTION 44 1

THEOREM 0.2: 8ypptxe (i) T has A independent fornzulae, (ii) T E TI, A = IT11 = IT!, (iii) There is a A-Kurep tree with x branch (e.g., A strong limit, or

cf A = No, x = ANO),

Then for p 2 A, I (p , TI, T) 2 min{2xy 2”).

It would be nicer to get such a theorem aseuming only ID( T) I = x >

The main aim of Sections 2 and 3 is A =lT1l.

THEOREM 0.3: Any unsulperstable T , has 2A norc-isommy.~hk Cardinality A, for any A 2 IT1 + N,.

of

We prove it also for pseudo-elementary 01- in many

The easiest case is:

(e.g., countable T, T,) . The proof is by cams.

THEOREM 0.4: If T i s uwwperstabk, A is regzrlar and > lT1l, then in PC( T,, T) there is a family of 2” nzodele of cardinality A, no one ekmedarily embedduble in anothw.

In the proof note that if M = UleA Ml (1 = 0, l), Mt (i < A) in- increaaing and continuous, Aregular IliKll = A > IIMf 11 then{i: Mf = Mt} is a closed unbounded subset of A. Hence if P is a property of pairs of models then {i < A: (My M f ) satisfies P} is determined, mod D(A) by the isomorphism type of M ; and in fact we can make P to depend on more information. We urn the indiscernible tree of sequences we have constructed in VII, Section 3. Choose for each S < A , cfS = w , an inmawing sequence of ordinals of length w converging to it, 76, and for B G { S < A: of 8 = w} let I, = @’A u {q6: S €8). Now from the isomorphism type of EM(1,) we can reconstruct S/D(h) .

The other cams have each one its specific trick, and the proofs are somewhat complicated.

If p < A 5 pNo, 2” 2A, the method of Section 1, on independent types, applies. If A is singular, but no previous m e appliee we choose a proper p < A s 2’, and make a construction similar to the first one, but A-fold. If A is a singular strong limit cardinal, we use games. For an elementary clw, T stable, A = IT], use Fk-primclry models over indiscernible trees, and repeat the previous oaaw (all this in Section 2).

442 THE NUMBER OF NON-ISOMORPHIC MODEL8 [a. WII, 8 0

Then for A = I T , I, T unstable we note that skeletons of EhrenfeuchG Mostowski models rn N,-skeleton like (Definition 3.1 or below), and if in an insomorphism of one such model onto another, the intersection of one skeleton with the image of another is big, the orders of the skeletons are not contradictory (Definition 3.2). Proving the existenoe of large families of pairwise contradictory orders, we finish exoept when A satisfies a strong set theoretic condition, which implies it is < 2h or is high in the sequence of 8,’s (this is 3.2, case I). Note that M1 =

(A) (a,: 8 E I) is X,-skeleton like, i.e., for every 6 E lM1l there is a h i t e J E I, such that if 8 , t ~ I - J , (VUEJ) (8 < W E t < u) then 63,, gnat realizes the same type.

(B) Hence if I = I, + I,, then for every 6 E M,, for some t , E I, ,

The m e left is T unstable, 1 T,I = A, AD(2”, A, A, No); we can assume T is countable. For understanding the proof let us concentrate on showing I(A, T,, T) > 1 for Aregular > N,. Let N be a (A, N,)-eaturatd model (see Definition VII, 1.5), N = Ul<ol N,: and M = EM(I) , I = I , +12,11 z A , I, s o*, and let ( a 8 : s € I ) be the skeleton of M (order by < , of come) andf : M + N the isomorphism, and 6, = f(a,), so (6, : s €1) is an KO-skeleton like sequence in N. We want to show for some 6, C 6 < b, for 8 E I,, and c b, N,, for some 5 < o,,

sequence (Aa: a < A), increasing, continuous, 1A.I < A, U,<” A, = INC[. We can h d 8 < A, such that UaEIl 6, E A,. AS (6,: 8 E I) is H,-skeleton like all except < lA,l + + 8, members of {6,: 8 E J} realizes the same type over A,, let 6: be one of them and J, = (8 E J: tp(a,, A,) = tp&, A,)}. Now it suffices to prove some consistent type extending {Z < 6,: 8 E I,} U {6, < Z: 8 E J,} belong to St+,. But we can define such a type using N,, A,, and 6*:

E M l ( I ) eatisfies:

reah08 the mme type for all 8 E I Satisfeg tl < 8 < t,.

- J : UnErl b, G NC, UIEj 6, E Nc, J c I , , I JI = A. In Sc+l there is a

p = {Z < 6: 6 E A,, Nc C St < Q U (6 < 1: 6 E INC] , and for every 6’ E A,, Nc C 6: < 6’ implies Nc C 6 < 6’)

(every b i t e subset of p is realized by all a,: 8 E I2 except hitely many). In the actual proof we have to take care for getting 2A non-isomorphic models, and making them K-compact, for suitable K.

In Section 4 the reader can relax; here we mainly use previous results to prove theorems e.g., on categoricity, we prove:

CH. WII, 8 01 INTRODUCTION 443

Tireorem. If PC(T,, T ) is categorical in A > 1T11, then T is superstable, without the f.0.p. and stable in lT1l.

We also give a partial solution to the problem: doea the categoricity of PC(T,, T) in h imply its categoricity in p ? Show its equivalence to a two-cardinal theorem with omitting a type. We get similar results when we replace “categorical in A” by “each model in PC(T,, T ) of cardinality h is homogeneous ”.

Then we use the construction of VII, Section 4 to prove results on universality, i.e. :

THEOREM 0.5: If T ie unawperetable, A is regzllar >2IT1l t h in PC(T,, T ) there are arbitrarily lwqe ( < A)-univereal not h-universal nzode2e.

On the other hand we shall show that if of h = No the conclusion does

We also characterize the class of cardinalities in which a theory hw not hold.

saturated models, by using various previous results.

THEOREM 0.6: T hua a eaturated model of Cardinality A iff X = ID( T)I or T ie h-stahle.

PROBLEM 0.1: Though we get a complete reeult for elementary classes, this is not true for pseudo-elementary clmms, when we look for K-compact models K < K(T) , and when we look for 2* pairwise mon- elementary embeddable models; it will be interesting to complete them. PROBLEM 0.2: It will be desirable to simplify and unify the proofs. A possible way is in the proof of 2.1 to try to replace the filter of closed unbounded subsets of A, by a filter on S,(h) or on the family of increasing sequences of members of S,(h), of length S (see Kueker [Kk 771 and Shelah [Sh 7681).

PROBLEM 0.3: In many cams we have parallel proofs; in 2.7 an F&,-constructible set, replaces the Skolem Hull. Also clearly different proofa uae different sets of assumptions on the construction. Maybe an axiomatization is in order.

+

PROBLEM 0.4: For uncountable T,, we know little on I(h, T I , T ) when ,LA = JD(T)I > IT,I. It is natural to conjecture that for h > ITJ, l ( h , T,, T) 2 min{2A, W } , but it is reasonable to try also an independence proof.

444 “HE NUMBER OF NON-ISOMOBPHIU MODELS [OH. WII, 8 1

VIII.1. InaepenaenCe of tspes

DEFXNITION 1.1: (1) I ( A , T,, T ) is the number of non-isomorphic models in PC(T,, T ) of cardinality A.

(2) IE(A, T,, T ) is the maximal p such that there is a family of M f E PC(T,, T ) , i < p, llMtll = A, such that there is no elementary embedding of Mt into M, for i # j.

(3) If we omit T , this meam T , = T .

Remark. The “maximal” in (2) is inaccurate, as there may be none, and we may get, e.g., 8, < IE(A, T , , T ) < K, for any n.

Remember D( T ) = Urn Sm(0).

LEMMA 1.1: (1) 2” 2 I ( A , T,, T ) 2 IE(A, T,, T ) 2 1 for A 2 IT,!.

IE(A, T,, T ) s IE(A, T i , T’). (2) If T s T’ E Tl c T,, then I ( A , T,, T ) I I ( A , Ti , T’) and

Proof. Remember TI is consistent, and any model of T , is isomorphic to a model with universe {i: i < A}, and there are only

n 2(Am(”) (2”)“ = 2” RELl

such L,-models where m(R) is the number of places in R (we consider here m-place functions as (m + 1)-place relations). The rest is even more trivial.

Remark. We will be mainly interested in I and not in IE.

Proof. By VII, 2.9(1), for every p E D(T) there is a model M , E

PC(T1, 2’) of cardinality A which realizes p and is stable in lT1l. (Replace T, by T, u {v(E): v E p}, E a sequence of individual constants.) Now M, r Mq induces an equivalence relation over D(T) . Each equivalence clam has cardinality s I T,I as I{q E D(T): M q M,}l I I{q E D(T): q is realized in M,}1 s lTll (by the IT,I-stability of MP). Hence ID(T)I is at most ITl! plus the number of equivalence classes (by cardinal arithmetic). As ID(T)I > lTll, ID(T)I is equal to the number of equivalence classes. So there are 1D(T)1 non-isomorphic

OH. n I I , § 13 INDEPENDENCE OF TYPES 445

models M,, so I ( A , T,, T) 2 ID(T)I. Let g(p) = {q: q E D(T), M , has an elementary embedding into Mp}. So g(p) c D(T) and again Ig(p)l s ITJ; hence, when proving (2), by theorem 2.8 of the Appendix there is S c D(T), IS1 = JD(T)I, such that p # q E S implies p $ g ( q ) , hence IE(k T,, T) 2 1{2M,:pEs)l = ID(T)I.

PROBLEM 1.1: Does (2) hold when ID(T)I = ITl[ + ?

LEMMA 1.3: If A 2 1TlIy p = I ( A , T, u T(A) , T(A)) > AIAl then I(A, T,, T ) 2 p; where T ( A ) = T u { ~ ( a ) : si E A, Cp[si]) (the a E A serve also i n d i v i d d constants).

Proqf. Every model M, of T, of cardinality A may be expanded to a model of TI U T ( A ) in < hlAl forms, and we can get each model of T, u T(A) in this way, from just one M , (up to isomorphism) hence

p = I ( A , Ti U T(A) , T (A) ) 5 I ( & Ti, T) + But p > N A l , so our result follows.

Remark. Notice that we cannot use 1.2 for 1.3 a~ the estimation in 1.2 is too weak. Hence the following concept is interesting:

DEFINITION 1.2 : (1) If S is a family of types over A (in L), T s T,, then S or (8, A) is called (T,, T)-free (or free in (T,, T)) if for every S' c S there are models M , of arbitrarily large cardinality, A 5 M , (of come M, r L 4 6) such that p E S is realized in M , iff p E S' (hence in every A 2 lTll + IS'l there is such a model).

(2) S is free if it is (T,, T)-free for every T,. (3) The pair (T,, T) has (p, A)-freedom if there are A, IAI = A, 8,

IS1 = p satisfying (1). If this holds for any T, with the same A, S we omit T,.

LEMMA 1.4: (1) If (T,, T ) irae (p, x)-free&Om, p 2 A 2 lT1l, 2A > Ax, then I(A, T , , T ) = 2A.

(2) If (Tl, T) ' has (p, x ) - f r e e h , A 2 p, 2' > (p + p,p, then I ( A , T,, T) 2 2y.

( 3 ) If (S, 0 ) is (T,, T)-free, A 2 lTll, thenIE(A, T,, T) 2 min{21SI,2A}.

Proof. For ( l ) , we estimate I ( A , T, u T(A) , T (A) ) , from below, (where (S , ,A) is (T,, 2')-free, IS,l = p, IAI = x ) by looking at { ~ E S , : M realizes p}, and then use 1.3. Part (3) is immediate as by 1.6(2) of the

446 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. VIII, 8 1

Appendix there are 2151 subsets of 8, no one is a subset of the other. We are left with (2); by assumption for every S E 8, there is a model M1(S) of T , of arbitrarily large cardinality, A E M1(S), each p ES is realized in M1(S) by as = a(p, S), and no p E So - 8 is realized in it. By VII, 6.3 proof there is a model Ma(&) of T , = T , u T ( A U {ap: p €8)) omitting each p E .So - IS, which is EM(A, 0) @ proper for (w, Ta). Let M3(S) be its L-reduct. By VII, 2.10 and 2.8(6) TP,(M3(S)) has cardinality 5 (p + I T,I)X which is < 2". As TP,(M3(IS)) depends on M3(S) only up to isomorphism and the number of possible 8's is 2', we finish.

THEOREM 1.6: (1) T has (A, p)-freedom if T is unstable and A < Ded p (i.e., there is an order of power A, with a dense subset of power p) .

(2) T has (A, p)-freedom if K < K ( T ) and there is a (A , +tree I , [In ASIC[ 5 p < [ I n AICl = A (e.g., if X , < K(T) , A 5 pwo).

( 3 ) T has (A , p)-freedorn if T has the independence property and p 5 A 5 211.

(4) (T, , T ) h (2b, X,)-freedom if lTll = No, T X,-umtable. It has (2%, 0)-freedcm when ID(T)I > 8,.

Proof. (1) Let J E I be orders, I JI = p , 111 = A, J dense in I. Let J c J, c I, and we can wume J, I are dense orders (by extending), and let I, be an order of type x. By VII, 2.4 there are (Z < 8) EL, and @ proper for (0, T, ) such that in M = E M ( J , + I , , @), M C 7is < at iff s < t . Clearly llMll 2 x which waa arbitrary. Let A = {a8: s E J } , S = {p8: s E I - J), where ps = {(Z < 7it)if(s<t): t E J } . In M ps is realized if s E J, (by 4); the converse is also true, for if si = 6(@) E My as s $ J, and J is dense, there are s(1) E J, 8 ( l ) - s(mod 8 ) for I = 1, 2, and s( 1) < s < s(2) hence (7i < a*,)) = (7i < si8(a)) so si does not realize p8. We only have to choose A independently of x, this can be done by VII, 2.6 or by:

CLAIM 1.6: If for every T I z T and x there are a pair (8, A ) , S E Um Sm(A), IS[ = A, IAl = p and for every s' E S a model My 11M11 2 x, M realizes p ES i f p €8' then T h (A , p)-freedonz.

Proof. We should prove that we can choose (8, A) independently of T, . Up to isomorphism (of Q ) there are s 2A such pairs, so if each such pair (8, A) fails for T,(S, A), s'(8, A ) and x(S, A) then (as T is complete) we can find T , z T , such that up to change of names of predicates and function symbols not in T, it extends each T,(S, A) (such T,, is consistent aa Q can be expanded to models of TI@, A) for any S, A by

CH. WII, 8 11 INDEPENDENCE OF TYPES 447

I , 1.12). For T, we get some suitable (8, A) for x = 2 h(S, A): possible (8, A)}; contradiction.

Continuation of the proof of 1.5. (2) Let I be the (A, K)-tree mentioned there, and x L A. T has a strongly uniform K-tree of the form ((q~~, ma): a successor, a < K ) and let @ be as in VII, 3.6( 1) and (2). Then let

A = {a,,: 7 E I n A<"},

8 = {p,,: 7) E I n A%}, p,, = {q~~(z , sinla): a < K successor}.

If s' c 8 let I(&') = x<' u (7: q E I n A", p,, €8') and M = EM(I(8') , @). Clearly lldlll L x, A G IMI, and if I),, E s', a,, E 1dl1 realizes p,,. If v E An ?(a,-) realizes p y , pv $ s' there is a < K such that q[Z] 1 a # v 1 a (or q[Z] has length <a). As M C pls(.i(E,), liyls) (/? = a + 1) for every i < x, letting p = (v 1 a)-(i) M C vs(?(iZ,,), 7ip) (by the indiscernibility of the indexed set {a,,: 7 E I(&")}). But ?(7i,) can satisfy such formulas only for finitely many i's, contradiction.

(3) Let x L p be arbitrary, T, z T, with Skolem functions, of course. As T has the independence property there are q(Z ; #) E L, and a,, such that for every w _c w {v(Z; ii,,)w(nsw): n < w } is consistent. So we can assume {a,, : n < w } is an indiscernible sequence. So as in VII, 2.4 there is @ proper for (w, T,) such that in M = EM(x; @) for w G x, {q(Z; a8)if(ssw): 8 E x } is consistent. Let A = {as: 8 E p} and {w(i): i < 2") be a family of subsets of p, such that for all distinct i,, i,, . . . , i,, w(i,) - U:=, w(il) is infinite (exists by 1.6(2) of the Appendix); and p, = {[v(Z; = +Z; ~ , a + l ) ] " a E * ' ~ : a < p} and 8 = {p,: i < 2u}. There are Skolem functions F,, = Fn(go, gl, . . . , #,,,-,) of (3Z) Ale,, [tp(Z; #,,)

+z; #m+1)1. SO Fn(aa(l), aa(1)+1, RzO), Ti,o)+ly . . .) realizes

{[d% aN1)) = l q ~ ( Z , ~ a ( l ) + d I : 2 < n)

(when, e.g., a(Z) # a(k) + 1 for 1 # k.) Let for i < 2", U G p, I Ul < KO 6,(U) = B',,(. . . ,a,, aaa+i , . . .)asw(t)n-u where n = Iw(i) n Vl . So for a E U; a E w, iff qJ[b,(U), a,,] = TqJ[b,(U), Let D be an ultrafilter over W = Sn0(p), such that for any finite w, s p, {w E W : wo s w} E D ; and let M* = Mw/D, making the natural identification of a and (. . .,a,. . . ) / D ; and for i < 2u let 6, = (. . ., 6,(U), . . .)&D. For 8' c S let M(S') be the Skolem closure of Idl] u {6,: p , E 8') in M*. Clearly A c M(s ' ) , and each p , E 8' is realized in M ( 8 ) . Suppose p , $ s', but 6 = .?(as, b,,,,, . . . , 6,,,,) realizes p,, I EX. By assumption on the w,'s, there is a $ UISn wj(l), 2a $ 8, 2a + 1 $8, a E wl.

-

-

448 TEE NUMBER OF NOW-IBOMOBPEIO MODELS [OH. n I , 8 1

We ca,n assume that for every U E W, 6[U] = T(~z,, 6,,,,[Uj,.. .) 80

if a E U E W 6[[u1=T(af) where i ~ { 2 p , 2/?+1: / ~ E w ~ ( ~ ) , ZGn} U 8,

hence 2a, 2a+ 1 $t; hence MCtp(.?(at), aZz,) = tp(?(af), aZt2.+J so {YE W: k&U], ZSu) = q(6[U] , Baa+,)} 2 {U: a E v) E D hence M* C q(b, aSU) E q(6, aSU+,), 80 6 does not redhe pfy contradiction.

(4) Suppose T, I> T hasSkolem functions, lTll = No. If ID(T)I '> No by 11, 3.16 there are formulas q, , (E) E L for r) E 2<" such that (3Z)q , (P) E

contradictory. We use theorem MI, 3.7, Exercise 3.1 on T,, q,, (and its notation), we let S = {p:: r) E 2"). Now (Is, 0) is (T,, T)-free, aa for any x and S' c Is let (Ml, . . . , b,,, . . . ),,<" satisfies the conclusion of MI, 3.7 where (in M,) {b,: i < w } is an indiscernible sequence over {an: r) E 2"}, and Ma an elementmy extension of M1 in which {bt: i < x } is an indiscernible sequence over {a": r ) E 2"). Then let M(Is') be the Skolem closure of {an: p: E S'} u {bt: i < x}. Clearly each p: E Is' is realized by a,, E M(S'), and by VII, 3.7(4) no p: $8' is realized in M(S'). If T is unstable in No we choose A, IS(A)I > IAI = No, and proceed as before with T(A).

T ; v Q 11 (W(V"(@ --+ VY(W E T and (Pv-<o)@), VV-<l>(Z) a m

Clearly 1.6 and 1.4 have many conclusions, e.g.

CONCLUSION 1.7: (1) If T b not eujperstuble, P o 2 p 2 h + IT,l, 2u > 2" t h n I (p , T,, T) = 2 Y .

(2) If T , z T , T , countable, T No-unstubk then h 2 P o implies I(h, T,, T) 2 2a'0y and 2" > 2'0 2 h implies I(h, T,, T ) = 2".

THEOREM 1.8: Suppoee T, 2 T are countable, T K,-unstable 2'0 2 h > KO, then I(h, T,, T) = 2".

Proof. We assume T, has Skolem functions. If 2" > 2'0 or ID( T)I > No the conclusion follows from 1.7(2), 1.6 and 1.4(3); if T is not superstable by the next sections (Theorem 2.1). So assume T superstable, No-UIIBt&ble, 2" = 2'0 and ID(T)I = No. The first two assumptions imply, essentially by 111, 6.1 that there am equivalence relations En@; jj) E L with finitely many equivalence classes (in a) En+, refining g,, and E,, r) E 2<" such that r) Q v, r) E 2,, implies En@,,, 4) but r ) # v E 2" implies 4 & ( Z V , 4). Let q,,(Z) = En@; 4) where 4 will be individual constmta of T,.

Using VII, 3.7 (to T,), and renaming, there is a model M of T, and a,,, r) E 2" satisfjdng (1)-(6) from that theorem, and by renaming we

CH. mII, 8 11 INDEPENDENCE OF TYPES 449

get h is the identity. We can dso wume (for (*2) by taking a subtree of 2<" and renaming)

(*) For every T&,. . . , Z,,,) E L, there is n, such that for every n 2 nf:

(1) There are in L, individual consbnta c? i < ni re- presenting the E,,-equivdence classes such that

~ I d V l Y - * * Y f), $1 E r,, (see VII, 3.7(5) for the F,,'s).

(2) If +j is a sequence of length m of distinct members of 2", and the same holds for 1, and there am k, n s k < w , and +jl, 1, such that Z(+jl) = Z(1,) = my +j[Q 4 +jl[Q E 2OP,

lEk(t(?ZGl)y T @ ~ J ) , then this holds for k = n, and any such +jl, 1,.

C[Q Q FJZ] E 2", and +j1[21] = Sl[Za] 0 +j[ZJ = F[Za] and

Remark. We can first take care of (2) then of (1). Note that M C E,,(sin, a,) (7, v E 2") 8 7 1 n = v 1 n.

Now for any 8 c 2" let M,(8) be the Skolem closure of {an: 7 €8) and N(8) its L-reduct. Suppose f : M ( 8 ) -+ M(8*) is an elementary embedding; 8, 8* c 2" uncountable. For any v E 8 there am T = T~~

n = n, and +j = +jv of length n such that M,(8*) b f ( a v ) = ?,(af). We can msume +j[Z] # +j[k] for 2 # k. As 8 is uncountable there is an uncountable 8, c 19 and t, no, k,,

+jo such that:

(**) For any v E 8,, T, = T , n, = no, and +jv[a ko = ijOIZ] 1 Eo and +jo[Zl] 1 ko # +j&] 1 ko for Zl # 2,. We can wume also that n, (from (*)) is 5 k, and that for 7 # v E 8,, h(7, v) > k,, where we let h(7, v ) = max{n:q r n = urn}, so q(n) # v(n) when n = h(7, v) .

Supposep # v E 8, h@, V ) = n, 80 B(8) C En@,, a,) A TE,,+~(~, , a,).

-En[f(q,),f(q,)l. Suppose for every Zh(qv[Z], qp[Z]) # n, so ~ , , [ l ] r n = qp[l] r n implies q,,[Z] r (n+ 1) = qp[Z] r ( n + 1).

If for every Z h(+j,[Z], ij,[Z]) < n then by ( ~ 2 ) 74,+lLf(a,),f(~,)] implies

Now define a sequence +j: if +j,[Q I n # +j,[Z] I n t h +j[Q = +j,[Q, and otherwise +j[Q = +jv[Q. Let 6 = .?(a,); notice that by the suppoai- tion above q[Z] (n + 1) = +j,[l] 1 (n + l), hence by (*1) M(Pi*) C E,,+,(6,f(aD)), hence M(8*) C lE,,+1(6,f(Bv)). Now 6 = ~( iZ6) , f ( iZ , ) = ?(a;") and for any Z+j[Z] = +j$] or h(+j[Z],+j,[Q) < n (by +j's definition)

450 THB: NUMBER OF WON-ISOMORJ?EIC MODELS [CH. WI, f 1

and ko < 12, +$41 t ko # +a1 t k, S V [ 4 1 t ko # .)?v[JaI t ko for 4 + la (by (**)). SO by (*2), 4 , , + , ( b , f ( G i Y ) ) implies TEn(6, f (a;)) . But kE,,[6, f (a,,)] A EnV(a,),f(7iv)] contradiction.

So we proved that for any v # p ~8,, for some 1 h(fjv[Z], + j , , [ Z ] ) = h(v, P I .

Choose infinite subsets w,, i < 2N0, of o, such that for i # j Iw, n wjl c KO, and let 8, c {r] E 2@: $13 = 0 when I $ w,}, be a set of cardinality A; and M, = M(8,) . Clearly IIM,)I = A. Suppose i # j and there is an elementary embedding f : M, --+ M,. By what we have proved till now there is an uncountable 8; E 8‘ such that for r] # v €8; there are r]’ # v’ E 8 j such that h(r], v ) = h(r]’, v’). NOW w = {h(r], v) : r ] # v E S ~ } is infinite (as 8; is) and w is a subset of w, (by the definition of a) and is a subset of {h(r], v ) : r ] # v E 8j} (by the above) which is a subset of wj (by the definition of wj) . Hence Iw, n wjI 2 IwI = KO, contradiction. so

I(A, T,, T) 2 IE(A, TI, T) = 2n~ .

But we have restricted ourselves to the w e 2N0 = 2” in the beginning, so we finish.

QUE8TION 1.2: Is thereaparallelfor 1.8 when 2No 2 A > ITl! > No? (Add i f n e c e s q ID(T)I > lT1l).

THEOREM 1.9:8uppoae

(*I 2wo > A > No, and there is no family of 2No aubaeta of A, euch of power A, the intersection of any two of which iajnite. (This is just rtot AD(2N0, A, A, KO) , )

If T , 2 T, T superatable but not totally transcendental A 2 lTll, them I(A, T,, T) L 2%

Remarks: (1) Baumgartner [Ba 761 proved the consistency of “ A = K, satisfies (*)” and of “ A = K, does not satisfy (*)” with ZFC.

(2) The msumption on superstability is needed only for proving the existence of the En.

Proof. Let p 5 2no be regular. As in 1.8 we can find epivalence relations E,,(%,g) (n < o) and a,, r] ~2~ such that M , kE,,(ii,,,av) iff 7 n = v n; in some model M, of T,. Assume for simplicity, l(ii,,) = 1 so a,, = a,,.

Let, for B c 2”, M ( 8 ) be the Skolem closure of {a,,: r] ~ r 9 ) . As in 1.8 choose w, c w for i c P o , w, infinite but Iw, n w,I < KO for i # j , and

CH. VIII, 8 13 INDEPENDENCE OF TYPES 45 1

let Sf E {q E 2O: $11 = 0 for I # w,}, of caxdinality A, Mt = M(Sf). So IIMiII = A, Y: a model of T,, and let M, be the L-reduct of M i .

Suppose I(A, T, , T ) < 2No; as p s 2 h is regular, for some i W =

{ j: M j M,} has cardinality 2 p, and let fj : M, j. bl, be the isomorphism (we use only its being an elementary embedding). Let, for j E W, A, = {f,(a,): q EL$}. Clearly IA,I = A, A, G lMfl where llMfll = A; the number of the A,'s is p, and for a # p A, n A8 is h i t e (otherwise this contradicts Iw, n w81 < So). If 2No is regular choose p = 2No and we get a contradiction. Otherwise combine our conclusion on all p and by 3.1 of the Appendix we get a contradiction to (*).

THEOREM 1.10: 8uppoee T s T,, ITl[ = A, and there is a family {tp,(Z): i < A} of A independent fomnulae of L(T) (i.e., for every $nite w E A and h : w j. (0, l}, T I- (33) AfEW tpf(3)h(f)). Suppose J is a subtree of 2sA (closed under initial eqm.enta) such that, letting J , = J n 2", for EL < A, IJ,I c A, and x = IJAI (for x > A , the interesting m e , this is a Kurep tree). T h

(1) For eweryp 2 A,IE(p, T,, T ) 2 min{2",2X}.Moreower i f (3u)(u < A < 21"1 A 1011<"(O) = loll), there are, for each p = p<"(O) 2 A, min{2",2X} L(T)-models, the reducts of ~(0)-compact models of T,, of cardinality p, m one elementarily embeddable in another.

(2) There are M, C T, , g,, E M l ( q E JA), zf E M , (i < A ) such that (A) For each formula P)(z~, ..., Z~~~),~,...,X~(~))EL(T~) there is

a, < A such that if v,, . . . , vn(,), ql, . . . , qn(,) E JA, p 2 a, and vl 1 fl =

71 rpbutforl # k, vl rp # vk lg, then

M1 C ~ @ v , , - * * 9 #vn(l), 21, * - * 9 zn(aJ tp&ql, * * - 9 #vncl), 21, - * - 9 zn(,)l. ( U ) {zl : i < p} is an indiscernible sequence ower {Yq : 7 E JA}. ( C ) For any v E J,, a < A, there ie tpv(Z) E {tpf(3): i < A} such that

( D ) For every term T(Z, ..., z,,, xl, ..., xm) there is a 01, < A, such that if rll, . . . , qn+, E JA, p 2 % and q, rp ( I = 1, n + 1) arepairwiee distinct, then ~(g,,, . . . , &, zl, . . . , z,) doea not realize I ) , , ~ + ~ , where

v Q 7 E JA implies Mi C V~&,J"(~).

p,, = {cpf(P)t: Ml != Vf@,It, i < A, t E (0, 1)).

( E ) For each K < A such that 2" = A choose a good ultraJlter D, ower K , and assume (Vu < A ) ( ~ U ~ < ~ ( O ) < A ) (and ~ ( 0 ) is regular). We can assume that i fS E JA, IS1 < K ( O ) , M; is the Skolem Hull of {q , , :v~S} U {q:i < K ( O ) } , V E J n - S then pv is not realized in MICs(O), where M i ( K < ~ ( 0 ) ) is increasing and continuous, M", = M,, M"! is isomorphic to (M,)"/D, ower M,.

452 THE WUWBER OF NON-ISOMORPHI0 MODELS [OH. an, 5 1

Remark. When does such a tree exist? If A is strong limit (i.e., a*) J = 2sA wi l l serve, 80 x = 2A. Clearly J c 2s* is not really ncessary, it suffices J c asB, cfg = cfh, y < g+ lJYl < A, x = IJ,I. So if of A = No, J = n,, A,,, where A,, < A = z,,<a A,, is sufficient, 80

x = A h > A.

Instead assume J is a A-Kurepa tree (i.e., a < A => I J,I < A), i t suffices to assume : there are Ji c J,(a < A), I u, < JiI < A for ,8 < A, and for every distinct ql, ..., qn E J,, {a < A: ql la, .. ., vn ra are in c} is unbounded (for part (E) we should replace n < o by K < ~ ( 0 ) ) . For x = 2A this follows from 0, (A regular), and if A = ACA it holds with x = A+.

Proof. ( 1 ) Follows by taking Skolem hulls of {tj,:qcq U { z i : i < p} for J c JA, using (2) (b) (the K(0)-compact case by (2) (E)), using 1.4.

( 2 ) Case I: There is K, K < A < 2".

Let 8 = {w(i): i < A} be an independent family of subsets of p (see 1.5 of the Appendix). For j < K let a, realize {q+(~)if~Ew(i)):i < A}, let I = {a, :j < K}, and let M , be a model of T, , a5 E (2M,(, llM,II = p and {z, :i < p} be an indiscernible sequence over U,,,it, (inM,). We define by induction on a < A, for each r] E J n 2" a filter D, over K, and a set a,, E 8 independent mod D,,, (see Definition VI, 3.1) such that 18 - 8,J < K + 11(7)1; and v 4 7 implies D, E D,, S, G S,. For Y E JA let D, be a completion of A D,, I (I to an ultrafilter. Let M , be a (x + A) + - saturated model of T,, No 4 M I , and for each r) E JAY a,, will realize Av(Di, B,,) where B,, = U {&: v < r ) (in lexicographic order) v E JA} U U5<r a, u {zt: i < p}, D,, = {{a,:j~ W): W E D,,}. It should be clear that the only nontrivial point is (and parts (C) (D) (E) are left to the reader) :

CLAIM 1 . 1 1 : Let A, K, S, I, M,, be as above, and let D,, . . . , D, be $filters over K, S, independent modD,, S, c S, and FEW,^, cpcL(T1). Then we can find fltm8 D:, and families 8: m h that

(A) D, c 0:; (B) Si C S,, IS,-Sil < K ; and for 1 = l ,n ,S t is independent

(C) There is tE{O, l } , such that { ( i ( l ) ,..., i ( n ) ) : i ( l ) , ..., i (n) < K,

mod 0: ;

MI t v[Gf,,), . . . , at,,,); ZJt} E D: x 0: x * * x 0:.

CH. VIII, Q 13 INDEPENDENCE OF TYPES 453

Prmf. We prove by induction on n.

induction on a d K , Dt,a, St,a for 1 d I < n such that For n = 1 this follows by VI, 3.3. For m = n + 1, we define by

(1) Dl E DlSB E Dl,, for B < a. (2) (3) Sl,, is independent Dl,,. (4) There is t(a) E (0, 1) such that

E Sl,p E S,, lSl-Sl,al < K for B < a.

{<i(i), ..., i (n)) :i(i), ..., < K , &?I kq[a,(,,, ..., a,(,,, 8a,f?~lf'u')

E Dl, , x * - * x D,,,.

We can do it by the induction hypothesis on n. Let D! = Dl,B (1 I I 5 n), and let Dk,Sk be such that D, C Dk,Sk E S,, (S,-Skl d K , Sk is independent modD;, and for some te{O, i}, {a < K : t(a) = t}ED;. Clearly this proves the olaim, henoe Claim I.

Case 11: A > No, not I; so h is strong limit. Here in the induction step- a, we have a set r, (a < A) of formulas of L, in the variables gfl ( r ) E J,) zf(i < w ) and set U, c A, s 1.1 + No, a model M! of T,, and sequenoes 6; ( r ) E J,, V s A - U,) and cf E My (i < A) such that:

(i) for every p)(gfln,, . . ., g,,,, z,, . . ., 2,) E Fa, and V(l), . . ., V(n) = A - U,, Mf k p)[6ri1), . . . , 6;>), o,, . . . , c,]; and M! C p)f(6fl)u(fev) for i E A - u,, v s A - u,,

(ii) {cf: i < w } is indiscernible over {E:: r ) E J,, V E A - U}. The connection between r, and r, (a < 8) is that p)(g,,,,, . . . , &, zl,. . . , z,) E r, the ql distinct, q1 Q v1 EJ, implies p)(gVl,. . . , &,, z,, . . . , z,) E r,. Again the point is the assertion parallel to claim. More exactly we want to add p)(Zfll, . , , , Zflnn, zl, . . . , z,) or its negation, extend U, by sx = 1.1 + No indices, and preserve (i) and (ii). Let N be the model (E1((2"+), E) expanded by individual constant for M? (assuming w.l.0.g. 11JfyII I 2')s ((7, $,): 7 E J,}, {(v, J', 6;): - u,}, {zf: i < A}, r. Our desire can be expressed by an (L(N)),+,,+-sentenoe, hence if it fails, it fails in some N , < N , llNlll = 2,, llNl n All = 2 5 and b s lNll A lbl s x =. b E N , . But then we can apply the claim, and find the truth value of p), and the subset of A - U, of cardinality 5 x we want, contrctdiction.

E Jh, v =

Case 111: A = No. Thie follows by 1.6.

THEOREM 1.11: Swppwe (*) from 1.9, A 2 IT,l, T stable not q e r - stable. Then I (& TI, T) 2 2Ko.

454 THE NUMBER OF NON-ISOMORPHIC MODELS [OH. WII, 6 1

Remurk. We a n omit "T stable" by 3.4 and replace 2% by 2" by 1.7(1), and replace "not superstable" by "not totally transcendental" by 1.9.

Proof. Let M = EM (AscD, @) where @ is as in 1.6(2) and w.1.o.g. we can aasume that if A = u {a,: q E A<@}, A, = UnccD a,,,, dim(p,,, A,, M') is A, where M' = EH(A<OD u {q}, @), p,, = stp(a,,,A,) for 7 E A@, and I),, is stationary, and let I,, c itl be an indiscernible set based over I),,. We can assume also that L(2') = UntcD A , ( A , finite and incredg) and

Iw, n w,l c No, and let Si be a subset of { ~ , ~ ~ 2 ~ : n # w , + r ] [ n ] = 0) of cardinality A, and let Mi = EM(A<(O u S,, 0). Suppose A < p 5; 2*0, p regular and

tp, ( ~ , , , E M ( A < ~ , = tp,,(av,v,qA<y a)) iffq rn = v rn. dhoose w, E o (i < 2N0) i # j

u = {j < 2%: N, 2 M,(,)} has cardinality 2 p, and let f, : 111, + M,(,, be the isomorphism. Let M,,,, = U,,ABa, B" increasing, 11BU11 < A . For each j~ V there is a ( j ) < A such thatf, maps A, ( q ~ 2 < " ) intoB"(-", provided that cfA > M,, hence for some /3, U' = {j E U: a(j) = /3)} has cardinality ~ p . For each j e U' and r ] ES, choose q ~ f , ( I , , ) such that tp(8, lBll) does not fork overf,(A,,). So for somej(1) # j(2) E U'

{6(1): 9 E B,,,)} n {5{(%): 9 E 8j,2)}

is infinite. But clearly for ~ E U ' , tp,,(?$B!) = tp(&Bl)iffq rn = v n, hence we get a contradiotion as in 1.8 and 1.9. So aasume cf A = X,, A = Znem A,,, A,, < A. For each n, j there is a(j, n) < A such that

l{q €8,: Ifj(In) n dla('sn)I 2 Xo}l 2 A,,.

We oan also find P(n) such that Un = { j E U: a(j, n) = /3(n)} has cardinality zp. So as before we get AD&, A, A,,, No), and aa thie holds for each n, p it is easy to prove AD(Po, A, A, X,), i.e., not (*) of 1.9.

QUE8TION 1.3: Can we in 1.4 deduce something about I E ?

EXERCISE 1.4: Prove in 1.9 and 1.11 that if 2No is a singular oardinal then IB(A, TI, T) 2 2% and if 2b is regular, x < 2w0, and in (*) there is no such family of cardinality x then IE(A, T,, T) 2 2Wo.

QUESTION 1.5; Can we prove in 1.8 that IE(A, TI, T) = 2A? (The remaining case is: T superstable, KO-unstable, 2N1 > 2'0, and ID(T)I = KO.)

QUESTION 1.6: Try to improve 1.7(2) to results on IE(p, TI, T).

EXERCISE 1.7: Suppose T has (A, p)-freedom by one of the cases 1.5 A 2 IT,\, A = 2 p = A". Then IE(A, T I , T ) = 2A.

Remark. Instead 1.5 it may suffice to assume that in Definition 1 . 1 , there is a model M , 2 A, and a P € M realizes p, {b , :i < w ) is indiscernible over A U {a, : p E S ) , and for each p, p is not realized in the Skolem Hull of A U {ag : q # p) U {b, : i < w).

EXERCISE 1.8: Rewrite the proof of 1.8 where you prove only that for any v + p E S1 for some lh(v, p) 5 h(lj,[l]), .ii,[l]) I; h(v, p) + 100, assume (*1) holds for even n's, (*2) for odd n. Is it simpler ?

EXERCI8E 1.9: Show that in Definition 1 .l, B and A can be chosen independently of the cardinality.

Our main theorem here is

THEOREM 2.1: I(A, TI, T) = 2"owidd that t h following condition holds:

(*I T is urcswperstable, A 2 ITII + K1 and at leaat one of the following me8 occurs: (i) A > I TI/; (ii) Po = A; (iii) A =

x,,, A,, A, < n, A> = A,.

Proof. By the subsequent theorems. If A is regular by 2.2 for p = A. Also if there is p < A, 2" = 2% 2.2 implies our conclusion. If there is p < A 5 pNo, 2" < 2Qhe result follows from 1.7(1). If none of the previous cases occur end there is p < A, 2Y 2 A, 2.3 gives our result. If p < A =- 2' < A, but A is singular, there is alwaya X, of A 5 x < A, x = KO or x a strong limit cardinal of cofinality KO. Noticing that Aof" Theorem 2.6(1) gives our result. We covered all possibilities, thus prove the theorem.

When (*) (from 2.1) holds there is a model No of cardinality ITII, proper for (us", TI), such that the skeleton of EM1(ws", No) is a uniform KO-tree of the form <(cpn, 1): n < w), vn E L (by VII, 3.6(2) and 3.5(2)); let cp,, = cpn(x; gn) = vn(J; 8). In Case (ii) we can aaaume

456 TEE NUMBER OF NON-ISOMORPHIC MODELS [a. VID, 8 2

N o is K,-compact; (by I, 1.7, and as we can replace N o by any elementarily equivalent model of the same cardinality). In Case (iii), by VII, 1.11 we can assume there are P, E INol, (n < w ) P, E P,+,, lNol = Un<o, P, such that N , = (No, Po, P,, . . .) satisfies: if a countable type over N, is finitely satisfiable in P,, then it is realized in P, (i.e., by a finite sequence from P,). In Cases (i) and (ii) let P, = INol, N , = (No, P o , . . .). In all cases let L, be a countable sublanguage of L(N,), such that P, E L,, and the formulas Re,cp,r EL, when 0 E L(w*”), tp E {tp,: n < o}, T E LE. We use those conventions in this section, except in 2.7.

THEOREM 2.2: Suppocre T is not auperstuble. (1) If A 2 p > lT1l, p regular then IE(A, T,, T ) 2 2’. (2) If (*) (ii) or (iii) ho& No < p 5 A, pregular, t 7 m I(A, T, , T ) 2 2”.

Proof. For every ordinal 8 < p cf 8 = No choose a (strictly) increasing sequence of ordinals 7s = $8) whose limit is 8. For every w E p let I ,

cardinal addition, so in (ii) and (iii) the third psrt disappears). Let ilfk = EM1(Iw, No) and M, be the L-reduct of Mt, (we use the notation of 1.1). By 1.3 of the Appendix there are pairwise disjoint stationary subsets of p, u, E (6 < p: cf 8 = KO} (i < p), and by 1.5 of the Appendix there are B(a) E p, (a < 2’) such that #(a) G 8(p) * a = p. Let ~ ( a ) = u,. Our family of models is {MWca,: a < 2’). To prove that it exemplifies our conclusion, it s d c e s to prove: supposing w , u c { & < p : c f 8 = No},w-uu-warestationary,andf:ilf,+ilfu is an isomorphism for (2), and for (1) an elementary embedding, we get a contradiction (“ u - w stationary ” is not needed for (1) and can be weakened to “A-w stationary” for (2)).

Clearly for every q E I , there are T,, E LE, F(q) = F, E p”” n I , and

be the KO-tl’Wp<” U {qd: 8 E W } u {(a): p 4- IT11 5 a < A} (p -I- lT1l-

I ( q ) = I, E {(a): p + IT11 s a < A}, En E No such that

Now we can define by induction on i c p, A, c INol, a ( i ) < p and X, _C { (p): p + ITl] 5 p < A} such that:

(A) X,, A, are increasing (by G ) and continuous sequence in i.

(C) a( i ) is a (strictly) increasing and continuous sequence, a(i) 2 i. (D) I f y E l , n uB<I/3Go, thenc9[Z]EXi,c9EA,, and F9[Z]~Up<a(o /3SW.

(B) 1-q < p, 141 < p, and if IT11 < tc, A, = INol.

CH. MII, 8 21 UNSUPERSTABLE THEORIES 457

(E) If r ) ~ a ( i + l)<* and f l IT11 =- % = %-<Y>*

(/3) i+ N $n<y> modX1 (in I,, so, in fact, in the ordered set

( y ) fi,,n^ N +<y> mod a(i)*".

It is easy to define a ( i ) inductively; for i + 1 notice that the demand is only that in the Skolem closure ofA,+l U { q , : v d w n ul<o(t+l)jqw} U X,+, there will be some < y elements, and as y > KO is regular, clearly such a(i+ 1) exists; for E) notice that I, is X-atomically stable for any 2.

Now clearly S = {i < p: j + + IT11 5 a < w. (6) +,,n^ = + V < Y )

Prooffor (1). (p > lT1l). Choose 6 E (w - u) n S. As 6 4 u, r), 4 I,, hence if u1 = ij,,,[Z] E p", then (as it is increasing)

there is at < 6 such that vt[n] < al e v1[n] < 6 e- n < nt where nl s w.

If v1 = ind[l] ~ p < * there is a, < 6 such that q[n] < a, ov l [n] < 6. Let a* = g(6) be < 6, 2 at and such that x1 = min{z E x d : 2 anna[l]} E X,. or x1 = a0 (exists as Xd = u{ < ,, X1). By the dehition of S there me 8', n; a* < a( i ) < 6, r ) d [ r n ] < a( i ) for rn < n but ?la[?&] > a(i + 1). Let p = v8 n, /3 = v8[n], then by (E) there are, for m < .w, distinct y(m), u* < a( i ) 5 y(m) < a(i + 1) < fl , such that ZDn<y(m)> - gDn mod Xi,

= ZDn<B) and Spn<v(m)> N ijD-<,,> mod a(i)$* and +p<ycm)) =

T ~ - < ~ > ; hence SDn<y(mn)) - 3D-<B) mod S,,, (by the definition of a*). Hence, by the indiscernibility of the skeleton M,~~,+l[f(~v~),,),f(~~-~ycm,>)l for m < w, hence Mw C g~,,+~[ii,,,; iiDA<y(m)J (as f is an elementary embedding); contradicting the definition of a uniform KO-tree of the form ((v,, 1):n < w). So we proved (1) as no sequence in the model realizes Q ) ~ + ~ ( Z ; ~ ~ - < ~ ) ) for infinitely many y's (see VII, 3.6(4)).

Proof of (2). As p 5 I T,I clearly Xi = 0. We define g as in the proof of (1). As (w-u) n SisastationarysubsetofyandforeachSinitg(8) < 6, by 1.3 of the Appendix for some a* < y , wo = { 6 : 6 ~ ( w - u ) n S, g(6) = q} is stationary. As the number of r ) E (a*)'* n I,, is <p; and if we partition a stationary set to < p parts, at leaat one is stationary (see 1.2 of the Appendix), there is a stationary w1 E w,, such that for all 8~ w1 : = T is constant (as Tv(8) E LE, LE is countable) ; ( F7(8)[Z]) r n is constant or always 4 (a*)s*; the similarity type of $,,(a) is constant;

458 T E E NUMBER OB NCYN-ISOMORPHIC MODEL8 [OH. VIII, 8 2

and there is a constant n(6) = n(*) such that ZMa, E Pno,(N,). Clearly we can find a closed unbounded 8, G 8 such that

(1) if a 7 E a<” and there is 6 E w1 such that 7 Q 7 6 then there is such 6 < a.

(2) if a E S1, 7 E a<@ and (6 E w1 : 7 Q 76) is s t a t ~ a n q then there are v E a<”, with arbitrarily large (<a) last element such that (6 < p: 7 - y Q vd} is stationary (the existence of such v’s follows from 1.3 of the Appendix).

(3) If SES, n wl,r]Ea<W,r]Q rs, then SEW,:^] Q q6} is stationary. So for each limit point S* of S,, cf S* = No, we can easily find

a strictly increasing 7 of length w with limit 6*, and S(n) < a*, 6(n) E

wl, and S(n) < P(n) < S(n + 1 ) where P(n) €8, ; such that 7 r n =

r]b(n) r n (define S(n), /3(n) simultaneously by induction on n w , using the properties ofS,). Choose also S > S*, SE wl. Then we can check that

are similar, by a*’s definition). Now choose E E P,(*)(N,) which satisfies every formula of L, all but finitely many Zv(d(n)) satisfy (a exists by N1’s definition). So clearly ~(7i~(,,,,,,,, E) realizes {rp,(E, f (8v,m)): m < w}. As in (2) f is an isomorphism, in M , {vm(E; Bqrm): m < w } is realized. Hence S*EW. As S* was arbitrary S, G w. So we finish the proof of (2) too.

?(&(,d)), Zv(d(n))) satisfies f (%$tI)): < (as Gvlmhfiv(d), FnrmhGMd(n))

THEOREM 213: Suppose (*) from 2.1, and that p 5 h is regular; h > IT11 * p > lTll; 2 Y 2 A, p > KO; and x 0 ) ( ~ [ n ] < p)} u (7: 7 E A”; n > O-r[n] < v[n+i] < p ; 7’s limit is 6 and S ~ t @ [ r ] [ O l ] } .

Let Mi = EM1(&,, No) and MG its L-reduct. Choose for each 6 < p, cf 6 = Ho a (striotly) increasing sequence of ordinals of length w, 78, whose limit is 6, and 7d[o] = 0. For a sequence 7 define q* such that 7 = (7[0])-7*. Let u,, i < p, be pairwise disjoint stationary subsets of (6 < p: cf 6 = No}, and {#(a): a < 2”) be a family of subsets of p, no one of which includes an intersection of finitely many others. (See 1.3 and 1.5 of the Appendix for existence). Let {ui: i c 2A} be a family of subsets of A, no one a subset of the other. Let a, be a sequence of length h whose range is u,: a < A, a E u,}. Our family is {ME,,,: i < 2”).

So it suffices to prove that iff is an isomorphism from ME onto Mg then for every i < h there are n < w ; j,, . . . , j, < A and closed un-

CH. n I I , 8 21 UNSUPERSTABLE THEORIES 459

bounded8 E p such that @[i] 2 @[jl] n . - . n @[j,,] n 8. For simplicity let i = 0, p > IT,\ and let f(a7) = T7(ti&(7),~7) for YE&,, (so i$ = F(7) E I,, Z,, E No) . Now define by induction on i < p, ordinals a( i ) and sets Xi such that:

(A) Xi is an increasing and continuous (by G ) sequence of subsets of h of cardinality <p.

(B) a ( i ) is a (strictly) increasing and continuous sequence of ordinals <P.

(C) If 7[0] = 0, 7 ~ 1 ~ n U,,,pSu then (F7[Z])[O]~X,, and (F#])*E B“” (this is possible as x < p * X’O < p) . We know (see 1.2 and

3 of the Appendix) that if w c p is stationary then if g is a function from w to a set of cardinality < p then for some t , {OIE w:g(a) = t} is stationary ; and if g is a function from w, g(a) EX, ;X, is an increasing continuous sequence of sets of cardinality < p then for some t, {OIE w:g(a) = t} is stationary. By successive use of this we can find a stationary w1 E 301 and a( *) < p such that :

(a) For d 6 E w1 T,,(d) == ?, Z,,(d) = c. (p) For all SE wl, v7(&) has the same similarity type. (7) For all ~ E W , for each Z,Bj = (FvO[Z])[O] is constant and EX=(*)

or always it # Xd, but min {fi E xd: f l > fij} is fixed (maybe as GO). Let

(8) Let 4 = (fiMd)[i])*. Then for some n,, vf I n , is constant and E a(*)s”, and vi[n,], if defined, is 2 8.

Now, &B in the proof of 2.2(2) we can find a closed unbounded 8 E p, such that for every 8 E 8 n Qj,] n. . n @[jn] for some 7 E pa, 7[0] = 0, 7 strictly increasing with limit S, {vn(~,f(lZvr,J) : n < w } is realized in Mu, hence 8 EW[O], what we want to prove (if p s lTll the changes are like the proof of 2.2(2) and we can have llN,II = JTJ’o).

pi: 0 n Xam be (jl, - * ’ , jn}.

DEFINITION 2.1 : For models M, N cardinal x and ordinal a we define a game GE;(M, N) [GI;(M, N)] between the players I and I1 as follows: in the @h move player I chooses i, < x and af E M i < i,, and then player I1 chooses bf E N, for i < i,. The play ends after a moves, and then player I1 wins if tp*({bf:i < ifl,/3 < a},f3,N) = tp,({uf:i < i,,

[TPP({bf: i < i,, f i < a}, 0, N ) = TPSX({af: i < i,, /3 < a}, 0, M ) ]

and player I wins otherwise. A strategy (of a player) is a sequence of functionsf, (fi < a) which “tells” him what to do (f, for the fih move) depending only on the previous choices in the play. A winning strategy

p < 4, B,M),

460 "HE NUMBER OF NON-ISOMOBPHIO MODELS [OH. WII, 8 2

is a strategy such that in any play in which the player "behaves" according to it, he wins. A player wins in the game if he has a winning strategy.

Remark. Sometimes we denote that player I chooses sequences Zf instead of elements.

LEMMA 2.4: (1) If there is an elementary embedding f of M into N ; then player I1 wins in QE,O(M, N ) .

(2) If M , N are isomorphic, pluyer I1 wins in QI;(M, N ) . ( 3 ) In the gamea from DeJinitim 2.1, at most one player wins. (4) If player I1 wins in GX$(M,N) he wins in GXI;(M, N) where

( 5 ) If X E {By I } , player I1 wins in QX;(N,, M I + , ) 1 = 1, 2 then /3 2 a, X E { I , E} .

player I1 wins in QX:(M,, Ma).

Proof. (1) In the pth move player I1 chooses be = f (af) . Also the other proofs are immediate.

To clarify the proof of 2.6, we prove first:

LEMMA 2.5: Suppose A = Ano 2 I Tl l , Mi are models of T of cardinality 5 A, for i < A. 8uppo8e T i s unswperetable. Then there is N E PC( T,, T) such thut player I Win8 in C#I&(N, Mi) for m h i c A.

Proof. Let g be a one-to-one function from {(i, a): i < A, si E lMil} onto A. Let g(g l (a ) , g,(a)) = a for a < A, and

I = A'" U {r] E A": gl(q[n]) = i for every n, and

g0(7[n1) E 1 ~ ~ 1 and {vn(s; ~ ~ ( q r n i ) ) : < w>

is (well defined and) omitted by M,}

and let N = E M ( I , N o ) (No-proper for (us", T,) and of cardinality lTll). Let us show how player I wins in GI$(N, Mi): for p = 0 player I chooses the sequence a,, r ] = qo = ( ); for p = n + 1, if in the mth move player I1 chooses 6m for 11c 5 n, player I now chooses a,,, r ] = v,+, = ( g ( i , 6O), . . . , g(i, 6")). If in such play player I1 wins, necessarily N realizesp, = {v,(z; a,,,): n < w } iff Mi realizes {v,,(s; gn): n < w}, (letting Y E A", r ] , 4 Y); then N realizes pv iff Y E I iff Mi omits {v,(z;gz(v[n])) : n < o} = {q,(z; 6"): n < o}, contradiction.

CH. VIII, Q 21 UNSUPERSTABLE THEORIES 46 1

THEOREM 2.6: S p e A 2 p = fl = 21, (VK < A ) ( K X s A) and A 2 lTll. Then:

(1) I ( A y Ti, T ) 2 A". (2) There are models Mh E PC(T1, T ) fOr h : p --+ A M h that if fop 8Orne

i hl(i) < h,(i) then in aI:(Mhly Mh1) player I wim.

Proof. Clearly (2) implies (1). By the assumptions on A we can assume No,Nl are as mentioned in 2.l(ii) or (iii) except that maybe lliV0II > IT,I (but llNoII s A). We can assume x = zncU xn, xto = xn or x = KO and let xn = n. We define by induction on a < A pairwise disjoint well- ordered sets JL (i < p), lJ6.1 = IaIx + p s A, and for a > 0 the universe of JL is the set of quadruples 8 = (a, 5, u, y ) where 5 < x and u is a subset of (J;)"" of cardinality < x and y < a, and let gl(s) = a, g, (s) = 5, g3(8) = u, g4(s) = y. For a = 0, we let J i = ((0, g, i, - 1) : 5 < x} and, to simplify notation, iet distinct Ji's have distinct empty subsets. The well ordering is arbitrary. For ~ E ( J ; ) ~ " let His(q) be the smallest set S such that ~ E S , and V E S , n < Z ( v ) , p ~ g ~ ( v [ n ] ) = s - p ~ S . Let His*(q) = {~[n]: Y E S , n < w } . Let DP(q) be the order type of T(q) = {p:His*(q) n J i # (4 for some i or gJs) = /3 for sEHis*(q)}. Clearly His*(q), His(q) have cardinality < x hence DP(q) < x+.

WeshalldefinesetsSI c xmfori < p,y < X+.Nowlet,forh:p+A,

and g d r ) ) E sr where y = DP(r))}, where we define ga(7) by ga(q)[Q = J h = &<pJk({)YandIh = uf<# ( Jk({) )c"~{r) : for~om~~ < /-br)E(Jk(t))mY

g,(r)[l]); and let Mh = E B ( I h y No) . SUppOSe hl ( i ) < h2(i) = a0 and We describe the winning Stl'abgY Of player 1 in aIj;)(Mhl, Mhl).

In the nth move player I chooses {a,,: r) E u,,} which is defined by induction as follows: let player I1 choose in the rnth move {&,: V,I E am}; and let 6,, = .fn(&m, E,,), F,, E Ih,y En E No, and let

v,,, = (J;,ct,)sa n {;,,[I]: 1 < I(;,,), E u,,,}.

Let uo = { ( (Uo , t, 0, hdi))): 5 < XO}, and

un+1= (7 -(<a09 CS,vn, h(i))) CS < Xn+1, VEUn}.

Clearly the strategy is well defined and the nth move of the player I depends only on what player I1 haa done in the previous moves. Let the history R of this play be (J {His(Sn[Q): r) E Uncm u,, I < I(; , ,)}; (Clearly lEIl 5 x) R* = U {His*(;,,[l]): r) E Uncm u,,, I < I(;,,)). So it suffices to prove we can define the Sr's so that those strategies me winning strategies for player I.

Suppose we have another such play, (where player I plays by the

strategy we mention) with hi, . . . ,Ek, . . . , H ' , instead of h,, . . . , En, . . . ,H. We call the two plays isomorphic if there are functions f : H + H' fY(y < p) with domain {a: JL n H , # 0 or a = g&) for 8 E Ha}, f*: H*+H*,gn:un+u6 such that:

(1) f, fa, g,, are one-to-one and onto H', Ha, u;, resp., (2) If 8 E Js, f(8) E J ~ , ( I ) and f, is inCreeg. (3) If s = (B, 5, u, a> E JJ then fb) = (fy(B), S,f*(u),fy(a)> where

(4) (f(s))[nl = f*(r)bI)Y f @ o ) = 4Y i = i'. ( 5 ) s0(((a0, 6,6, h,( i )>>) = ((a;, 5, B , h ; m > , and

(6) If r) E u,, f*(%$I) = q l where p = g"(r)).

CM,,), E;(o)- * - * h-'

f* (4 = {f* (0: t E 4.

gn+l(rlh<(a0, 5, V,,, hdi)))) = gn(Vr<<4Y 5 , 4 , Y(i')>>.

(7) If q(Z) EU,(~) Z s n then; denoting p(Z) = g,,,(ll(r)(Z)); EM0,^. - - (8) f* r J; : J; -+ JJYu) is order preserving ; moreover if sl, s2 E J; n

cp(,,) realize the same type in the L2-reduct of N,. n-

Domf,, then

&{&Y I{t EJ?: 81 S t < 82}1} = &{&I, [{t EJSI0:f*(81) 4 t < f*(82)}l}.

Clearly, by the dewtion of the Ih's, the winner in two isamorphic games is the same (as the La-reduct of iVl is 8,-homogeneous). It is also olear that the number of isomorphiam types of such playa is s 2x = p. Let {at: j < p} be a set of representatives. We define by induction on j, a set I', of non-oontradictory "requirements " E 8r, $ 8r, euoh that II', - I'ol s lj I + x and I',+, c'ensure"' the victory of player I in the play @.

is eventuaJly zero}. If I', is defined, let us use for at the notation in the definition of the strategy of player I. Chooae r ) E n,,<, x,, suoh that r ) does not "appeaz" in I', (there is one by cardinality wnsiderahions). Let r)o be defined by yo 1 n E

Ua, and ( . . . , g2(r)O[m]), . . . ) = r), and let p = {V,,(Z; 7in,r,,): n < o}, and

Choose, if possible, 8 t ' s so that they saw the requirement of I', and SO that q is well defined and *, in a6,, say by ~(6, 5); note 7 0 1 71 E Ihl- Then define, when y = DP(q0) I',+ 1 = I', U {q $ &} U @ E 8t: for some m < Z(3), Z(C[m]) = o, p = ga(F[m]), f l = DP(G[m]) and P[m] E Uaeh (J:)<,}. There is no contradiction tw we can aaume, in the third term of the union above, that 6 = i =s fl < y beculuse if Z(f[m]) is w, g,(fi[m]) E 8f then ( ~ [ m ] ) t Z E Uk<, uk for every Z < w [otherwise we wn make P[m] eventually zero but stil l T ( ~ , E ) will

Let To be (7 E&: r ) E xm,

q { ~ n ( ~ s &o,n): n < w}, and Y = DP(to), i = B ~ ( ~ [ o I ) *

OH. WII, 8 21 UNSUPERSTABLE THEORIES 463

r e h e q] hence by the strategy definitions, Hie*(r)) z Hb+(i@]), a. = max T(q) 4 Hk*(c[m]), so /3 < y.

So F,+, ensurea q is realiz;ed in H,,,, but p is not realized in M,,, hence player I wins. But if there are no such #$'a, let F,,, = F, u (7 E For limit 6, F, = Uje6 F,; so Fu shows us how to define the 8's properly:

s; = {?j:{T/€AS;}Eq}.

Pro$ The proof is similar to that of 2.2.

By 111, Exercise 4.13 we can find a,, E

(1) ForqEpm,vEpn, (la > 0) C 9 7 , , ~ , , ; t Z v 3 i E v 4 r ) .

(2) For q E pn, the Q,, + ,(Z; (3) For r) cpSm, tp(tZ,,, A,') does not fork over A: where A: =

U {G,,,,,: ta < Z(r))}, Al, = u (8": v # q but not q Q v}.

Let, for S < y, cfS = w , 7, be an increasing sequence of length w of ordinals with limit 6, and for w E {S < p : cfS = KO} let M , be a F&-constructible model over A, = Ck of cardinality A, where CL={cT7: ~ ] E U ( " or 7 =yd , ~ E W , &<a}. Let the construction- sequence of M , be {ar:i < i(0) < A+}. (See table in IV, 2 and IV, 3.1). Let A: = (a?: i < a}, and let E A, U A: be a finite set over which tp(a,",A, U A;) does not fork. So if (Va < a)@," c Cb, U A",, 6 < p, cf 6 = o, 6 4 w, r), € p S w , q, t n, E S S W but n, = o or r), 1 (n,+l)$Scu for Z < m ; then C = U { a v : v Q ~ z ~ n z , Z < m , or v = r ] , n, = w } is finite, and tp(Eq,Cb,) does not fork over C, where r] = (q0 , ..., T ~ - . ~ > , hence, by 111, 4.13, and as B," c Cb, U A;, also tp(cTq,Cb, U Ab,) does not fork over C. Now by IV, 3.2 if ~ E A , U Ag there is finite B c A, U Ab, such that tp(b,A, U Ab,) does not fork over B, and there is ~ E , U ( O such that B n A, c aq, %nd there is finite C as above; so tp,(6 U B, Cb, U A;) , hence tp(b, C: U A:) does not fork over C U (B n (Cb, U A;)) . So for every ~ E A , U AL tp(b,Cb, U Ab,) is F&-isolated. Ae in 2.2 it s&ces to prove that i f f : &Iw -+ Mu is an elementary

embedding, then for some closed unbounded8 s p, w n 8 E u n 8. By renaming we can aasume f maps A, u A5 onto A, u At (see IV, 3.3), hence 8, = (6: 6 < p, f maps Cd, u A: onto 0: u A:}, 8 = (6 ~ 8 ~ :

for r] ~ y ~ " and pn E L such that:

i < p are pairwiae contradiofory.

464 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. nII , 8 3

6 a limit point of S,} are closed and unbounded. If 6 E w n S but 6 $ u, then 6 = .f(iZnc6J will give us a contradiction, as tp(6, C$ u Ad,) splits strongly over any CE u A4, for any a c 6 (because {ii,,-(t>: 6 I i c p} is indiscernible over Ck u A&, also {f(iZ,,-): 6 I i c p} is indiscernible over Ci u At for 6 E 8).

EXERCISE 2.1: Suppose A = lTll = (VK < ~ , ) ( K ~ O < K,) A 2 = = 2%; /3 < x+ or 2”3l I 2%; T not superstable, then I(A, T,, T ) 2 2’.

PROBLEM 2.2: Uniformize the proofs (maybe use the filters from Kueker [Kk 771 and Shelah [S 761 Section 3).

CONJECTURE 2.3: IE(A, T , , T ) = 2A for T unsuperstable, A > IT, I. EXERCISE 2.4: Show that for A regular > IT], T not superstable the partial order (P(A), G ) can be embedded into {M: M C T, llMll = A} quasi-ordered by elementary embeddability.

EXERCISE 2.5: For each cardinal A = P o , show there is no Boolean algebra M, of cardinality A, such that any other Boolean algebra M of cardinality I A has an embedding in Mo, preserving countable intersections. (Hint: See 2.5, Th. 12, Grossberg and Shelah [GSh 831.)

EXERCISE 2.6: Generalize VII, 3.7 and 1.7 to the cme IT,] c 2b, assuming MA (Martin Axiom).

VIII.3. Saturated models and the case A = ITl[

DEFINITION 3.1: The indexed-set (4: 8 E I) (I an index-model) is called K-skeleton-like in (a model) M when: (1) It is indiscernible. (2) For any E E M there is J c III, I JI c K such that if By 8 E I ,

B N t mod J , then for any p E L(M) M C p[@, E] E p[?if, El.

DEFINITION 3.2 : The orders I , , I , are called K-contradictory if there is no model M y nz c w , an antisymmetric formula in L(M) [ E < j’j](Z(E) = 1( j j ) = m), an order J with lower cohality 2 K and sequences iZ8 E &I for 8 E I , u I , u J (we wume for notational simplicity I , , I,, J are disjoint) such that:

OH. =, 8 31 SATITBATED MODELS 465

(1) (a*: 8 E I1 + J ) , (g8 : 8 E I , + J ) are K-skeleton like in M . (2) For 1 = 1, 2, s, t E I , + J , M t= [a, < iLt]u(*rt).

LEMMA 3.1: (1) 8wppose a,, M are d e b of T,, K is regular 1M1 =

M,. If the type each a, realizes over lMol U {a,: j < i} ie F~-isolaterl, or even F~-isolated, then (a8: s E I ) is ~-8keketOn like in M .

(2) The skeleton of EM1(I , @), or of EM1(I , N ) are ~-8kehdt-m like, for every K 2 8,.

(3) If (a8: s E I ) is the skeleton of M = EM1(I , N ) , M , t k subntodel of MAID (D ultraJilter over A) whose universe is {h/D: for 8ome J E I , IJ( < K , {h( i ) : i < A} E dcl{gS:sEJ)} (clearly it exists), then ( g S : s € I ) is K-skeleton like in M , and M , < MA/D(dcl in L,).

l d l o l U {a,: i < a}, 4 E lM,l for 8 E I and (a8: 8 E I ) i8 ~-8kehton like in

Proof (1) Prove for CE POI U {a, : i < p} by induction on 8. (2) and (3) immediate.

THEOREM 3.2: If A 2 lTll + K + , A<n = A, T umtable, K regular then there are M , E PC(T,, T ) for i < 2A such that M, h cardinality A and

(1) For i # j, M,, M , are not i8omorphic. (2) M , ie K-wm‘paCt, and ~-hmmgeneoue and if ID(T)I 5 A, M , ie

(3) M , is the L-reduct of M t , M i ia ~-c~ml ) (cc t , ~ - h m m g e n ~ u ~ ; and i f K-8dUrded.

ID(T1)I I A, &O K - 8 d W ’ d d .

DEFINITION 3.3: (*)! means that there is a family of 2” subsets of A, each of mdinality A, the intersection of any two is < K.

Remark. This is just not AD(2”, A, A, K ) (see Definition V I I , 1.11). A related property appears in 1.9. On independence results concerning it see Baumgwtner [Ba 761.

Proof. Let 0. be proper for (w, T,), such that the antisymmetric formula, [Z < 83 E L orders the skeleton of EM(w, @). We assume the assumptions from VII, Section 2.

Case I : not (*)!. By 3.3 of the Appendix there axe 2A pairwise K-contradictory orders I,, a < 2“; each of cardinality A. Let, for a < 2A, i 5 A If, be an order isomorphic to the converse of I,, and sf, E I,. Let, for a < 2”, J , = z,sAIf,. Let N: = EM1(J,, @), SO llNiII = A (T, has

Skolem functions). Clearly (as: 8 E J,) is the skeleton of N;. Let N i < M i , M i satisfy (2) and (3) from the theorem, but such that (as: 8 E J,) is still K-skeleton like (this can be done by 3.1 and VII, 4.4), and h = IIMi(I (by cardinality considerations this is possible). Suppose the number of non-isomorphic M,'s is < 2A and p 5 2A is regular, where Ma is the L-reduct of M i . Then for some M S = {a: Ma z M) has cardinality 2 p and let fa : Ma + M be such an isomorphism. Let & = f,(a8) for s E J,, a E S. So (g : s E J, ) is K-skeleton like in M. Let W, = {@: 8 = sf , i < A}. Suppose a # p; a , p E S , [ W a n W,l 2 K; then, as h is well ordered, there are for < K i (6) < A, j ( f ) < h such that 8&) = 6tCe) where a(& = @), t ( f ) = g e ) , and i ( f ) , j ( f ) are strictly increasing.

h t &I) = SUp{i(f): f < K} 5 h, 8(2) = SUp{j(&: 5' < K } I h. NOW clearly

are r-skeleton like in M. Looking at Definition 3.2 this clearly proves I,, I, are not K-contradictory ; contradiction.

So a # /3~8* IW,n W,I < K ; also IW,l = A, and all those W, me subsets of ]MIrn for suitable m, which has cardinality A. If 2A is regular, choose p = 2", so we have proved (*)f, hence finished, otherwise use 3.1 of the Appendix to get the same contraxiiction.

Owe 11: h = p+, hand p me regular. Let I, be the set {(i,j, y) : i < A,

j, > j2, or i , = i,,j, = jz, y1 < y2. We identify (i,j, 0) with ( i , j ) . For every I c I, let M,(I) = EM1(I, @), M,(I) = Mo(I)A/D where D is a regular, good ultrafilter over A (exists by VI, 3.1). Let d,((i,j, y ) ) = {i,j,y},andforJ c I , ,d , (J) = (JseJd,(s).ForeveryI G I 0 , I t / D ~ M 1 ( I ) let dl(h) = {s E I: for some i < A, h(i) = T(@, 8 minimal, and 8 = t[Z] for some 11, (note w.1.o.g. Cis uniquely chosen). Now for I G I,, let M2(I) be the submodel ofM,(I) whose universe is {h /D : Jd,(h)J < y}. By 3.1(3) M,(I) < Ml(I) m d (a,: 8 € 1 ) is p-skeleton-like in Ma(I) . It is also Clem that M,(I) is K-saturated because if J c I, IJI < p then M 2 ( J ) = M , ( J ) < M 2 ( I ) is even y-saturated (by VI, 2.3 and VI, 2.11). In the same way we can prove that for such J, if p is an m-type over M2(I ) finitely satisfiable in M 2 ( J ) , Ipl d h then p is realized

j < A, y < p} ordered by: (il, jl, yl) < ( i s , j,, 72) iff il < ia OT il = i a ,

in N,(J). Let 93 = (R(k), IS), and define 93, for a 5 p such that:

OH. VIII, 8 31 SATURATED MODELS 467

(1) p a l l = A, %<9. (2) If a E 193,l and a has cardinality 5 A (in R(k)) , then b E a; e-

b E I%l* (3) @ n : B (4) L, I,, D, Mo(Io) , J H M l ( J ) , (a8: 8 E I,) all belong to 9,; and

I E 9 1 .

(6 ) In fact, 93, depends on I so when confusion may arise we write 8, = S,(I), but sSo(I) = SO(I0).

Let for a 5 p, M i ( I ) be Ma(I) as interpreted in 3,. Clearly Ni(1) < M a V ) and JfiV) = U;<a J f h ( 0

Let N;(I) be the L-reduct of Mg(I). Notice that when cf 6 2 K, Ng(I) satisfies conditions (2) and (3) from the theorem.

For w E A let I (w) = {(i, j , y ) E I,: i $20 * j < p} . Suppose wl, w2 c (6: 6 < A, of 6 = p}, w = w1 - w, is stationary and there is an isomorphism f from N t onto N{ where Nf = N;(I(w,)) (for I = 1,2) . We shall get a contradiction, and this is sufficient t o prove the theorem in this caae (as in the proof of 2.2).

Let 68 = f(a8). We can find a stationary w' E w, and 5 < p such that for a E w', 6(,,,) E% and U, = {i < A : t?(a,t) E@} is a stationary subset of A. Let, for a E Ml(I0) , d2(a) be a set J c I of minimal cardinality such that d,(lr) c J where a = h/D (we can aasume the function

I%+il, I%l = U i < d 19d.

d , belong to So) . Also, for E Ml(IO), da(a) = Ul da(a[Q). SO for each E dla(Io), Id&)] < p. Let d&) = (a < A: for some 8, (a, 8) E &(a)

or (8, a) E da(a)}. SO clearly, for a E Ma(Io), d,(@ is a subset of A of cardinality <p. Hence by a double use of 1.8 of the Appendix, there are w", U&, f o and V,, V such that:

(1) V c A, I VI < p; w' E w', U: c U,; and w', U; are stationary, for aEw" and V, c A , IV,l < p.

(2) Let? @ E w", a < @. (i) d3(b<a,O>) n da(6<~,o>) E V , V a n Vn E V , and V E VaY Van

(ii) if i E dg(6<,,,>), i 4 V then i 2 a, (iii) if i ~'d3(6<, ,0>) , then i < 8. (3) L e t a E w " , i , j E U ~ , i < j.

p = V ,

(i) &3(6<rr.i>) n d3@<,,,>) = Va, (5) if f E d3(5<a,t>), f 4 V,, then t 2 i, (iii) if f E d3(6<,,;>), then i, f < j. (4) If i E U;, then i > sup V , and i > a + a, and a > p. ( 5 ) There are sets B , E ~ $ + ~ (a < A ) , B,GN2, IB,l < A ; P E W '

8 < a =- 6<,,,, E B,, and B, = Used B, (this ody because 9, ~93,+1).

468 THE: NUMBER OF WON-ISOMORPHIC MODEL8 [OH. WII, 5 3

(6) a E w", c E B, =- d3(c) c a.

Cliooee 6 E w" which is an accumulation point of w", (so bigger than lo, and than p). So 6 E wl, 6 # wa, cf(6) = p and (6, i) E I(w1) o i < A

Define J = J, u ((6, lo + n): 0 < n < w} u {(a, 6, lo + n): n < w,

a E V , a E wa}. Clearly I JI < p, J E I (W2) .

(7) l o > SUP(V n p), l o < p.

but (6, i ) E I (Wa) o i < p. Let J o = {(i,j, y): (i,j, y ) E 1 0 ; i,j, 7 E V } .

Let (for a < 8 )

A = {a: U € N 5 , , d 2 ( U ) E { (a , i , y ) :a = s*i <&, y < &}},

A,= {a E A : d&) c a v ( A - a)}, p = {'p($; E): E E A , 'p E L, and for all sufficiently large a < 6,

'% ' d b < K , O > ; '11, PO'p I A K '

As f is an isomorphism, (b8: 8 E I (w, ) ) is p-skeleton-like in ivg, so p is a complete type over A. We now show that p is realized in N2(I(w2)), andeveninM,(J).As IJI < p, IpI s h,its~cestoprovepiehitely satisfiable in H,(J); so as p = u a < d pa, pa inoreasing it suffices to prove any finite q s pa is realized in it. By the choice of 6, p there is B E W " , a < f l < 6, B > p such that 6<8,,) realizes q and let it be (ho/D, . . . , hn.,-JD) where dl(hl) s cEa(6<<B,o)) for Z < n. NOW define hb,. . ., hkql such that: If h(i) = T(@) (minimal 8) then x'(i) = .t(iZj) and t satisfies the

following (here x( i ) = (hl( i ) : 1 < n)): (a) t - 8,

( y ) if 8[m] E J,, then t[m] = arm], (6) if B[m] = <i, j, y>, i > 8, then i[m] E ((6, to + k): 0 < k < a}, (8) if 9[m] = (i, j, y>, i < B (so i E V), 9[m] # J, (so /3 s j < 6) then

t[m] ~ { ( i , 6, lo + k): k < w}.

Clearly this can be done, and 3 - i mod {(i,j, y): i = 6 j < to, and i < 6 * i s a} hence if 6 = (hb/D, . . . , hk-JD) then 6 O realizes q, and by the above 6 c M 2 ( J ) .

SO by previous remarks there is 6 ~ M , ( l ( w ~ ) ) realizing p. As (6#: 8 E I(wl)) is p-skeleton-like in N;, l B d l < h, there is &1) < h such that all 6<d,') ([(I) 5 i < h) redhe the same tSpe (in N$) over &. h t

(B) Z E J ,

q = {E < z: E E B d , N& k E < 6<d,E(l)))

{z < E: A , tp(E, Bd, N!l) = t P ( ~ < ~ , ~ ( l ) > , B d , %)}*

CH. VIII, 31 SATURATED MODELS 469

It is easy to check that q E p (by (6 ) ) so 6 realizes q, and q E 8, + , . So, as 99 C "in M2(I(wa)) there is an element realizing p"

also CZJ,+, satisfies this, so some 6' E Nb+1 realizes q, so (&: 8 E I(w,)) is not p-skeleton-like (as U; is an unbounded subset of A, a d i E U; =- 6<d,i> E A). Contradiction. Cme 111: A = A, is a limit cardinal, or A = A,+ , A, singular; and for some p < A,, 2' = 2A. We can choose regular p < A, p > K, p+ # of A,, 2'= 2A. The proof is similar to the previous case. W.1.o.g. h =

Define I,, M,(I) , M , ( I ) as in Case 11. Let, for J E 10 p r l ( J ) = {a: for some 8, (a, 8) E J } and M,(1) will be the submodel of M l ( l ) with universe {h/D: pr,[d,(h)] has cardinality e p , d,(h) has cardinality sp}. 23,, M;(I) , are defined aa in case 11. For any function g : p+ + A, such that each g ( i ) is a regular cardinal > p+ , < Al, let

lEM(P, @)I.

I(g) = {(i,j) E l o :j < g(i), i < p'}.

Now we suppose g,, 9, are such functions, and for every x .c A, {i < p+: cf i = p, gl(i) > g , ( i ) > x} is stationary, and f is an isomorphism from iVt onto N t , and get a contradiction, and this is sufficient [where iVF = N;(I(gI))].

Easily we can find 5 < p, B E B,, p+ < IBI < A, and S = {a < p+: 6<,,,> E B, cf a = p}

is stationary, and let i3 = pa: a < p+} where pa is increasing (so 8, 2 a). Let w' = (6 < p+ : cf 6 = p, gl(S) > ga(6) > IBI} (so w' is stationary) and let w", V be such that (exists by 1.8 of the Appendix):

(1) w" is stationary, w" E w'.

(3) If a E w", i ~ p r ~ [ d ~ ( 6 , ~ , , , ~ ) ] , i 4 V , then i 2 a. (4) If a 6 w", i E V , then i < a. (5) Let B, = B n {c: c EN;, p l [ d 2 ( c ) ] E a} so B, E 23,+,, B, is in-

( 6 ) If 6 E w", a < 6, then 6<8a,o> E B,. Choose 6 E w", 6 an accumulation point of w" (so cf(6) = p, gl(S) >

ga(6) > IBI). Let J, = {(i,j): (i,j) €IO, i E V; and for some a E w", a < 6, (i,j) ~d,(6<~,,,>)}. Now there are 6, c gJ6) and U E gl(S),

Let J = Jo u ((6, (, + n): n < w}. Clearly IJI I p, Iprl(J)I H,). Let A = {a: u EN$, d,(u) n ((6, i): So 5 i < ga(6)} = 0}.

470 THE NUMBER OF NON-ISOMORPEIU YODEL8 [OH. m, 8 3

A,, p , pa are defined as in Case 11, and so is the existence of 6 M2(I(g2)) which realizes p (only part (e) falls). The resi is like Caee II- we prove q is realized in N$ and so get the contradiction.

So it suffices to prove that Cases 1-111, exhaust all possibilities. Suppose not (*)R, and it suffices to prove that for some regular p,

p+ I A, 2y = 2A. By 3.2(1) of the Appendix AK = 2A, so if A I 2K, 2'5 = 2A so p = K will suffice. If A > 2=, by 3.2(2) of the Appendix for some X , K + < x+ < h , ~ " 2 h so x" = 2A) and 2X = 2A. So we finish.

COROLLARY 3.3: If A 2 I T,I + N,, T uwtable, then I(A, T , , T) = 2A.

Proof. Take K = No in 3.1.

COROLLARY 3.4: If A 2 lTll + N,, T not Bulperatuble, then I(A, T ) = 2 A .

Proof. By 3.3, 2.1, 2.7 and 1.7(1).

THEOREM 3.6: 8 m e A 2 p 2 lTll + K + , Aca = A, K < K(T), K , p areregular;andx < p=+x<IC < p.8upposealsotlurtp > lTll&x ITJ, then no M, is elementarily embeddable in M, for i # j . (3) A¶, ia K - c o T ~ ~ & , ~-liomogeneous, and, i f lD(T)l < p, ~-edureted.

8imilarly for w. Proqf. For simplidty we aasume p > IT,!. Using VII, 3.6(3) and 3.6 we can find @ proper for (us", T,) such that the skeleton of EH(osa , 0) has the form ((cp,, 1): a < K a successor). For 8 < p, cf 8 = K choose qs e p K increasing with limit 8, and I, = psa U {qd: S E w}, and let Nt, = EM1(I,, a), Hr a IFb,-primq model over NL with constructing sequence {ur: i < A}, eatiafying (3). (For proving (2) when A s 2ITd we ehouldaeaumethatiffhetypeofaroverBp 3: { a v : r ) ~ I , , , } u { u ~ ; j < i } is rAv(DfO, dd Bfo) ,

Let Hw be the L - d u c t of i& f : MW + H,, an isomorphism (for 2) elementary embedding). For (2) let p = A and the proof is aa in 2.2. For (1) there is V c A, I Vl = p such that f maps {itn: r) E I,] u {ap: i E p) onto {a,,: q E I,,} u {ur: i E V).

Let V = {i,: a < p}, Then there is a closed unbounded 8 E p such

< K, then I{p: B f o = B}I < A.)

OH. Vm, 8 41 CATEQOBICITY, SATURATION AND HOMOQENEITY 471

that for a E S f maps {a,,: r) E I,, r) E UBeK 8'") u {a?: i = i, E V , 8 < a} onto {a,,: 7 E I,, r ) E Ubcr p'"} U {a;: i = i, E V , 8 < a}. The rest is like 2.7.

THEOREM 3.6: 8-e A 2 p + IT1 + K + ; A<' = A, K < ~ ( 2 ' ) ; K , p are regdur, x < p =- x c u < p. Then T has 2' n o n - k m p h i c models of cardinality A, which are ~-lumtogeneowr, K - C O ~ Z ) ( C C ~ , and i f ID(T)I s A also K-saturated.

Proof. Similar to 3.6 and 2.7.

CONJECTURE 3.1: (1) If A = A"K 2 lTll + K + , K < K(T) , K is regular, then there are 2" non-isomorphic models M E PC(T,, T ) of cardinality h satisfying (2) and (3) from Theorem 3.2.

(2) Moreover, if h > I T,I, no one of them is elementarily embedded in any other.

EXERCISE 3.2: Prove conjecture 3.1(1) for A strong limit cardinal > D ~ . (Hint: look at 2.6.)

EXERCISE 3.3: Prove conjecture 3.1(1) when there is p, pcK = p < A, 2' = 2".

EXERCISE 3.4: Prove conjecture 3.1(1) when K < K ~ ~ ( T ) and A ie regular, or p < A s p", 2fi < !P.

PROBLEM 3.6: Axiomatize the proofs in Sections 2 and 3 (to include a180 1.7(1)).

Remark. Clearly in 2.6, 6 and 1.4, 7 we have less than in, e.g., 2.2. Of course, it will be d d a b l e to use fewer axioms in each proof.

VIII.4. Categoricity, saturation and homogeneity up to a cardinslity

THEOREM 4.1: Suppose PC(T,, T ) i s categorical in A. (1) If A 2 lTll + K,, then T k8tabkandif A > 2No lrae nd the f.c.p. (2) If A > lTll then T k eulperstubZe without the f.c.p., and dabk in

every p 2 lT1l.

Proof. (1) T is stable by 3.3; hence hae not the f.c.p. by VII, 3.4. (2) T is superstable by 2.1. By VII, 2.7 there is M E PC(T,, T) of

cardinality A, stable in lT1l. If T is not stable in ITII, there is M E

PC(T,, T ) unstable in ITl!, 11M11 = A, contradicting the categoricity. So T should be stable in I T , I, hence by 111,O. 1 stable in every p 2 I T, I . By VII, 3.4, T is without the f.c.p.

THEOREM 4.2: (1) If T is not mperstable, then for every A > lTll there is a nvn-I T,I +-model--- model in PC(T,, T ) of c a r d ~ d d y A ( 8 0 i t k n o t I T , I + - l u r m o g e n e o u e , ~ - ~ ~ ~ ~ ) . SM below.

(2) If T i8nOteulpW8tabk, IT11 < p 5 A,p*@rY (VX < p)(xWO 5 A), then there are rnodels Mt of T , of cardinality A, for i < 2y

the L-reduct of Mi,

cardinality < p doea not t-kpnd on i.

thd (i) for i # j, the L-redwt of 211: cannot be elernentady ernbedded into

(ii) the set of ieommphhni typerr of elementary submodels of Mi of

Remark. M is A-model-homogeneous if M, < M y lldl, 11 < A, f : dl, Ma, Ml < N , < My llNIII < A, then for some f ' extending f, and N,, M2 < Na 4 N , f ' : N , Na.

Proof. (1) We use the terminology of 2.2. Dehe a(i) for i < I!l',l+ as follows: a(0) = 0, a(i + 1) = a(i) + i , a(&) = Uleaa(i). We SSY u = /I + 9 i f u [ i ] = /I + q [ i ] for any i < l(u) = l(q). Let

I , = A<" u {a(in-l)+ ... +a(i,)+7ja: s < a(i,) < ... < a(in-l) < ITII+},

I , = IT,J<" U(9a: 8 < p l l + } Y Ml = EM1(PY No).

As a(i) is an inrreaaing and continuous function (6 < I T,J + : there is 9 E Io, r ) E a', sup,, ~ [ n ] = 6) is not stationary (each a( i ) does not belong to it). As in 2.2 this implies that M1 1 L cannot be elementarily embedded

into Mo L. Let, for 7 < lT1l+, I : = I 1 n y s m , 131: = EM1(I i ,No) . Clearly every elementary submodel of M1 of cardinality 5 I T,I is an elementary submodel of some Mi, but Mi a n be elementarily em-

those two facts imply No

YEP for some p < a(i)}. The rest is like 2.2 and 4.2(1).

bedded into Mo, by the embedding induced by a,, H t5crtr+l)+n. Easily, L is not I T1 I +-model homogeneous.

(2) For w c p let I, = A'" U {vd: 8 E w} u (I. + a( i ) + 7: i < p,

CH. n I I , $41 CATEGORICITY, SATURATION AND HOMOGENEITY 473

THEOREM 4.3: The following assertions on the cardinals, p, A, x, where A > x, p > x, are equivalent:

( 1 ) If ILI I x, L(T) E L, p a 1-type in L, P a one place predicate in L, and T haa a nzodel M omitting p , 11M(1 = A > I P(M)I, then T hae a

(2) If T c TI, (TJ < x, PC(T,, T ) is categorical in p, then PC(T,, T ) is categorical in A:-

(3) If T c T,, ITl! 5 x, and every model in PC(T,, T ) of cardinality p is h g e n e o u s , then every model in PC(T,, T ) of cardinality A is hornogeneowr.

(4) If T E T,, ITl] I; x, p a 1-type, IpI s x and every model of T , of cardinality p omitting p has homogeneous L-reduct then every model of T , omitting p of cardinality p haa hmnogeneozls L-reduct.

mo&el N omittingp, llNll = p > IP(N)I.

CONCLUSION 4.4: If A = ad, ( V C ~ < S)(a + S(x ) s S), A, p > x then (1)-(4) from 4.2 h7rolds.

Proof of 4.4: By V I I , 6.3, ( 1 ) holds in this w e .

Proof of 4.3. (1 ) =+ (4) Suppose M1 is a model of T,, omitting p, of cardinality A, with a non-homogeneous L-reduct M. So there are K < A, a, E M (i K) b, E N (i < K) such that

tp*((b,: i < K), M) = tp*((a,: i < K), M)

but for no b E M

tp*((b,: i < K ) ^ ( b ) , Y) = tp*((ai: i I; K), M).

Let M* = ( M , P, P, c) where P = {a,: i < K} (one place relation), P(a,) = b, for i < K and P(a) = c for a # P, c = a,. So by ( 1 ) there is a model N* elementarily equivalent to M*, ((N*I( = p > (P(N*)( and omitting the types p and

Y = { (vY~,.. - - Y J [ A P(Y:) -+

i

(because omitting p and q is equivalent to omitting r = {~(z) v #(z): ~ ( z ) €21, #(x) E q}). Clearly (by q) N is not homogeneous. So we prove (4).

(4) * (3) Immediate.

474 THlD NUMB= OF AON-ISOMORPHI0 MODlDLS [UE. mI, 5 4

(1) 3 (2) Suppose PC(T,, T) is categorical in p. By 4.1 T is stable in every X 2 IT1[, hence in p , hence by 111, 3.12 T haa a saturated model in p, so by I, 1.13 PC(T,, T) has a saturated model in p, hence every model in PC(T,, T) of cardinality p is saturated. Suppose M1 is a model of T, of cardinality A and its L-reduct H is not saturated. So there me A E lMl, andp €&'(A, M); p is omitted by H, IAI c A. For every 'p(z; 5) E L let R, = {a: 'p(z; 7i) ~ p } , and P = A; let M* = (M, P, . . . , R,, . . .),EL. By (1) there is N* elementarily equivalent to M*; IN*] = p > IP(N*)I and it omits

Q = W5W,(B) -+ 9J@, 0)l: 'p E L}.

Clearly N* L E PC( T,, T) and it is not saturated as it omits p' =

{'p(z, a): N* t R,[7i], 'p EL, E IN*I} (p' is consistent as N*, H* a m elementarily equivalent). Contradiction. So every ME PC(T,, T) of power A is saturated. Hence (by I, 1.11) PC(T,, T) is categorical in A.

not (1) 3 not (2), not (3). Suppose L, T, p, P form a counterexample to (1) and M is a model of T omittingp, 11M)1 = A > IP(M)I, and let I, = {vi(z): i -c x}. Let P be a one-to-one function from 1M1 x onto M - P ( M ) , with converses P,, Pa [i.e., for a, b E IMI, c E lM1 - P ( M ) , P,(P(a, 6 ) ) = a, P,(P(a, 6 ) ) = b, P(Pl(c), Pp(c)) = c]. Let Qi =

{a E

Th(M*), L* = {Q: i < x}, T* = Tf n L*. Clearly eat& Q, hee cardinality A or is empty, and 13.21 - Ui<,Qi = P(M). Be IP(Mf)l c A = IMf I, the L-duct of Mf is not saturated nor homogeneous. (If P ( M ) = {hi: i < K), let at = a, = b,, and then we m o t find 8 euitable b,.)

Suppose now Nf is a model of Tf of cardinality p. Being a model of Tf, each Q,(Nf) is empty or has cmdimlity IINf 11 (use P, Pl, 3'2). By aesumption Nf realizes p or IP(Nf)l = A. In eaoh 0&88 A = Nf - Ui<,Qi(Nf) has oardinality A (in the first ( ~ ~ 8 8 , if a realizee p , as {P(a, c) : c E Nf} E A; and the second ( ~ ~ 8 8 as P(Nf) E A). As Tf has elimination of quanti6ers, it is easy to check that the L-reduct of Nf is eaturated, hence homogeneous. So T?, T* provides the countmexample for (2) and (3).

: a 4 P ( M ) and M I= 'p,[P,(a)] for j c i, but M I= 7v,[Pl(a)]}. Let Mf = (M, J', FI, FS,. . .,Qi,. . . ) {ex and Lf = L(M*), Tf

Be we prove (1) * (4) * (3) * 1, (1) (2) * (1) we hi&.

THEOREM 4.6: T?MM i8 Q 4 1 Ml of Tl of CMClinaljty p which is ( < A)-univemd, but its L-reduct i8 not A-univereat, if:

( 1 ) T unsuperstable, 2IT1l < A, 2<A < p ; and A is regular or An, = A ; or

Ca. WII, 8 41 OATEOORICJITY, SATUEATION AND HOMOUENEITY 475

(2) T = T , is stable but not superstable, IT1 Q A , 2cA Q p ; and A is regular or hNo = A.

Proof. Our model M , will be the model constructed in VII, 4.6. Clearly it ie ( < A)-universal. We should show only that its L-redud,

h f o is not A-universal. (1) If A is regular, the proof is similes to that of 2.2. If P o = A,

o h m @, cpn aa in the proof of 2.2, and let No = EM(Asm, @), so N is a model of T of oesdinalify A, henoe it s d m s to show No is not elementarily embedded into M0. Suppose f is suoh an embedding, and 8, = f ( i i , ) for q E As@. W e mn 6nd finite w, c p suoh that v Q r )

w, c w,, We now define by induotion q,, = q(n) E An, suoh that r),, Q qn+l and

- b, = q $ . o , - * * Y Z,,k(,)), 4 . 1 E J f & , I ) , w, = {i(r), 0: 1 < k(7)).

(7 E An+': 7, = TMn+l), k(q) = k(~n+d;

01, * - * > ( ih+l , 0)s * * * > mod WMn);

i&, ZM,,+l),l realize the same type over M1, moreover if

a =I u {z,,~,,: i c ny j 5 k(7 1 i), i(7 1 i, j) = ~ [ n l } ,

then 7i"Z,,1, anZM,,+l),l rea.lizee the same type over lM1l}

haa oardidity 2 No. (Of o o m , 2, # 1, * i(q,Z1) # i(q, la)).

Let 7 E A@, q,, 4 q for every n. Then {v,,(Z, 8,,,,): n < w} is realized by b,, but oannot be realized in Mo by the dehition of q, oontradiction. (2) A similar proof.

LEMMA 4.6: (1) There are complete countable t h r i e a T I z T , T un- etable, 8uch that: if M E PC(Tl, T ) i8 ( < A)-univered, A etrong limit, cf A = KO then hf ie A-univered.

( 2 ) There are complete countable theories T , 1 T , T etable but not ezclper8tableY 8uch that: i f bl E PC(Tl, T ) iS ( c A)-universa2, A etrong Jim$, cfA = KO then M is h-universal.

Proof. (1) Let T be the theory of the rational order, and T 1 eays that any two intervals are ieomorphio. We leave the proof to the reader.

(2) Define the relation E,, over w@: 71,,v o 9 r n = v 1 n. Let M = (wcu, 1 0 , 4,. . ., 4,. . .). Let P,,(v, p) = ( ~ [ o ] , . . ., v[n - I), p[n], p[n + 11,. . .), and MI = (M, Po, P,, . . .). Let T = Th(M), T , = Th(M,). We leave the proof to the reader.

476 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. WII, $ 4

THEOREM 4.7: The folknning cunditiom on A, T are equivalent: (1) A = A < A + ID(T)I or T is stable in A. (2) T laas a saturated nzodel of cardinality A. (3) If T, z T, lTll 5 A t h n there is a saturated bl E PC(T,, T) of

cardinality A.

Proof. (1) * (2) If A = A < A + ID(T)I, the proof of I , 1.7 shows that (2) holds. If T is stable in A 111, 3.12 shows that (2) holds.

(3) + (2) This is immediate. (2) + (3) This is I , 1.13 (in its proof we use (2) * (1) only, so no

vicious circle arises). (2) (1) Let M be the saturated model of T of cardinality A. So

clearly A = llblll 2 ID(T)I, and if IAI < A, I,S(A)I 5 A (as A2 is universal w.1.o.g. A E My now bl is saturated). Now we prove by cams.

Case I : T unstable, A < 2<”. By 11, 2.2(6) there are Q, and 7 E A>2 such that for each r ) E A2 {~ (z , 7i,r,),[al: a < A} is consistent. Now we define 6, E bl by induction on 1(7), such that if q0 4 7, 4 - - - 4 7, then G,(0)hGMl)h- - - a,(,,), and 6,(0)hb,(l~n- . --6,(,,) realize the same type. (This is possible as the only requirement on 6, is that it realizes a certain type over U {6,: v Q r)}. The type is consistent by the induction hypothesis, and is realized aa M is A-saturated). So for every 7 E ”’2, some c, E M realizes &, = {I&, ii,ra),caJ: a < l(7)). If l(7) = l(v) then c, # c, [for if a = min{a: y[a] # v[a ] } , p = 7 a = Y a, then ~(c , , G,,) = ~ c p ( c V , ~ J ] . So for every a < A, h = llMll I ( c , : 7 ~ 2 ” } = 127, hence A 2 2<”, contradiction.

- n-

Case 11: T unstable A 2 2<A, A regular. It is etley to check A = A<”, so we finish.

Case 111: T unstable, A 2 2<A, A singular. It is easy to check that p < A - 2” < h (otherwise 2~+cPA > A , ,u + cf h < A ) . So it suffices to prove (as a saturated model is universal by I, 1.9(3)).

LEMMA 4.8: 8uwo8e A 2 I T,I, A 8trong limit cardinal and cf A c K ( T ) . Then thre i8 H E P C ( T , , T), lldlll = A, mch that no elementary ez- temion N Of bl O f Cardinality is (Of h)+-&UTded.

Proof of Lmmu 4.8. Let K = of A. We can define suitable M and ii, E M for q E x > A y such that for every 7 E “A, q, = {Q@, Gv)u(n9v): v E x>A} is

CH. VIII, 8 41 CATEGORICITY, SATURATION AND HOMOGENEITY 477

consistent (by 111, 7.6, 7 and the definition of the independence “property”). Suppose M < N, llNll = A, so let IN1 = U f e e A f , !A,! < A, hence I.S(Af)I s 2141 c A. Now defme by induction r),, u, for a s K such that a c and tP(av(a+1), A,) = tp(liv(a+l), A,), hence no element of A, realizes

a < K } is cqvcrc1, hence consistent, but is not realized in N &B IN1 = (JaerC A,. So N is not K+-saturated.

=- r)“ Q 78, l(v,) = a, Z(r),) = a, r), Q

F J h %)b+l)) A T ( Z , %+l)). Clearly Q = M . 9 av(a+l))9 lQ)(Z,

Case IV: T stable, A<K(T) = A. So A = + ID(T)I. If T is stable in A we finish, otherwiae, by 111,5.16 it follows that A c 2n0, and for some countable A, IS(A)I 2 2n0, contradiction.

Case V: T stable, A<c(T) > A. It suffices to prove the following lemma.

LEMMA4.9:SuppoaeA 1 lTll,Ax > AandK < K ( T ) < o o o r T h m t h e atrict order property or K < KrOdt(T). Then there is M E PC(T,, T ) , 11M11 = A auch that ru) elementury extension N of M of cardinality A ia K + -Saturated.

Remrk. In fact if K < K ( T ) < 00 or T has the strict order property, then K c mdt(T).

Proof of Lemma 4.9. We can aasume K = minb: A# > A}, hence A = A<’. So there is H E PC(T1, T ) , tp, E L, and a,, E bl, for 7 E ‘“A such that:

(1) {tpa(it; av)il(v*V): a < K , r ) E “A, a successor} ie consistent, for every v E “A.

(2) If r ) E “A, i c j < A {~.I,+,(?z; i~ , - (~>): 5 E (4, j}} is inconeistent. (This is by 111, 7.7, 111, 7.6(1) and (5) . )

Suppose M i N , I(N(1 = A, and for ~ E ‘ A let p , = { F J ~ ( z , @ ~ ~ ~ ) : a < Z(r)), a a successor}.

If N is K + -saturated, each p n is realized in N; but the p i s me pairwise contradictory. So IlNll = Arc contradiction. So we finish the proof of Theorem 4.6.

THEOREM 4.10: .Suppo8e A 2 lTll + K,, and every model M E

PC(Tl, T ) of wrdiml i ty A ia K,-homogeneoua. Then T is swperatable.

Proof. Our assumption implies that every M E PC(T,, T ) of cardinality

478 THE NUMBER OF NON-ISOMORPHIC MODELS [OH. VIII, 8 4

2 A is K,-homogeneous [Because if (a,: i II w), (bi: i < w) is a counterexample, i.e., tp,((a,: i < 0)) = tp,((b,: i < w>) but for no b, E M tp*(<a,: i I; w)) = tp,((b,: i I; w)) and M is the L-reduct of M,, M , a model of T,, then by any elementary submodel M i of M , of cardinality A such that a,, b, E IMiI, we arrive at a contradiction]. But there is a strong limit cardinal p > A of cofinality No; so by 4.8 T is stable, and by 4.9 superstable (remember I, 1.9(4)).

CONJECTURE 4.1: If PC(T,, T) is categorical in A, A 2 lTll + N,, then T is superstable (but see [Sh 801).

PROBLEM 4.2: Can we improve 4.2(2) (i.e., weaken the conditions on p) ?

PROBLEM 4.3: Close the gap between 4.5 and 4.6. That is characterize the cardinals A, p , x such that: if I T I ] 5 x, T 5 T,, T unstable [stable but not superstable] [has the independence property], then there is a model M , of T , of cardinality p which is (<A)-universal but its L( T)-redwt ie not A-univereal.

PROBLEM 4.4: Characterize the T, A, K such that:

(*I For every model M of T of cardinality A there is N, M < N , IlNll = A, N K-saturated (note 4.7,4.8, I, 1.7 and Exercise 4.5) (but see [Sh 80al).

EXERCISE 4.5:Suppose pn = p, p I; A s 2’, T -- Tina (see 11, 4.8). Then T , h satisfies (*) from Problem 4.4. [Hint: See the solution in [Sh 811.1

PROBLEM 4.6: (1) Characterize the cardinals p, A, K , x such that for every unstable T, T c T,, ITl] II x, there is a model M E PC(T,, T), lldlll = p, M is (< A)-universaI but not A-universal, M is K-saturated but not K + -saturated.

(2) Replace “unstable” by K c K(T) < 00, or by “has the independence property ”.

EXERCIBE 4.7: Solve those cases of 4.6 which can be solved by inessential changes in the proofs here. E.g., assume G.C.H., p, A, K are successor cardinals.

CONJECTURE 4.9: Remove the exception in 3.4.

CONJECTURE 4.10: In 3.5 waive the condition (Vx < A)(x‘“ < A) .

CATEGORICITY AND THE NUMBER OF MODELL3 IN ELEMENTARY CLASSES

IX.0. Introduction

In this chapter we return to elementary classes so in some respects it is a natural continuation of Chapter V. Part of Section 1 is devoted to categoricity theorems, eta, which are,

in a sense, a summation of previous reaults with some additions. We characteripe the countable theories categorical in No (1.6). We prove that for countable T, T is categorical in some h > No iff T is categorical in every h > No iff T is totally transcendental and for zi E 1421 v ( M , a) 2 No * I&lZ, a)! = 114211 ( ~ e e 1.8 m d 1.4).

For not neceeearily countable T: for some A, T(A) is categorical in some X > T(A) iff for some A, IAI 5 2ITI, T(A) is categorical in every h > IT(A)I, iff every 2~T~-universal model of T is saturated, iff T is superstable and undimensional.

If those conditions fail, for every h 2 2’, p 2 2ITl T has a p-universal not p+-universal model of cardinality A (see 1.14).

We prove also that T is categorical in some h > I TI iff T is categorical in every h > IT], and that if T is categorical in IT1 > KO it is a definitional extension of some T’ E T, IT‘I < ITI. Note that the property “T is categorical in I TI +” is not absolute. The properties “T categorical in N,,” for countable T, and “T(A) is categorical in I T(A)I + for some A ” are absolutes by the proofs of the previous theorems.

We also prove that if every model of T of cardinality h is IT1 +-

universal, T is I TI + -categorical; (see 1.16) and if every model of T of cardinality ho > IT1 is homogeneous, then: every model of T of cardinality > A( T) is homogeneous, and T has a non-homogeneous model in A when IT1 + N, s h s h(T) (see 1.16).

The method of those proofs is as follows: by Chapter VIII we get the 479

480 CATEOORICITY AND WUMBEE OF MODEL8 [CE. Ix, 6 0

results on unstable and unsuperstable theories, and by V, Section 2 on stable non-unidimensional T. So we concentrate on unidimenaional theom, and by V, Section 7 we find weakly minimal formulas cp(s, a). By the unidimensionality for no M 4 N , M # N is 7i E 1611, cp(N, a) = cp(Y, a), so bi is “almost prime” over a u cp(M, a), w h e w it is eaay to handle cp(M,Zi) by the weak minimality. For using this we need theorems assuring, for a suitable A, the existence of M, A E 1M1, ~ ( b l , a) E A, see 1.1, Exercise l.lO(2).

In Section 1 we deal with two more problems. We prove that if IT1 < A(T) (=sup(lS(d)I: Id1 5 ITI}) then for every p 2 ITI, I ( p , T) 2 min{2’, 2*(m}. In VIII, 1.7(2), 8 we have proved this even for PC(T,, T) but we demand T, is countable; the most diflicult caa is

The second result is that for superstable T, and if, e.g., IliIfll > ITI, 6 < llblll+ , then bl is the union of a strictly increaeing elementary chain of length 8 (see 1.3). (For stable T this may fail, sea Exercise 1.2.)

In Section 2 we try to compute I ( A , T) in some cams. We try to count the models by their dimension (or dimensions).

We prove that for T totally transcendental, if IT1 = K, I K,, T not categoricalin ITI+,thenI(N,, T) 2 IS + lI .I(X,, T) 2 + 11 + No except, possibly, when T is countable, categorical in KO (the idea is that T is not unidimenaional by 1.8, so we have a freedom to define one dimension and for appropriate T, also the finite dimensions are possible). We also prove that for superstable countable T, not categorid in KO, I (Ko, T) 2 KO (the idea is that if p E P ( 0 ) is omitted by the prime model of T, there is a model M,,, in which {aI: 1 < n} E p(iIf,,) whioh are prime over I,, = {ar: 1 < n}, when I,, is included in an infinite indiscernible set baaed on p, and dim(p, M,,) is n. In V, Section 6 we deal with multidimensionality. We prove multi-

dimensional theories have many models (in 8, > IT1 at leaat 214). For non-multidimensional T we prove they have quite few F&(m- saturated models. For superstable T, we get here a sharper result, of course for non-multidimensional T. We also get a better structure for the models. Every Fio-sctturated models is (in VQ) F;f,-prime over u, J,, each J , baaed on 0, Av(J,, u J,) semi-regular, and the Av(J,, u J, ) am pairwise orthogonal (we see somewhat more by the inductive definition of the J,’s).

In 2.4 we get a similar theorem for T totally transcendental, for all models, or for the N,-compaot models.

2’’ = PO, ID(T)I 5 !TI.

CH. I x , 5 11 SUPEXSTABLE THEORIES AND OATEQORICITY 48 1

IX.1. Superstable theories and categoricity

THEOREM 1.1: Suppose p = tp(zi,, A) does not fork over B c A, p is stationary, PI < K(T) < 00. If A' is Fi-constructible over A then p is not realized in A', moreover p has a unique extension in Sm(A') provided tluct:

( * ) A Foreweytyper E p , 1.1 < handC E A , ICl < h + K J T ) and finite A , there is 7i E A which realizes r guclc tW

P ( p , A, 2) = R"[stp(a, C*), A, 21 for some C*, B V C c C* c A .

FACT 1.1A: If p = stp(ao,A) satisjies ( * ) A for some B , h or even is just $finitely satis-able in A , then p is statiosmry [because for every E E FE"(A), E = E(z, y, F ) , choose by induction on 1 < w , ~ E A realizing {-E(z, a", 6) : m < 1, -E(ao, a"l, b)}, so for some 1 E(a0, a', b), hence p I- E( Z, a', b)].

So we can omit " p stationary " from the assumptions of 1.1.

Proof, This is proved by induction on the length of the construction; for limit length, or length zero it is immediate. For the successor step, note that if$ has a unique extension pr E Sm(A') then n e c e d y pr does not fork over B. So clearly it suffices to prove the following two claims. For simplicity we assume p € S m ( A ) . Note that as p is stationary, R"(p, A , 2) = R"(stp(ao, A), A , 2) for every A.

CLAIM 1.2: Swppoae A E A', p c p1 ES"(A'), p1 does not fork over A (and the conditions of 1.1 holds). Then ale0 B, A', p' satiefv ( * )A.

CLAIM 1.3: Suppose B, A , p are as in 1.1 and tp(6, A ) is F;-isolated. Then p has a unique extension in Sm(A V 6).

Proof of 1.2. Let T' E p l , lrll < A, r1 is over C', C1 E Al lC1l .c h + K,(T), and A be finite, and let T' = {pi(%; a,): i < a < A}. By IV, 4.2 (with B, A here for C, A there) for each i there is a formula y$(Z; 6f) E p such that for every a E A and 8 p,(Z; a) k B(Z; a) implies $,@; 6,) k 8@, a). SO for any a E A h,@; 6'1 + pt[a; a,] (use 8 = (3 # a)). Choose C, B E C c A , ICl < h + K,(T) such that tp*(C1, A) does not fork over C, and 6, E C, and Rm(p, A , 2) = Rm(p 1 C, A, 2). By the hypothesis, (*)A holds, and clearly T = {#'(Z; E i ) : i < a} E p and C satisfy the

482 OATEaORICITY AND NUMBER OF MODEM [OH. IX, $ 1

assumptions of (*)! Hence there is 2 E A c A1 realizing r such that for some C*, C G C* c A.

P ( p , A, 2) = Rm[stp(a, C*), A, 21.

By the choice of C, tp(C1, C* u a) does not fork over C, hence over C*, hence tp(a, C* u C1) does not fork over C*. So by 111, 4.2,

P[stp(a, c*), A, 21 = ~m[~tp(a , C* u cl), A, 23.

We can conclude that (again by 111, 4.2),

Rm(pl, A, 2) = P ( p , A, 2) = IP[stp(Ti, C* u C1), A, 21

and as E A mlizes r, by the choice of the t,hi(Z; 6i)'s it realizes rl, so we finish.

Proof of 1.3. Suppose not, then some realizes p but tp(a, A u 6) forks over A (remember p is stationary). So (by III,4.2) for some hite A and formula cp(f, @, a), 6, a, 'a E A and

As tp(6, A) is FR-isolated there am q c tp(6, A) Iql < h and C s A, ICI < h such that q is over C, and no type over A extending q forks over C. We can aasume q is closed under finite conjunctions, and that for any 6', a', Rm[9)(x; 6', Z'), A, 21 < n* (by 11, 2.9). Let r = {(3g)[cp(f, a, E ) A #(g, a)]: #(@, a) E q}, so Irl < A, r is over C, and clearly r E p. Hence by (*)" there are a*, C*; C c C* c A, a* realizes r, a* E A and n* = P[stp(a*, C*), A, 21. Clearly q1 = q u {tp(a*, 5, Z)} is consistent and let 6* realize it; aa 6*, Z], and P[tp(Z, 6*, a), A, 21 < n* = P[stp(a*, C*), A, 21 it follows that tp(a*, C* u 6*) forks over C*, hence tp(6*, C* u a*), fork over C*, so also over C, contradiction.

THEOREM 1.4: B u w e M ia K-compad, cf 8 2: K 2 ~ ( n , and at least one of t h following conditha

(i) t h e ia p = p<" m h tluct 181 + IT1 s p < 11M11, (ii) t h e ie an indiscernible eet I c M, 181 s 111,

(iii) t h e i s C c 1611, and I r JMl independent over C, and IT1<" + p]<x < 181 S 111.

Then we can jEnd a strictly increaeing e k d y chain of K-compad mode2eM,,i < 8,suchtluctM = (J1,,M1.

Remark. Theorem 1.4, Example 1.4, can be adopted to prove the non-existence of Jonsson models.

OH. Ix, 8 13 SUPEBSTABLE THEORIES AND OATEQOBIOITY 483

First we prove Claim 1.6.

CLAIM 1.6: (1) 8wppo8e A, (i < 6) i8 increasdng, U,<dA( C A, cf 6 2 K ( T ) .

Then we can $nd B, (a < 6) 8 ~ h that B, is increaeing A = u,<d B,

(2) Moreover, if A i8 the universe of a K-IXWP~& model, then the B,'e are and tp*(B,, uj<d A,) doea ?d fork over A,.

too, provided that for emh i

(*I For every type r over any C E A,, 1.1 < K , ICl < K , for any a E Uj<d A, realizing r and any $nite A , there is a, E A, rd i z ing r m h t h d

R[stp(iZ, C), d, 21 = R[stp(li,, C), d, 21.

Proof of 1.5. (1) Let us define B, inductively: B, is maximal subset of A such that U,<, Bj u A, E B,, and tp&, u j < d A,) does not fork over

u # < d B,. For every i, aa c $ B,, neceesclrily B, u {c} does not satisfs our requirement, so tp,(B, u {c}, u,<d A,) forks over A,, hence tp(c, B, u u,<d A,) forks over B, u A, = B,, hence for some j < 8, tp(c, B, u A,) forks over B,, hence tp(c, Bj) forks over B,. Aa K(T) 5 of 8, we get a contradiction.

(2) We suppose A is the universe of tc-compact model, and prove that B, is too. So let r E SnZ(B,) be an Fe,-isolated type, which is not realized in B,; and let I3 E A realize it. For every 2 E u j < d A,, A, and stp(2, A,) eatisfy the requirement (*)" from 1.1 on A , p reap. (for any suitable B). By 1.2 we can replace A, by B,, and by 1.3 tp(& B,) I- tp(& B, u a), hence tp(& B, uZ) does not fork over B,, hence tp(t5, U f < d A,) does not fork over B,, hence B, U 17 contradid the maximality of B,.

A,. SO we have to prove O ~ Y that u f < d B, = A. SUPP C E A -

Proof of 1.4. We prove Case (i), and leave (ii) and (iii) to the reader.

We can easily find a strictly increaaing sequence M, (i < 6) such that llM,ll 5 p, M , is K-compact, M, E M y and for every m-type r over any C E lM,l, Irl < K , ICl < K, and any finite d, for m y li E lMl realizing r there is iz, E 1M,1 realizing r such that

R[stp(7iy C), A, 21 = R[etp(Zi,, C), A, 21.

Now apply 1.6(2).

484 CJATEQORIIOITY AND NUMBER OR’ MODELS [CH. Ix, 8 1

EXERCIBE 1.1: (1) Show that in l . l (*)A we can replam “and finited” by “and finite A, there is a finite A 2 A,, and” (and 1.1, 1.2 and 1.3 holds).

(2) Show that also we can replace “Rm(p,A, 2 ) = P[stp(Ti, C*), A, 21” by R”(p, A, KO) = Rm[tp(a, C*), A , KO] ” and remember R(tp(6, I?), A , KO) = R(stp(6, q, A , KO).

(3) Show similar asktione on 1.4.

EXERCIBE 1.2: Find a countable stable theory T such that T haa a model of power A, which is not the union of a strictly increaaing elementary chain of length p, p regulaz when at least one of the following holds:

(i) there is such countable T’ (not necessarily stable), (ii) An = A, or (iii) of A # p < A, and x < A =- f‘ < A. [Hint: Let M = ( A u “‘A, P”, Q”, . . ., Fn,. . .),<, where PM = A,

QM = @A, F,,(q) = q[n]; FJa) = a. T .= Th(M) does not depend on h and have elimination of quantifkm. So let, e.g., A = An and construct a model N, IP(N)I = = A, such that for every striotly in& sequence (At: i < p), A, G P(N), lAtl s p there is some a E PN, suoh that if c E Q ~ , Fo(c) = a, Pl(c) € A t - U,<( A,, then {Pn(c): n < o} n (U,+A,-- UlliA,) # 8. This suffices for the relativized version : if N = U i < M N L , Nt increasing, then V , < . , P N G A , . To get the full version we have to use a more complicated M: it is the disjoint union of P?, P,M = A , Py+, = “(Py), with all the projection functions.]

PROBLEM 1.3: Charwterize, or at leaat investigate the possible { ( A , p): A 2 p, p regular, A 2 I TI, and every model of T of oardinality A is the union of a strictly inoreasing elementmy chain of length p}.

THEOREM 1.6: The fouoWing are eqvuivalent for a countclbk T: (1) T w categorical in No. (2) For euch m, D,,,(T) k j n i t e . (3) Every p E D( T) is Fh,-ieolaterl. (4) Every model of T of m r d i d i t y KO is atoncic (i.e., F&,-atomic). (6 ) Every d of T of m r d i d i t y No k 8aturated.

Proof. (1) - (3) If p E D( T) is not Fk,-iaolated T has a countable model which realizes it, and T has a countable model which omits it (by IV, 6.3).

(3) s (2) Let F,,, = {p(Z): Z = (zo,. . . , z,,,-,> and ~ ( 2 ) k r for some r E D,,,(T)}. So by (3) {7v(Z): v(Z) E T‘,,,} is inconsistent, so there is a

CH. Ix, 8 13 SVPER8TABLE THEORIES AND CATEOORICITY 485

finite subset I'k of I', which is inconsistent; clearly IDm(T)I I II'kl

(2) * (3) For every my p # q E D, choose Q~,,(Z) ~ p , such that 7(pp,,(Z) E q. Clearly p E Dm(T) is isolated by A {tpp,,(z): q # p , q E Om( T ) } (which is a formula by the finiteness of Dm( T ) ) .

< 8,.

(3) * (4) By definition. (4) * (1) A particular case of IV, 5.2(2). (3) =- (6), (6) * (1) Trivial.

THEOREM 1.8: The following conditions on a totally transcendental T are equivalent:

(1) T is categorical in at lea& one h > IT]. (2) T is catq- in every h > ITI. (3) Evey model of T of cardinality > I TI is eaturated. (4) T ie undimensional. (6) I f M < N, M # N, B E M y Q(X, 8) is not algebraic, then Q(M, a) #

Q", a). (6 ) T h e is no Q ( X , 0) m h tirat for every h > p 2 I TI thre is a

p-eaturated model M y 11M11 = h and E My IQ(M, 7i)l = p. ( 7 ) Working in aeq, there ie a etrmgly eemi-minimal formula ~ ( x ) , such

that for every M y M is prime over Q(M), Q alnmt over 0.

Proof. (1) =- (4) Immediate aa V, 2.10(1), Definition V, 2.2 says not (4) implies not (1).

(4) * (6) Immediate. (6) =+ (6) By V, 6.14. (5) * (7) Let fi(z, a) be a minimal formula, end N be a satureted

model 7i EX, and let p = tp(a, M) not fork over 0, and not algebraic. In the prime model N over PI U {a} there is CE +(N, @)-/MI (by ( 5 ) ) hence p , tp(c, M) rn not orthogonal, hence by V, Exercise 4.10(3) there is E such that stp(a/E, 0) is semi-minimal, (working in P) and there is in it a strongly semi-minimal p(x), and there is a F E M such that +(M, @) is included in the algebraic closure of F U ~ ( ( 5 ~ 9 ) . There is no infinite indiscernible set I PI over f i (M, a) (otherwise the prime model over wl U b , b realizing Av(I,(NI) contradicts (5)) hence in Meq there is no infinite indiscernible set over q$kleq). Let M be a model.

Let Neq be FMo-prime over cp(Meq), Neq < Meqy hence by (6) Neq =

Meq, hence for every 8 E M- tp(8, v(Meq)) is F~o-isolated and as in M there is no infinite indiscernible set over ~(61"); by the characterization of prime models (IVY 4.18) 211" is prime over Q ( M ~ ~ ) .

486 UATEQOItICITY AND NUMBER OF YODELS [UH. Ix , 5 1

(7) == (3) For eaoh bl, 11M11 > IT1 olearly: If p is a type of Wrdinalitg < p = 11 MI1 , finitely &isfiable in 'p(Mea), then p is realized.

(*)

Let Nap be FM-prime over 'p(Mm); then clemly B = 'p(Nw) = 'p(Mm), and we know by (7) that Nw and Map am prime over B, hence isomorphic over B, hence Mw too is saturated, hence M is saturated.

(3) =- (2) By I, 1.11. (2) 3 (1) mvial.

CONCLUSION 1.9: If IT1 < 2w0, T categorical in at leaet one h > IT1 then T i s categorical in e v e y h > ITI.

Proof. By VIII, 4.1 T is stable in IT1 hence by Definition 11, 3.1 and II,3.2 T is totally transcendental, hence the previous theorem applies. LEMMA 1.10: Swppae T ie atable and unidincensional.

(1) ~~anyJi .ni teA,modeZM,ancE(A,m)-ty~pover IM1,1p1 < 11M11, p ia realized in M.

(2) The conclueion in (1) implies thd T doer, not h v e the f.0.p. and if EM $ N , 'p(z, a) i e nd algebraic then 'p(M, a) # cp(N, 3); and for

Proof. (1) Suppose p , a (A, m)-type over M, is a counterexample. We can assume p E @(A) , A c M, IA I < llMll , and for each 'p(Z; g) E A let

For every set A , we define a theory T,( A ,), consisting of the following (i) T. (ii) {aB: < a} is a non-trivial A,-indiscernible set over &. (iii) {'p(Z; b,, b,, . . . ): 'p E A , b,, . . . E &, k#,(b,,, . . . ; a,)} is consistent;

a, E &. (aB, a, serve as additional individual constants, & as a new one-place predicate.)

For every finite A, TJA,) is consistent: we expand M to a model of T,(A,) by interpreting & aa A, a, aa itsee and for uB-by 11, 2.19- there is in ilf an infinite set indiscernible over A. As T,(L) = U {TJA,): A, s L finite} there is a model M1 of TITI+(L), and we can assume M1 is ITI+-saturated IlMlll = 2ITl. So we can find a ITI+- saturated model M of T, A c M, q E @(A) is omitted by M, and I = {a,: a < ]TI+} is indiscernible over A, and 11M11 = 2ITI. Hence the g,, + -prime model over A u I omits q. Let J be such that J u I is indiscernible over A, IJ1 = (2ITl)+; so clearly also the FfTI+-prime model over A u J omits q. We find that T has a I T 1 +-saturated model

every 'ps for 8Ome k, I'p(Jf, 2 k, * I'pW, all = pq.

#,a a@) define P f 'p-

(iv) The type described in (iii) is omitted.

OH. Ix, $11 SUPERSTABLE THEORIES AND OATEOORIOITY 487

which is not saturated, and has cardinality ( 2 9 + (as IAl < (IM(I < 219; hence T is not unidimensional.

(2) The firat phrase follows by VI, 4.4, 6, the second by V, 6.14. For the third phrase note that if there are a, EM,, n < ~ ~ ( M , , , i z , , ) ~ < KO, w.1.o.g. llM, 11 > 2Ho and by an ultraproduct get a contradiction to (1).

CONCLUSION 1.11 : Bqpoee T is iwper8tuble and unidimwbnal. Then:

( 2 ) For every tp(3;jj) and n there is a formula $ ( j j ) euch that

(3 ) For every model M of T there are tp, and ii E A m h that tp(x, 6 ) is

(4) If T ia totally t r a m d , in (1 ) and (2 ) we can r e p h 00 by

(1) RyE = 53, L, a) < w .

R[tp(%; a), L, 003 = n if k$[6].

weakly minimal.

KO, and in (3 ) ccweakly minimal" by "minimal".

Proof. By 1.10, V, 7.6, V, 7.10, and compaotness.

LEMMA 1.12: ( 1 ) B e e T is iwperstuble and unidi-, O(x, 6 ) is a form&, 6 € A and

(*)rc If p i13 a 1-type over A , p k O(z,6), IpI < K , t h 80m.e

c E A realim p (for K = KO we can a88um p = { ( ~ ( x ; 6)}, 6 E A) .

Then for 13orne K - w T ~ ~ & Fc,-mt&ible rnodel M over A, O(M, 6 ) E A

(*)$

then for every c-prim nw&l M over A , O(M, a) E A .

Proof. (1) It suffices to prove that if tp(b, A) is Fc,-isolated, for some q E tp(b, A), 1q1 < A and for no extension q E B(A) of q is R(q,, L, 00) < R(q, L, a), then A u b satisfies and b $O(C, a) (i.e., O(C, 6 ) n A =

This is because we can find a K-compact M, lMl = A U {a,: i < a} such that tp(a,, A,) is as above, where A, = A u {a,: j < i}. Now we prove by induction on i that A, satidles (*)Ic and O(C, zi) n A, E A. The induction step is by the above mentioned ammrtion. For limit stages notice that by the dehability Lemma 11, 2.12 A satisfies (*)n iff A eatisfies and B u (A n O(6, 6 ) ) satisfies (*)rc. Hence the property " O(Q, 6 ) n A E A,, E A and A satisfies is preserved by union of increasing chains of A's.

(2) If w8 a88um inete4bd

if p i8 an F:-type over A , p k O(x; ii) t h 80112e C E A realizes p ,

e(c, 6 ) n ( A u b)).

488 CATEGOFLICITY AND NUMBER OF MODEU [OH. Ix, 0 1

SO let B c A, IBI < K , Q E p B, 1q1 < K where p = tp(b, A) be such that q has no extension q1 in &A) for which R(q,, L, 00) < R(q, L, 00). We can aasume b $A, hence q cannot be realized in A, hence b $ O(@, a). Now suppose A u {b} does not satisfy (*)rs, SO there is a type r over A U {b}, lrI < K , r t- O(z, a), but r is not realized in A. Let c realize r , then tp(c, A u b) forks over A (otherwise by IV, 4.2 r is realized by some c' E A), hence tp(b, A u c) fo rb over A , hence by 1x1, 1.2(4) n =def R[tp(b, A u c) , L, a] < R[tp(b, A ) , L, 003 I R(q, L, 00).

So we can find iZl E A and 'p such that I=& c, a,] and R['p(z, c, B1), L, 001 = n. Let 'p*(y, Z) be such that R[cp(z, c', a:), L, 001 = n if€ C'p*[c', a;] (by 1.11). We can assume q is closed under conjunctions and let

r* = ((3Y)('p(yt 5, ad A 'p*(Y, a*) A e(z, a)): 'p*(y, a*) E g}.

So r* 1 O(z, a), Ir*l < K, r* is consistent (c realizes it: choose b as y), hence some C * E A realizes it. So q ~ { ' p ( z , c * , ~ , ) } is consistent, R['p(s, c*, a1), L, 003 = n, and this contradicts the choice q.

(2) A similar proof, using IV, 4.2.

CONCLUSION 1.13: If T is supemtable and unidimen&nud a E M, e(z, a) is not algebraic and IMI, O(z, a) sat6fy (*)= then M is K - c o ~ ~ ~ ) ( c c ~

[(i)! then M is E-~aturated].

Proof. Otherwise let N be an FZ-constructible, over M, K-compact, such that 8(N, a) E 8(M, a) (exists by the proof of 1.12). So N # M , and we get a contradiction to 1.11.

CONCLUSION 1.14: The following pqertk8 of T are q u i d : (1) T is superstable and unidime7tsional. (2) For some A,,, every A,,-univemal model of T is saturated. (3) As (2) for &, = 21'1. (4) For some set A , IAI I; h(T) I; 21'1 (h(T) 6 the first cardinal in

which T is stable) T(A) 6 categordcal in every h > h(T). (6) For some h , p , A 2 2#, p 2 21'1, T haa no p-universd, nun-

p+ -universal 4 of cardinality h (for e2llpe*8table T , h > p 2 A( T ) is enough).

Proof. Trivially (3) * (2) s (6), and by VIII, 4.7, VIII, 4.1 (4) * (3). Assuming (6), by VIII, 4.6 T is stable and even superstable, and by V, 2.10 (as &,-saturated implies &,-universal) T is unidimensional; 80

(6 ) * (1).

CH. Ix, $13 SUPERSTABLE THEORIES AND CATEQORICITY 489

So assume (1) and we shall prove (4). Let A = 1M1, where M is a saturated model of T of cardinality h(T) (exists by VIII, 4.7). It suffices to prove that any N 2 M is saturated. By 1.11(3) for some Ti E M, and 8, 8(z, T i ) is weakly minimal. Now if r is an F&&p over IN[, which is omitted, and r k 8(z, a) choose rl E ~ ( I N I ) , r 5 r,; so r, is not algebraic, and O(z, T i ) E rl, hence rl does not fork over Ti hence (using IV, 4.2) r is realized in M , contradiction. So N satisfies of 1.12; hence by 1.13 N is F:(,,-saturated; hence (as T is unidimensional, and stable in every h 2 h(T)) N is saturated.

THEOREM 1.16: 8uppose T i s &egm*cul in 80me h > IT1 or every model of T of cardinality A (for some h > I TI) is 1 TI +-universal. Then T i s categorical in every p > I TI, and every model of T of cardinality > I TI ia saturated.

Proof. If T is categorical in h > I TI every model of T of ca,rdinality A is necessarily I TI +-universal, and this implies T is stable in I TI (by VII, 2.9(1)) and superstable (by VIII, 4.5) and unidimensional (by V, 2.10). Now it suffices to prove that every model M of cardinality > I TI be saturated. So suppose M is a counterexample, and by 1.11 there are

E bl and 8 such that 8(z, a) is weakly minimal. As M is not saturated, itisnot ITI+-saturated (byV,2.10) by 1 . 1 3 t h e r e i s q ~ S ( B ) , a ~ B r M , 8(z,a)Eq, 1B1 s IT], M omits q. As 18(M,Ti)I = IlMll > ITI, there is a set I c 8(N, a), indiscernible over B, 111 > IT!. Let J be any indiscernible set over 1M1 such that I u J is indiscernible, I JI 2 h and let A = acl( 1M1 U J). If we prove A omits q, and we prove t b existence of N; A c N, 8(N, a) G A, then we have a non-ITI +-saturated model of cardinality zh, hence there is such a model of cardinality A. As T is superstable and stable in I TI, hence in A, and by VIII, 4.7, T also has a saturated model of cardinality M , of power ITI+. If M , < N we can see that every type T over N, Irl < ITI, O ( x , & ) ~ r , is realized in N (either i t is algebraic or i t is realized in M,) , contradicting 1.13.

By 1.12 in order to prove the existence of N, it suffices to prove only the following:

(i) A = acl( 1M1 u J) satisfies (*)& of 1.12. For suppose E E A, 'p(z, E) I- 8(z, T i ) : if & q E ) forks over li, it is algebraic, hence 'p(&, E) c A (as acl(ac1 B) = acl B by 111, 6.2(4)). If 'p(z, E) does not fork over Ti, it is realized in M , by 111, 4.10.

(ii) No c E A realizes q. Otherwise there are a,, . . . , a, E J , 6 E M , and 'p such that 'p@, a,, . . ., a,, 6) I- tp(c, U J ) (by IV, 2.4(1)).

490 UATEOOBICITY AND NUMBEB OF MOD= [OH. Ix, 8 1

Clearly there is a finite I , c I such that J u (I - I , ) is indiscernible over B U I , U 6. Choose distinct a;, . . . , a; E I - I,. Then ~ ( z , a;, . . . , a:, 6) I- tp(c, B), hence some c' E M realizes q, contradiction.

QUEBTION 1.4: Suppose T is superstable, not IT1 +-categorical. Does T have a non-N,-univemal model in every wrdinality ?

THEOREM 1.16: Buppme h > IT1 and every nzodel of T of cardinality

$r8t cardinality in which T i8 8table): h i8 laornOgeneo~r, OT at l w t I TI + - ~ ~ v M o w . T h ( w h A( T ) M th

(i) euey model of T of cardinality > A( T ) ia Iwmogeneowr, (ii) iflTl + 24, 5 A I h(T), then T h a anon K,-lwmogeneowr modez

Proof. By VIII, 4.10 T is superstable. If T is not unidimensional, T has a model M, 11M11 = A, and a maximal indiscernible set I E M, 111 = No; 80 M is not N,-homogeneow.

Now B U P ~ IlMll > h(T), M is not homogeneous, so it is not p+-homogeneous, for some p < IlMIl, p 2 h(T). Choose a weakly minimal O(z, a), a E M , and a h N < M such that = p, and if c E O(M, a) - IN!, then tp(c, IN!) is realized by r p + elements of Af (possible BB each such tp(c, INI) is minimal), and there are b i ~ N ( I = 0, j s p or 1 = 1,j < p). tp*((b!:j < p), 0) = tp*((bf: j < p), 0) but for no bj E M tp*((b,O: j 5 p), 0) = tp*((bf: j 5 p), 0).

Now choose M,, N < M, < M, llMIII = p+, and for every c E e(M, a) - INI, tptc, INI) is realized by r p + elements of M,. Now choose hl > p+, hl = ad, 6 is divisible by (21'1)+ x W; and let M, be a rulturated model of T, IIM,II = A,, M, < M,.

Let A = 1M,1 u {c EM,: tp(c, INI) is realized by some c E B(M,; a) - "1) and it is easy to check that A satisfies (*)E(o from 1.12(1) hence there is a model M,, A c M,, B(M,, a) G A , 80 IIM,II = A, hence by VIII, 4.4 M, is homogeneous. So there is M,, M, < M, < M,, 11M411 = p+, M4 is homogeneous (possible &B T is stable in p+) . Clearly each c E 8(M,, a) - IN1 realize over IN1 a type realized by p+ elements

Now it is e&sy to find an elementary mapping P, Dom P = IN lu QM,, a), Rng P = IN1 u O(M,, a), P }NI = the identity. Now it is not hard to extend P to an elementary embedding P, of M, into M,. By 1.10 Rng P, = IM,l, hence bl, is homogeneous too; contradic-

tion. So we have proved (i). For (ii) note:

of caraid* A.

of MI.

CH. E, $13 SUPER8TABLE THEORIES AND CATEOOELIOITY 49 1

LEMMA 1.17: If IT1 + N, s A s A(T) (=the$rst cardinal in which T is stable), No = K(T) < K < A, then T has a model MA of cardinality A , in which there is an indiscernible set of cardinality K , but m more.

Proof. Let I be indiscernible, I I I = K, and let M be F&,-primary over I , 80 clearly 11M11 = A(T) and let N < M be such that I c N , IlNll = A. N is aa required (see IVY 4.9).

QUEBTION 1.5: Can we in 1.16 replace “ITI+-homogeneoue” by “ &homogeneous ” 1

EXERCIBE 1.6: Show that any counterexample to Question 1.6 ie supemtable and unidimenaioml.

PROBLEM 1.7: Can we in 1.16 replace homogeneous by model- homogeneous ?

CONCLUSION 1.18: If T is countable, and every model of T of cardinality N, ia Iwmogeneous, then T M c d q h c a l in K,.

Remark. We can replace 8, by any A, KO < A s 2Mo.

Proof. By 1.16 A(T) < K,, so T is K,-stable. By 1.16 T is also unidimensional, so by 1.7 T is categorical in 8,.

THEOREM 1.19: If T is categorical in I TI > No, then T i s a & $ n i t i d exteneion of some T’ s T, IT1 < IT1 (see 111, 6.14).

Proof. By VIII, 3.4 T is superstable, so by 1.17 n e c e d y A(T) < IT1 but ID(T)I s A(T); so we can apply 111, 6.14.

PROBLEM 1.8: Suppose IT1 > KO, IT1 5 ID(T)I; prove T haa a non-univeml model of cardinality I TI.

EXERCIBE 1.9: Suppose cf A = KO, and (Vp < A)(2” < A) (e.g., D ~ ) , hence 2” = P o .

(A) If IS”(A)I > IT1 + IAI 2 A then IB”(A)I 2 2” (instead A strong limit it suffices to wume that for

IA‘I + IL’I < A, 1BC(A’)I s IT1 + IAI

and get IS”(A’)I 2 P o ) . [Hint: Note that if 8 c Sm(A), IS1 > IT1 + A‘ c A , L’ c L(T),

492 0ATEQOR"FY AND ITUMBER OB MODELS [W. Ix, 0 1

IAI, p < A, and we define 'p,(Z, a,) by induction on Z(q) for 7 E "'2, such that, when

7 ) ~ JV = {v: I ~ P E B : ba < z(V))(cp,la(~,a,la)Ep)l > + IAI),

then qn(0), qn(l) E W; and then I W n '21 2 p and even q E W n "'2 implies I W n {V E "2: 7 Q v } I 2 p.]

(B) Suppose A < IT1 < 2Ay JD(T)I = 12'1 then for every L' E L(T),

(C) If T is as in (B), IT1 regular, then there is an m-type p over 0, IpI < A, and p has a unique extension q in D,(T) which is not Fi- isolated. (Hint: Otherwise, act like in (A), where " = 12'1 " replaces

(D) Prove that if A < I TI < 2A, I TI regular, then T has a model of cardinality I TI which omits some type (from D( T)) hence is not universal (except when ID( T) I < 1 TI hence T is a definitional extension of some T' c T, IT'I < ITI).

ID(TnL')I 5 IL'I + A.

" > IT1 + IAl".)

DEBWITION 1.1: Let TVf(8 , A) where 8 = 8(z, a) is a formula over A, means that any Fz-type p over A, B ~ p , is realized by some c E A.

EXERCIBE 1.10: (1) TVL(8, A) if€ TVho(8, A) and TVt,(6, 8(A, a) U a) for stable T (use 11, 2.12).

(2) If A, is increasing (i < a) B(A,,a) = 8(Aa,7i) and TV;(e,A,), then TVL(8, Ute, A,) provided T is stable.

(3) Suppose TP,(8, A) K 2 K(T), and N is an Fc,-constructible model over A. Then 8(M, a) c A when one of the following holds:

(a) for every finite A, C c A , ICl < K , a type p over C, 1p1 < K , 8 €13, and c E B(C, a) realizing p , there is ct E A realizing p such that

&tp(c, C), A , 21 = mtp(c'Y C), 4 21,

(8) every equivalence relation ~ ( s , y, a), over 8(C, a), E A, hae < KO or > IT1 equivalence class inp(A) whenever p is a l-type over A, IpI < K,

6 E P (use Exercise 111, 4.11). (4) If T is unidimensional and stable, TV~o(fly A), then every

equivalence relation 'p(z, y, a) over O(C, a), E A, has < KO or1 8(A, a)] equivalence classes in B(A, 7i). (6) If T is unidimensional (and stable), K 2 K(T), TVL(8, A) ,

lB(A,a)l > IT1 then for any E"c,-constructible model N over A, e(iv, a) E A .

CH. I x , 8 13 SUPEBSTABLE THEORIES AND CATEOORICITY 493

EXERClsE 1.11 : (1) Use Exercise 1.10(3)(p) to get another version of V, 6.14 and V, 6.16.

(2) Show some condition (as (a), (6)) is needed in Exercise l.lO(3). [Hint: M = (WI,P",Py, Q", Qy, F,M, c y ) ~ o ' 2 , E y 2 , T = Th(M) where :

Let

P" = "'2 u ("2 x w ) ,

Q M = w 2 ~ w ~ w ,

W( = P M U Q",

Py = { v : v e r ] or v or r]e veW>2} u { ( p , n ) : r ] 4 p e W 2 , n < w } ,

Q y = { ( p , n , m ) : r ] ~ p e ~ 2 , n < w , r n < w } ,

F,M is a partial one-place function: its domain is Q",

p r k P r k = v r k , P r k + w, { ( p d p = v,

F,M(<p, n, m ) ) =

I$" = r ] (which is in P").

Building an automorphism of saturated models we can prove that T is superstable. Also for some model N of T, IlNll > IPMJ (remember F , is not one-to-one even on (for any n) the set {v} x {n} x w which we can "blow up"). Let A = "'2,O = P(x) , now for every aeQe, for some V E " ~ , F ~ ( U ) E P - A . Lastly TVNo(B,A) holds [note that it suffices to prove this for the reduct to every finite sublanguage, where we have enough automorphisms].]

EXERCIBE 1.12: Through the following series of awrtiona we reprove 1.18.

(A) If every model of T of cardinality h is homogeneous I TI < 2h, h > ITI; then for every p , CR(p, 2) < a0 (see Exercise 11, 3.8) (use

(B) If for every p , CR(p, 2) < 00 then there is an infinite set of indisaernibles, and for every A, there is an F",,-atomic model M over A. (C) Use (B) to show that under the assumptions of (A), T has a non-

8,-homogeneous model M of T, which is N,-saturatied, hence

(D) Show that under the hypothesis of (A), there is T, z T,

EM(h)) .

IlJfll 2 IWQI. I TII = ID( T)1 Suoh that PC( T,, T) ie a h g o r i d in A.

494 OATEOORIOITY AND NUMBER OF YODBLS [m. Ix, 8 1

EXERCISE 1.13: ( 1 ) If T is countable and K,-categorical prove that every model of T is F%-saturated (see 111, 6.16).

( 2 ) If T is totally transcendental, every N,-saturated model is F&,- saturated.

EXERCISE 1.14: Find a totally transcendental T such that for some modelMdT, llMll < IT1 = (D(T)I.[Hint:LetIbeatree,(B,:i < A} be a set of branches of I, let L = {P1: i < A}, Pi a one place predicate, and T = Th(M), where = I , P,(M) = B,.]

EXERCISE 1.15: Suppose h = IT1 = heA is regular, then the following conditions are equivalent.

( 1 ) AU Fi-compact models of T of power IT1 are isomorphic. ( 2 ) Every p E D( T) is Fi-isolated. (3) Every FA-compa& model of IT is A-saturated.

THEOREM 1.20: Suppo8e T ia eu~perstable and A > IT1 b the $rat cardinality in which T b atable. Then for every p 2 I TI + K,, I ( p , T) 2 min{P, P}.

Proof. Let x be any regular cardinal I TI c x 5 A, and choose a formula ~(z, a) (existing by 111, 6.16) such that:

(i) For some A, E A , IAI s ITI, and I{p ES(A): tp(z,a) E ~ } I 2 x . (ii) a* = R(&, a), L, a) is minimal (for 'p's sa t i smg (i)). Choose

a model Mo, llMoll = ITI, T i € lMol, such that if B , A , n , p are finite, B 5 lMol, p a finite type over B, and for some c realizing p , n = R[stp(c, B), A , 21 then there is such c E Mo. Choose also I indiscernible over IMol, I I I = KO. Note that by the regularity of x and the minimality of a* :

(*) If a E A, IAI < x , then { p ES(A): ~(z, a) €21, R(p, L, a) < a*} has cardinality ex.

Now there are two possibilities.

When Possibility A occurs, there is a set A*, lA*l = A(q*) (see D e w -

CH. Ix, $ 1 1 SUPERSTABLE THEORIES AND CATEOORICITY 495

tion 111, 5.1, Lemma 111, 5.1, 3) such that every extension of q* in #(A *) which does not fork over ais stationary ; and I ( ~ ) E S ( M ~ ) : q* C_ p , p does not fork over = 2h('J*). We can assume A* 5 M , and that for any E(z, y, iz) E E(A) and G realizing q*, there is an indiscernible set I c A*, b d on Borne q, z q* u {E(z, c, B)}. If Possibility B occurs let A* = a. Now we define by induction on a < x, pa and a, realizing pa such that (a) ~(z, a) € p a ES(A*), R(pa, L, a) = a*, and if Possi- bility A holds q* c pa, (/3) no extension of pa in S( lMol u I u {al: i < a}) forks over iz. Now pa exists as the number of p's satisfying (a) is 2 x (check each possibility) whereas by (*)

[ { p €#(A*): p satisfies (a), but not @)}I < x because lMol U I u {Bf: i < a}I < x so we can choose an appropriate pa and also a,. Now w.1.o.g. for every a, no extension of p , in S( woI U I U {a, : i < x, i # a}) forks over A* [if Possibility A holds, as pa is stationary p , = tp(a,,A*) I- tp(@,, wo( U I U {ai : i < a}); if Possi- bility B holds, we choose by induction on a, y, < x and a,,Jy < y,) such that (i) holds, and we replace: (ii) by (ii)' no extension of pa in S(wol U I U (ap,y:/l < a, y < yp) fork over A*, and (iii) realizes p, ,{ i~, ,~:y < y,} is independent over woI U I U {@p,r:,8 < a, y < yp}, and every p~Sl(!.Ilf,I U I U {ap,y:p < a, y < yp}) extending p , is realized by some a,,y, this is possible and it, = a,,o are as required- if qeS(wol U I U : p < x, /3 # a} is an extension of p , forking over A*, for some y < y,, q I- stp(a,,,, A*) and by IV, 3.13 applied to FZ we get a contradiction]. Clearly, I is indiscernible over

So let p 2 12'1 + N,, and I, be an indiscernible set over IM,l u {af: i < x}, I E I,, II,I = p. Clearly for every a, no extension of pa in S(lMol U I, u {al: i < x, i # a}) forks over iz. For every 8 c x let M; be any F&-constructible model over lMol u I, u {al: i E S}, so by 1.1 M$ realizes pa iff a E S (for a 4 S if Possibility A holds, each pa is stationary, if Possibility B holds, show that for each c realhing pa, stp(c,A*) is omitted). Clearly 1M;l = p + 181, thus we have proved that (T, 2') has (x, IA*l)-freedom. All this depends on the parameter x which we suppresss up to now.

deP

P O I u {a, :i < XI.

Let K = mi.&, A}; and we split the proof to cams.

Cme (i). There is a regular x 5 K , 2X = 2". Apply the above construction for x + 12'1 +. If Possibility B holds, for some k < No, (T, T) has (X,X)-freedom hence by VIII, 1.4 I ( p , T) > 2X = 2" = min{2c, 2"). So,

496 CATEOORICITY AND NUMBER OF MODELS [CH. I x , f 1

suppose Possibility A holds. If 2x > 2IA'I again by VIII, 1.4 101, T) z 2x = min{2', 2'5), 80 aasume 2x = 2IA'1. Now IA*I = h(q*) 80 if h(q*) > KO, by III,6.13'sproof ID(T)I 2 2A(a') = 2X = min{2A, 2') > IT1 hence by VIII, 1.2 I ( p , T) 2 lD(!l')l 2 2"q" = 2IA'1 = 2 X = min{2", P}. SO we can assume h(q*) 5 No (hence h(q*) = No), so 22 = 2u0, IA*l = No, but 2' = 2% hence IT1 K < 2Wo. We can also assume lD(T)l < 21, hence by 11, 3.2. JD(T)I s 12'1. By 11, 3.15 as h > ITI, in fact h = 2n0, hence K = p < P o . We shall deal with this in Case (iii).

Caee (ii). Not Case (i). So K is singular and x < K * 2x < 2'; let K = zf ~ ( i ) , of K < x(i) r: K , x(i) regular, and increasing. For each i there me asf), Mx(f) , pX(,)(x; FZX(,)) as in the construction above and choose &lo, I, llMoII = of K .t ITI, Ufedk M,, E Mo, as before. Now we define inductively on a 4 K elements a,, euch that if I,<, x(j) 5

has a unique extension in 8(lMol u I u {a,: i < a)). Now let I,, be an indiscernible set over lMol u {a,: i c K}, I c I,,, II,,l = p, and for each 8 G K let M i be any F&,-constructible model over lMol u I,, U

{as: E 19). As before iKi realizes pa 8 a E 8 (by 1.1) hence (Ip, 2') has ( K , cfK+ITI)-freedom as 2cf"+lTJ < 2" (provided that IT1 < K )

by VIII, 1.4 I (p , T) 2 2" = min{2A, 2 p ) . Suppose IT1 = K . We can assume ID(T)I 2 h (otherwise we return to Case (iii)), and as h > I TI, we can work with x = I 2'1 + , so if 2IA*1 < 2ITl we can finish as in Case (i). Otherwise, D(T) 2 2A*(*) 2 21A'I = 2ITI, so by VIII, 1.2 we finish.

a < x ( i ) then Rrn(PU, L, 00) = as0 I= VX(f)[%, ZX(f)I = tp(a,, IMol)

Caee(iii). ID(T)I 5 IT1 s p < 2W0,2" = 2~o.Herewechoosex = IT]+, and &;a) w before; by VIII, 1.3 we can Mume FZ is empty, so 'p = p(z). By the choice of 9, if $(z, 6) 1 p(z), #(z; 6) forks over 0 (or even R[#(z; 6), L, a01 < a*) then for every A, 1{p ~r9(A): #(z; 6) ~ p } l s IT1 + IAI, hence by 11, 3.2 R[#(z; 6), L, No] < 00. Choose q, p(z) E Q E D,(T),suchthatforsomeA,IA~ = IT1 andj{pES(A):q s p}] = 2b; by 111, Section 6 MI h*(q) 5 No, hence h*(q) = KO, there are E, E E(0) such that En+, refines E,,, and for each c realizing q, q U {E,,(x, c ) : 7c < W } is stationary. Choose b, (Z < W ) realizing q, such that for every b realizing q, and n, for some I > n, I=E,[b,,b], and {b, : I < w } is independent over $3. Now we define by induction on I, a natural number n(Z) and elements a, (7 E l2) realizing q such that

(i) if q # v E '2, then I=4JMl)(urr, av), (ii) if v Q 7 , 7 E '2, v €82, then kE,(k)(an, av),

OH. IX, 5 21 LOWER PARTS OF THE SPEO!CRUM 497

(iii) {bi: i < w } u {a,,: r) E It2} is independent over 0, (iv) if for each r) E ‘2, I.ln0,(a,,, a;) and {bf: i 2 } )

= tp(( ..., a; ,... ),,sta,{bi:i < ~ } U { U ~ : Y E ~ ~ ~ } )

(for (iv) use ID(T)I I IT1 < 2No). Now for each r) E O2 choose a,, such that {bI: 1 < a} u {a,,: r ) E 012} is

independent over $3 and a,, realizes q U {En( l ) (x , a7,1): 1 < w } . For each S c “2, (81 = p, let M,(S) be an Fgo-prime model over

A, = {bI: 2 < w } u {a,,: 7 ~ q , and let B, = {c E HI(&; c realizes q, and tp(c, 8) is realized in MI(#) by 5 I TI elements for some 8 E A,} 80

A, E B,. Now let M ( 8 ) be a Fg,-constructible model over B,. The proof is now like that of VIII, 1.8, when we observe the following, whose proof we leave to the reader.

Let c depend on C (here always C u {c} s q(6)) if for some 8 E C, in the Fg,-prime model over 8, tp(c, a) is realized I I TI t’ imes.

Fact 1: If tp(c, C) forks over 0, then c depends on C. Fact 2: This dependence relation is transitive and of finite character. Fact 3: Any FLI,-type over B, which forks over 0 is realized by some

Fact 4: For any Z, {stp(c, 0): c realizes q, c depends on 8} is countable. CEB, , hence by 1.1 I{cEM(S) : c realizes q}l = B,.

M.2. On the lower parts of the ~pe~trum

Remark. We have used here, for p€S”(B), B c A, “a stationarization of p over A ” as “a complete type extending A not forking over By’.

THEOREM 2.1 : (1) If T ie euperstable, countable and not categorical ie KO, then I ( N o , T) 2 No.

T) 2 la + 11; and i f T ie totally tranecendental except w h a = 0, T categorical in No, then I ( N , , T) 2 la + 11 + KO.

(3) If T ie totally trawcendental, IT1 = N, (and of c~uree IT1 s ID(T)I),P 2 a a n d T i e n o t c a t ~ ~ i n I T I + , t h e n I ( N 6 , T ) 2 1s + 11 and, except when a = 0, T categorical in No, I ( N 6 , T ) 2 + 11 + No.

(2) If IT1 = 8, (and of course IT1 I ID(T)(), then

Remark. On IT1 I ID(T)I see 111, 6.18.

498 UATEGOE1ICITY AND NUMBER OF MODELS [CJH. Ix, # 2

Procf. (1) If JD(T)I > KO, the conclusion follows by VIII, 1.2. So aasume ID(T)I 5 KO, hence by IVY 5.1 T haa a Fi,-prime model M,. As T is not N,-categorical, there is a p E Om( T) which is not isolated, hence M , omits p ; and let p , E Bm(M0) be a stationarization of p. Define inductively ii,, (n < w ) suoh that p,, = tp(si,,, 1M,1 u U,<,, 8,) is a stationarization of po (hence {ii,,: n < w } is indiscernible over lMol); and let N, be an FL-prime model over Ul<,, ?Zl (exists by IVY 5.1(4)). Now we shall prove that Nn omits

r l : rl a stationarization ofp over U ii, I<n

Otherwise there is cp(E, a,, . . . , a,,- k r , ~ ( 5 3 , a,, . . . ) does not fork over 0. So cp(Z, iio, . . . ) does not fork over lMol hence some Z E M,, reaJize it (by 111, 4.10(1)). But N o omits p, hence for some #(if) ~ p , C,#[C] hence t(3Z)(cp(Zy a,, . . . , a,,,) A 7y5[Z]), contradiction. So in N,, In = {a,, . . . , a,,- 1} is a maximal set independent over 0 of sequences realizing p.

Hence by V, 3.13(2),

&(P, 0, Nn) 5 l I n l = n s W(P) ~ W P , 0, N n )

and by V, 3.9(1) w(p) is hih. We can conclude that {n: N,, r Nn0} is finite; hence I(No, T) 2 No. (2) The assertion I&, 2') 2 Ice + 1 I , follows by VIII, 3.4 if T is not

superstable, by 1.20 if T is superstable, not stable in IT1 = Ha except when a = 0, and then the aesertion is trivial. Now if T is superstable, T stable in IT1 (see 1.17, IVY 4.9) for every /3 s a T has a maximally KB-saturated model of cardinality K,, hence I(Kay T) 2 la + 11. So now we assume T is totally transcendental, and IT1 > KO OT T not categorical in No, and have to prove, in fact, I&, T) 2 No. As IT1 I ID(T) and by 1.6, for some m D,(T) is infinite, hence aa in 1.6 some p E Dm(T) is not isolated, hence M,, the prime model of T , omits p; and M, exists (by IVY 3.1,lO and IV, 2.18). We can proceed aa in the proof of (1) but maybe llN,, 11 < I TI. If for some 6, the prime model over 6 has cardinality ITI, we define B,, such that tp(a,,, lMol u 6 u Ul< , 4) is a stationarization of po, let N,, be a Fh,-prime model over 6 u Ul<,, then I,, = {al: 1 < n}, is, in N,,, a maximal set independent over (by 0) of sequences realizing p , hence by V, 3.13(2) and V, 3.15(2),

d W p , 0, N,,) 5 lI,,l = n s w(p) dim(p, 0, N,,) + w(6,0)

hence as in (1) I( Ha, T ) 2 KO.

OH. IX, 8 21 LOWER PARTS OF THE SPECTRUM 499

So suppose for every 6, the Fko-prime model over 6, has cardinality < \TI; we can also assume, w.l.o.g., 0 < a < w (by (1) and the first part of (2)). For each non-algebraic p€D(T) , let I, be an indiscernible set based on p of cardinality N,, I, = {.", : i < Na}. Let M, be Fko-prime over I,, so clearly J(r€D(T) : M, realizes r}l = /{TED(!!') : for some n, r is realized in the Fko-prime model over ui<n$}l < N,. Also the number of algebraic q€D(T) , is d llM,II < N,. As N, d (D(T)I, clearly there are 2 N, non-isomorphic Mp's, so we finish.

(3) By 1.8, T is not unidimensional, hence by V, 2.10 for every y 5 8, T has a maximally Kt,-saturated model of cardinality K,, hence I(&, T) 2 Ip + 11. So m in (2) it suffices to prove I&, T) 2 K,.

As in (2) let M , be a F&-prime model of T, p E D,,,(T) omitted by No. We can assumep is stationasy. (By 111,2.9, for some&(%, y) E FE(0), p u { E ( q a)} is stationary for any a realizing p ; we add the names of the 1F-equivalence classes as new relations. As eaoh model can be so expanded in finitely many ways the proof is suffioient.) As in (2) we choose appropriate 6, choose as before a,, N,(n < w) , and then find N t , 6 V U l < , al c N$ llNtll = K,, such that no iZ E N: realize.^ a stationarization of p over 6 U U,<, a,, or at leaat dim(p, 0, N t ) < KO and the rest is aa in (2). So what remains are the choices of 6 and Nt;.

Case A. For some 6, in the Fbo-prime model over 6, Ms there is an infinite indiscernible set I.

Clearly we can find an idnite indiscernible set I, c INnl, and by 111, 3.6 we can assume it is indiscernible over 6 v Ul< , 4. Let J , be an indiscernible set over 6 U Ute,, a,, I,, c J,, lJnl = K,, and let N: be Ft,,-prime over J , u 6 U Ulen Z l ; clearly IlN~Il = K,, and every type over 6 U U l < , 8l realized in N;, is realized in N,; so N i is aa required.

Case B. There is a non-algebraic type orthogonal top. Let q be such a complete type, p ~ S ~ ( b ) , such that q is stationary; choose {q: i < NB} be independent over (6 u Ulea, a,, 6), c, realizing q, and let N; be Fk,-prime over 6 U Ul<,, Zl u {cl: i .c 24,). If rj is any stationarization of p over 6 u U1< al u {cf : i < j}, it is weakly orthogonal to the stationarization of p over the same set (by V, 2.1(2)). So r, has a unique extension in Sm(6 u Ul<,, til u {cf: i < N,}) so m4 does not fork over 0, hence as in (1) it is omitted by N i . As clearly llNtII = K, we finish.

Case C. Not Case A nor Case B. By 1.8 T is not unidimensional, so by V, 2.10, there are a minimal type r , and a type q which are orthogonal.

500 CATEGORICITY AND NUMBER OB MODELS [OH. Ix, 5 2

As K ( T ) = KO, we can assume that r and q me complete types over some 6, and they are stationary; and the stationarization of p over 6 is not weakly orthogonal to r nor to q (by not Case B) and as T is totally transcendental, some ~ ( z , 6) E r is a minimal formula. Let J be an indiscernible set over lNnl based on q, and let Nf; be Fi,-prime over lNnl u J. Clearly the stationarization of r over N , has a unique extension over I Nnl u J , hence is omitted by Nf;, hence rp(N:, 6) = t#,,, 6). If dim@, N:) = KO, then dim(p, 6, N i ) = KO, so there is {an: n < W) exemplifying it. As tp(P, 6), r are not weakly orthogonal, there are c, E acl(2 u b) - acl (6), c, E rp(Nf;, 6 ) ; and clearly {c,,: n < w } is independent over 6, hence indiscernible, contradiction to not Case A.

EXERCISE 2.1: In 2.1(1) replace “superstable” by “stable and K ~ ~ * ( T ) = 8,’’ and show N,, g N,,, ifF n = m.

THEOREM 2.2: 8wppose T is countable and categorical in K,. (1) If T is not categorical in KO, then I(K,, T ) = KO. (2) Every 4 e . l M of T has m i n i m l and prime proper extension N

(min iml means for no N’, M $ N’ $ N ; prime means that for each N’, M < N’ implies N can be elementary embedded into N’ over M ) . Note N is necesearily unique.

(3) Suppose T is not categorical in KO. There are countable models Ma (a I W) of T such. that:

(i) Every countable model of T is isomorphic to one of them. (ii) They are pairwise nun-isomorphic, M , is saturated and each M ,

(iii) M , + , is the prime minimal proper extension of M,, and M , =

(iv) Ma can be elementhry embedded into M , iff a I /I (for a, /I I w).

is homogeneous.

Un<, M,, and M , is the Fk,-prime model of T .

Proof. (1) By VIII, 4.1 T is totally transcendental, hence by 2.1 I@,, T ) 2 KO; the other direction follows by (3)(i).

(2) Let M be any model of T , and let d(z, a) be a minimal formula, a E M (at least one exists by l . l l (4 ) ) . Let b E O(Q, a) - [MI, and N be F&,-prime over M u b.

If M N‘ then by 1.8(5) O(M, a) # O(N’, a), hence there is CEO(N’,E) - d ( M , a ) . Let B’ be the following mapping: P t =

the identity, P(b) = c. Clearly P is elementary, and we can extend it to an elementary mapping from N into N‘ as N is F&-prime over /MI U b. So we have proved the primeness.

OH. Ix, 5 21 LOWER PARTS OF THE SPECTRUM 50 1

Now suppose M $ N' $ N; so again choose c E 8(N', 8) - [MI; hence tp(c, I M I u b) forks over [MI, hence tp(b, [MI u c) forks over IMI, hence b E acl(]MI u c) hence B(N, Ti) E acl(lM1 u b) E acl([M] u c) E

IN']; hence by 1.8(5) N = N'; so we prove the minimality. The uniqueness follows as in IVY 4.20.

(3) Define Ma by induction: let M , be the F:,-prime model of T, and B(z, Ti), 8 E M,, be a minimal formula. If M , is defined let b, E 8(Q, 8) - M,, and M,+, be F&-prime over ]M,l u b,, and let M , = Un<, M,.

By 1.8(6), n = dim(tp(b,, a), M,) c No ( M , is not saturated, as T is not No-categorical) so w.1.o.g. n = 0. Clearly M , is the F&,-prime model over 8 u {b,: Z < n}.

If M is a countable model of T, we can assume M , E M , and let I c 8 ( M , 8) - acl 8 be a maximal set independent over 8; and clearly M g Ma where a = III , so (i) holds.

Clearly (iii) follows by ( 2 ) . Now let us prove (ii); by (iii) if H, g M,, n c k then for every 1

M,+, g Mk+, (proof-by induction on Z), hence every M,,,,, is isomorphic to some M l , 1 5 k , contradiction to what we proved from (1) by (i). Hence M , (n c w ) are pairwise non-iaomorphic. By 1.8(6) it is easy to see that if 8(z, 8) is minimal, I E &Cry Z)-ad Z is independent over 8, 111 = KO and 8 u I E [MI then M is KO-saturated. Heme M , is saturated, and any countable model into which No can be elementarily embedded is saturated. So from (ii) we have to prove only the homogeneity of M,. Assume Til EM,, tp(8,, 0, I M J ) = tp(Z2, 0, [M,1) and we shall prove that M , has an automorphism P, #'(til) = 8,. We define by induction on k Mf, < M,: Mb is an F&,-prime model over Ti,,

If Mf, is defined it is not saturated (as M , is not) and, by ( 2 ) if Mf, # M , there is a prime minimal proper extension ML,, of it, Mf, $ Mf,+ < M,. If for some I , Mf, is defined for every k again U k < o Mf, is saturated, hence M, ie saturated, contradiction. So let Mf, be defined just for k I k(Z); clearly there is an isomorphism F, from MA onto M i , Po(Til) = Ti2 , and we can extend it inductively to the isomorphism Pm from M i onto M i . If k(1 ) = k(2 ) we finish. Now let MA g ill,,(,) hence Nb g M,,,,, hence (by (iii)) Mf, g L W , ( ~ ) + ~ , so k( 1) # k( 2) implies M,(,) +,(I) M,,(,) + r(2), contradicting, the part of (ii) we have proved.

So only part (iv) remains, for a = w we have already proved it. Now suppose M a g N < M B defines inductively N,: N o = N . N , + , < M B is a prime minimal proper extension of N l . Tf N, is defined for every 1,

wl < M,.

502 CATEOORICITY AND NUMBER OF MODELS [CH. Ix, $ 2

EXERCME 2.2: Extend 2.2(2) to any ITI+-categorical theory when llMll > IT1 or M is FiO-saturated or M extends such a model.

EXERCISE 2.3: Suppose T is totally transcendental IT1 = K, L N,, and let M be the F&-prime model of T. Then (1 ) o (2) =s (3), (1) =- (4), cf X, > KO implies (4) * (1) and when N, is regular they are equivalent.

(1) M is not K,-saturated. (2) Some p E D( T ) is not F&-isolated. (3) For some m IDm(T)I L N,. (4) M is not Np-saturated.

DEFINITION 2.1: Let I(A, F, T ) be the number of non-isomorphic F- saturated models of T of cardinality A.

When F = F;T, let I=(& K , T ) = I(A, F;, T) .

MAIN THEOREM 2.3: Let T be superstable not multidimensional (see V , 6.8) then there is a cardinal p = N D ( T ) (=the number of dimensions of T ) and a group B = a( T ) of permutations of p , such that:

(1) 1 I p I 2ITI andevenp I; A(T),andwhen T iscountable,p I KO

(2) p = 1 iff T is unidimensional. (3) If X, > A(T), or K, = A(T), a - p+ p 2 w , Ia(N,, N,, T ) =

(4) If Na = A(T), 0 5 a - ,9 < w , p < KO, then Ia(Na, N,, T ) =

( 6 ) If 8, < A(T), then Ia(N,, K,, T ) = 0,

o r p = 2No.

la - /3 + I l f i / l a (see below).

Z:=B IY - p + l I P / B - where f/G = { f / - G : f : p -+ x, 0 E Range f } ; for 0 a group of permutations of p ; x, p possibly jinite and - is the equivalence relation, f l - fa iff for some 0 E B, f l = faa.

THEOREM 2.4: Suppose T is totally transcendental, p = ND(T). There are functions hl, ha : p + {HE: No I N, I ]TI+} 8 ~ ~ h that hl(i) = h,(i) is KO or a successor except when cf[hl(i)] = No, hg(i) = hl( i )+ and hl, h2 preserved by the permutations of a, satisfying:

(I) The following cardinals are equal or both ,finite, when, N, 2 IT1 + K , , p > 1:

OH. I x , 8 21 LOWER PARTS OF THE SPECTRUM 503

(2) If in (1) they are$nite, every model is Fto-saturated as h,(i) 6 K, for every i. If for no i, K , = h,(i) < h2(i) , then every K,-compact model of T is FGB-saturated, hence It&, NP, T) = I“&, K,, T) and 2.3 applies. In the remaining case necessarily a < o, and if K, > ITI, easily

I t (K, , K,, T ) = n K,/G(T) - n (K1- 1)/G(T)* t <r t <lI

If /& = 1, N, > IT1 then I(Hay N,, T) = 1; if /& = 1, 8, = IT1 =

ID(T)I, then I t ( sa , K,, T) = KO.

(3) I f a 2 /3 + w , OT a > 8, p 2 No, 8, 2 ITI, then It&, N,, T) =

la - p + 11”.

Proof. The proof is broken to several stages, which we may use later. In fact, they contain information not summed up in the theorem. For 2.3 let x = Q and for 2.4 x = t , but we concentrate on the first. 8trzge A. By V, 1.13( 1) the relation of non-orthogonality among regular

types is an equivalence relation. Let {pi: i < p} be a set of representatives and ND( T) = p, and let I, be based on p,. Every automorphism P of Q induced a permutation HF of p: HF(p, ) = pf iff P(p,), pf are not orthogonal. The set of permutations HF of p is a group we call a = a( 2’). As T is not multidimensional cleaxly p s 2ITI. Now in order to get the precise results we shall find “nice” p,’s. Btuge B. For any cp(s,a) there is k, such that for any model M,

I{ltp(M,a)l:&~M)I < ttV (by V, 7.5 for the trivial K). By VII, 3.4, V, 5.8, 9. T does not have the f.c.p. ; and for any suitable K (see V, Section 7) and weakly K-minimal p(x; a) the conclusion of V, 7.6 (2) holds.

The proof is easy by the non-multidimensionality of T. 8Qe C. We shall define by induction on a formulas pa@; fja), h(fj,),

&, and a natural number %(a) for a < a. < p+ . Let

w a = {(9Jt(% alh WJ>: Y < 4 80 by V, 7.1(2) we can find a K, satisf+g C1-6 from V, Section 6, Wg = W,, 80 R,-minimality, etc., axe defined. As T is superstable, either every formula is K,-algebraic, or there is a non-K,-algebraic formula ~ ( x ; 7i) with minimal R(cp(s; a), L, a), hence p(x; a) is weakly

504 CIATEGORIGITY AND NUMBER OF MODELS [CH. Ix, 8 2

K,-minimal. If T is totally transcendental we can assume p(x; a) is R,-minimal. By V, 7.7 (possibly after changing p) there are formulas #(g), 1 such that:

(i) B is an equivalence relation on #(a) with n(a) < X, equivalence classes.

(ii) k#p'] implies p(x; a') is K,-weakly-minimal and, for totally transcendental T, p(z;B') is even K,-minimal. Also there is a", kE[a', a"], tp(a", 0) = tp(Si, 0).

(iii) If k+[aJ, then p ( ~ ; 7il) 5 Ka p ( ~ ; a,) iff kB[Sil, aa]. Let pa, #,, E,, n(a) be p, # , E , n reap. Now a. exists aa clearly

p + , bemuse if I, G p,(B, a,), C#,[iZ,] HKa(I,) = 00, then the 1,'s a. are pairwise orthogonal.

Clearly each (p, #) appear at most once, so a, < I TI +. Stage D. We define by induction on a, a, = ?i,,on. . .n~,,n(,)-l

is weakly K,-minimal when x = a, p"(s; a,) = Vl<S(f#) pa(z, S . l ) such that

(i) q,(z, (ii) kA\l<n(u~ #[7iu,,l A A \ l < k < n ( a ) lBa(aa ,b aa,k),

(iii) tp(G,, A,) is F~,,-ieOlatd where A, = U {as: fl < a}.

Clearly this is possible by IVY 2.18, (ii) of Stage C and (i) above. Now for each a we choose a maximal family of pairwise orthogonal, regular, infinite indiscernible sets I ; (y < y(a)) such that

(iv) 1; E pa(&; a,), I; based on a,, (v) H&) = 00, SO I; is indiscernible over Us<, ps(By as) U

(vi) if each p,(x, Ti,,l) is &-minimal then y(a) = n, and If c

Clearly the I f s are pairwise orthogonal, and regular. Let us work in P, and let A* = aol(A,,) 4nd p t = Av(I;, A*). Stage E. Let N be any FGo-saturated model of T. By the inductive

choice of the am's we can assume A* E P I , and for every a,y, let J; c N be a maximal indiscernible set over A* based on p; .

Let M E N be Fc0-constructible over A* U u {J; : y < y(a), a < ao}, and we shall prove N = M. We prove by induction on a that pa(N; a,) G M. If a is the first for which this fails choose c E @(N, a,) - 1M1. Clearly M <Ka N (by V, 6.4) hence IZKa(c, !MI) = 00, hence tp(C, !MI) is not orthogonal to some p;, hence if x = a by V, 1.12 Av(J;, 1611) is realized in N, contradicting the maximality of I;. If x = t, T is totally transcendental so for some 1 < y(a) = n(a), c E p,(cE, and c realizes Av(If, ]MI), as p,(x, a,,l) is K,-minimal we get again a contradiction.

Stage F. By Stage E, the function h$ defined by h$(a, y ) = IJtI = dim(p?, M) determines M up to isomorphism (remember the p; are

v,(Q, au.1)-

OH. I x , 8 21 LOWER PARTS OF THE SPECTRUM 505

stationary (by 111, 6.9) and pairwise orthogonal). Moreover no regular type is orthogonal to all p; (otherwise let M be IZ'~+-snfurated, tp(c, lM1l) such a type, Ml an F&-prime over 1M( u {c}; then necessarily not all J t are maximal in Ml). So necessarily by Stage A, p = C{ly(a)l: a c ao}, and the p; are a set of representatives.

We can conclude I"(N,, X,, 2') 5 la - 81" + HE. Stage G. The two first assertions of 2.3( 1) are immediate. As for the

third as a. < IT1 + = N, it suffices to prove for each a, Iy(a)l 5 KO or ly(a) = 2Ko. Choose N countable, G,,EN, and let S = {~ES(N) :gY(x, G,) ~ p , p is not K,-algebraic} ; we know each p E S is regular, hence non-orthogonality is an equivalence relation on S, and y, is the number of equivalence classes. Now S is a closed set (in the natural topology on S(N)) and non-orthogonality is a Bore1 equivalence relation on 8, hence by a theorem of Silver (see e.g. [Sh €441)

Stage H. Suppose p < No, then we can find a* E A* such that each p; does not fork over a* and p; 1 a* is stationary, and let us rename the p; 1 a* as {pi: i < p}, 'and we allow ourselves to write p, = p&, a*). Now for every sequence (A,: i < p), A, 2 No there is a unique, up to isomorphism, model M = M(si*, (A,: i < p)) such that dim(p,, M) = A,. But we have here an arbitraxy choice of a*. However it is eaay to check that M(a*, (A,: i < p)) M(a*, (A': i < p)) iff there is 6*, tp(a*, 0) = tp(6*, 0) and a permutation u of p such that A, = Xu") and p&, a*) is not orthogonal to p0(&, 6*) (because tp(6*, a*) ie always Q,,-isolated) and 111, 3.6. (The proof here does not work apparently for p 2 No aa it is not clear why B* is F$-constructible over A*).

Iy,l < EC, or Iy,l = 2'0.

It is also eaay to check that

11M(a*, (A,: i < p))II = 2 Ai + A(T). i < u

This clearly proves 2.3(3), (4) and (6). To make the proof work for p 2 KO (and a - /.I small) we use a

better canonization. In stage I assume x = a. Stage I. We repeat stage C more carefully, working in P a . We define

by induction on a, formulas e,(z) (with no parameters) ordinals ?(a), types GES(aclp), g S c g(aeQ), such that (letting W, = {(O,(z), ):/.I < a} and K, is defined by V, 7.1(2) to satisfy C, 1-6 from V, Section 6, Wzo = W,) using V, 7.14:

N, M y N are P&(n-aaturated, then O,(N) = OJN) iff (i) If Af !?w) = #(N) for every Y < Y ( 4 Y

506 CATEBORICITY AND NUMBER OB YODELS [CH. Ix, 8 2

(ii)

(iii) For A E g(E), i #j, we have low,;(A.acl~) = 0. For a model M let h;(M) = lowp(qi(M),j3), which is equal to

dim($, q, M) (as q;(M) c aclKo(G(M) 6 @ using V, 4.18 or V7.14(3)). Now we shall prove that for Fi,-saturated M,, Ma, they are iso-

morphic over acl0 8 A;(Ml) = A;(M2) for every a < ao, y < ~ ( a ) . The only if part is clear. For the if part we define by induction on a 5 a. an elementary mapping Pa, Dom Pa = u B < a 8,(M1) u acl 0 onto uB<a 8,(Ha) U acl0, Pa acl0 = is the identity, increasing with a. For a = 0, a limit, there is no problem. For a = + 1, choose by induction on y < y(a), E ~ ( M ~ ) U aclf3, realizing tp(q, acl g), such that t p ( q , DomF, u u{e :i < y } ) is FEo-isolated, extend F, to an ( M I , Na)-elementary mapping Pi,

is K,-minimal, @ is P;-simple and @(aeq) c aclKi(C(EeQ) U EJ, where = cP(c} and c, are orthogonal if i # j. Hence

Dom Pi = Dom P8 U u {Z;: y < y(a)}.

Using A$(Ml) = A ; ( M g ) , $ K-minimal, G(Ml) c aclKu($(Ml) u Z?) and the parallel fact for Ma, and (iii) we can extend PL to Pi, an (M,, bla)-elementary mapping extending Pi , Dom Pj = Dom PB U

u,$y(MT), Range F i = Range F, U u , q 3 M 2 ) ; you may wonder what to do when q? has small dimension, then e.g. use V7.14(3) or Example X, 5.3A (or [Sh 88a, 1.41). LetN,<M, (N2 < M 2 ) be Fio-prime over Dom Pi (Range Pi), FZ an isomorphism from N , onto Ngextending Pi, and Pa = P; 1 (UfrB 8,(Nl)) is onto Ufrs 8,(N2). So cleaxly Pa is as required (as B8(N,) = 08(M,)).

It is also clear that for each sequence A = (A;: y < y(a), Q < %> of infinite ctudimls, there is an %,-saturated model M = M(X), A;(M) = A;, and 11M11 = A(T) + za,y A; and M is F&-satumted iff K B 5 A; for any a < ao, y < y(cc). The only remaining question is when M(X) z M(p) . Notice that any automorphism of Qeq preserve the BU(z)’s, but may take @ to another type. If P: M(X) M @ ) and w.1.o.g. E; E M(X), then f(<) should be not orthogonal to some A; = p ~ u t y ~ , and P f acl0 determinefJy). The rest is easy.

Stage J: Now we return to the case z = t , T totally trmcendental, and we repeat stage I, using now V, 7.14(2), using EY,, instead Fgoy with no special problems.

The only point that remains is the possible values of A;. Let I;*o be an indiscernible set on 7ia based on p;, 1; = {a;,t: f s w}. Let Wl(a, y ) be the h t cardinal K such that for every n tp(a;,,, 7i, u {a;,,: I < n}) is

CH. IX, f 21 LOWER PARTS OF THE SPECTRUM 507

Fk-isolated and hp(a, y ) is the first K such that tp(at,,, lia U {a;,l: I < w})

is Fk-isolated. For (1) we do not divide by - G , as we can easily prove that the result is the same : as T is totally transcendental, each ~ ( a ) is finite; so the equivalence relation on p : i -j iff for some automorphism F of 6, B’(I,),I, are not orthogonal (see Stage A) has only finite equivalence classes, and i - j =- h(i) = h ( j ) , and if we take the product (n,,,) on a maximal set of non- N -equivalent ordinals we get the same result. Also, the fact that the model should have power K, and not less does not influence the computation.

EXERCISE 2.4: In 2.4 replace “totally transcendental” by “Aa(T) I K,, T superstable”. Also replace it by A,(T) = No, and investigate Ia(Na, KO, T ) .

EXERCISE 2.5: (1) For fi > 0, Ia(Na, x,, T) = Ia(Na, K,, T). (2) For countable K0-categorim1 T,

Ia(Ka, K,, = It (Na, x,, T ) .

EXERCISE 2.6: (1) In 2.3, prove that for countable T, la - f i + ll@/U = la - fi + 11” except, possibly, when p and a - f i axe finite; and

(2) Show that for every x, finite p, and a group U of permutation of U, for some T, IT1 = X, ND(T) = p, U(T) = a.

(3) In (2) we can also have arbitrary h,, hp (when they satisfy the conditions mentioned in 2.4 and hl = ha).

(4) Show that if there is a Boolean algebra B, with s x elements and p ultrafilters (this occurs, e.g., whenever p I x or p = 2”, or p = f for some K I x), then for some T, ND(T) = p, IT1 = x.

(5) In 2.4(2) complete the case K, = IT1 < K , , ~ , K ( < KO (look at the proof of 2.1(3)).

Ir(.>l 2 No * Il44l = 2%

CHAPTER X

CLASSIFICATION FOR Fi;SATURATED MODELS

X.O. Introduction

We introduce here two dividing lines: having the dop, and being deep. They are meaningful for stable theories, but a t present important for superstable theories. The dop (dimensional order property) says that the relation: “there is an indiscernible set I realizing the type p = p( ( zo , xl, . . .),ai, a,) with dim(I,M) > IT1 ” (where ai E M ) can define an arbitrary two-place relation on {at : i < a}. This is very similar to the order property from Chapter 11, only the order is not defined by a first-order formula (hence, does not imply the complexity of pseudo-elementary classes) but i t implies T has 2A pairwise non-isomorphic Fio-saturated models of power h > h(T). The positive property equivalent to “not dop” is that for F:r(,,-saturated M , < M , , M , such that {Ml ,M2} is independent over M,, the F:(,,-prime model is F:r(,,-minimal. This is developed in Section 2. Its importance is revealed in Section 3 where we prove the decomposition lemma for superstable T without the dop : any Fio-saturated model is Fio-prime, Fio-saturated over UTGINv, where (N, : 7 €1) is a non-forking tree of Fgo-saturated models, I E ‘“’a, for some a, is closed under initial segments (non-forking trees means tp*(N,,,,, U { & : ~ - ( a ) $ v } ) does not fork over N,).

Why do we assume T is superstable ? Because we use regular types to show that the tree “exhausts” the model.

Note that the class of Fio-saturated models is very nice from our point of view. The saturation demand is weak, local (e.g. preserved by increasing unions), it is still easy to prove for unsuperstable the existence of many non-isomorphic models, and superstability implies the existence of an Fio-primary model over any A .

The existence of such a decomposition is a structure theorem, e.g. it bounds the number of pairwise non-elementarily embeddable Fie-

508

CH. x, $11 PRELIMINARIES 509

saturated models. Thus, the question naturally arises as to whether I may contain an infinite branch. If this is the case, T is called deep, if not, T is called shallow. Shallow (superstable without the dop) T has quite a few Fio-saturated non-isomorphic models, whereas deep T has 2A Ffto-saturated non-isomorphic models of power A 2 A ( T) + N,. In Section 4 we investigate the depth (which measures how far are we from the deep case), which really is the depth of a type, and get the upper bound. In Section 5 we prove that any deep T has many models. For simplifying the proof, in Section 7 we introduce “trivial regular types ” (when the dependency relation is trivial : tz depends on I iff it depends on some &E I). In 7.2 and 7.3 we investigate them : if T does not have dop, Dp@) > 0, p regular, then p is trivial, and we have better canonization theorems (e.g. if stp(a,A) is not orthogonal to a trivial p, then for some E almost over A, stp(tz/E, A ) is c13(p)-simple with weight 1) .

In Section 6 we compute exactly the number of non-isomorphic Ffto-saturated models of T of power A, when T is superstable, without dop, shallow of depth 2 w . Note that for countable No-categorical T every model is Fi0-saturated, so we can (essentially) compute I(A, T). However, whereas usually the depth can be any countable ordinal, here it is finite. This is as if T is superstable and K,-categorical, it is No-stable (see 111, 5 . 1 7 ) . Hence by Cherlin, Harrington and Lachlan [CHL 861 R(x = z , L , No) < w , so by 4.8 Dp(T) < w .

X.l. Preliminaries

In what follows, I, Jare sets of sequences from 6, I, J are sets of indexes (ordered sets, or trees or sets of sequences of ordinals, usually).

(Hyp) In this section T will be stable.

CLAIM 1 . 1 : If A E B, ( 1 = O , l ) , tp(B,,B,) does not fork over A , peSm(B0) is orthogonal to A , then it i s orthogonal to B,.

Proof. W.1.o.g. we work in Eeq and w.1.o.g. B, = aclB,, A = aclA. Choose any qeS(B1). Construct an infinite indiscernible set I of elements realizing q such that tp,(I,B, U B,) does not fork over B, and Av(I, I) is a stationarization of q. Similarly, choose a set J of

510 CLASSIFICATION FOR F g o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, Q 1

indiscernibles over B, realizing p . Since tp,(I,B, U B,) does not fork over B,, and tp(B,,B,) does not fork over A, by 111, 0.1 tp,(B, U I,B,) does not fork over A, hence tp,(I,B,) does not fork over A. As ~ E S " ( B , ) is orthogonal to A by V, 1.5(1) tp,U,B,) is orthogonal to A. Hence tp,U, B,), tp,(I, B,) are orthogonal, hence by V, 1.2(1) (since we work in Eeq) weakly orthogonal, hence by V, 2.7 Av(I, U I), AvU, U.J) are orthogonal, so p, q are orthogonal.

DEFINITION 1.2 : For A G B E C we say B < A C iff for every CE C, tp(c,B) is orthogonal to A.

LEMMA 1.3 : (1) Let N C M C A, M < A, M and N are F; -saturated, K 2 K,(T) and M' is F:-prime over A, -then M < NM'.

does not fork over A , and tp,(C, B,) does not fork over B, then B, < A o C,. (2) If B < A C , A G Ao,B C Bo, A, EB, E Co, Co = C UBo, tp*(C,Ao)

Proof. (1 ) By V, 3.2 and V, 1.2(3). (2) By 1.1 it is easy.

CLAIM 1.4: Let K 2 K,(T). If N is Fz-saturated, p € S m ( B ) is regular, stationary, not orthogonal to N, then p is not orthogonal to some regular q E S" (N) *

Proof. W.1.o.g. p € S m ( M ) , N c M and M is Fz-saturated. Let C E M , (C( < K , p does not fork over C, p C stationary. Let A E N, IAI < K ,

tp,(C,N) does not fork over A, tp,(C,A) stationary. Choose by induction on i < w elementary mappings f,, Dom ff =

A U C, fi /'A = id, stp,( ff(C), A) = stp,(C, A), and f o = id, Rang( f,) E N, and for i < w , i 2 2, tp,( f , (C) ,M U uj<f f,(C)) does not fork over A.

Let p1 = . m r (A u c)). Clearly for i < j, p, is orthogonal to p, iff p o is orthogonal to p, (by

indiscernibility ) . Extend the domain of fz to N U C by fixing N-C. By V, 3.4, p,,p,

are not orthogonal (p,f,f(p), A, B there, stand for p,, fz, p,, N, NU C here). Hence p,, p , are not orthogonal. Now p is parallel to p,, not orthogonal to p,, parallel to the stationarization q of p , in Sm(N), q regular. So we finish.

CH. x, $11 PRELIMINARIES 51 1

DEFINITION 1.5: Let A C B, p€S"(B), then we say that p is almost orthogonal to A if for every C, s.t. tp@, B) does not fork over A and every 6 realizing p, tp(6,B U C) does not fork over B.

CLAIM 1.6 : (1) Let A C B, then tp(5, B) is almost orthogonal to A iff tp(6, aclB) is almost orthogonal to A if tp(6, B) is orthogonal to A.

(2) If A G B, p = tp(6, B), then p is almost orthogonal to A iff for any C, tp(C, B) does not fork over A implies stp(6, B) I- stp(6, B U C) iff for any C, tp,(C, B) does not fork over A implies stp(a, B) I- stp(a, B U C).

(3) Let A E B, p = tp(6,B), then p is almost orthogonal to A iff for every &, tp(a, B) does not fork over A implies tp(a, B U 6) does not fork over A.

(4) For every a, tp(a,B) does not fork over A implies tp(6,B) I- tp(6, B U n) iff tp(6, B) is almost orthogonal to A and in aeq, tp(6, B) t- tp(6,B U aclA).

(5) If A = E B, M is Ff-saturated, p€S"(B) is Ff-isolated, A 2 ~ , ( 7 ) , then p is almost orthogonal to A.

(6) If A = WI E B, M is Fi-compact (i.e. A-compact), pcS"(B) is Fi-isolated, then p is almost orthogonal to A.

CLAIM 1.7 : If N is Ff-prime over $3, A 2 K,(T), IAl < A, cf A 2 K(T) and M is Ff-prime over NU A, then M is F,"-prime over $3.

Proof. We concentrate on the case h = No, so A = a. W.1.o.g. M is Ff-constructible overNU A. By IV, 4.9(3), IV, 4.10(2), IV, 4.18 we can find I c N , I an infinite indiscernible set, Av(I, N) = tp(@,N), and Av(I, U I) I- Av(I,N). Now IIIII = A as N is F,"-prime over $3 (see IV, 4.9(2)). By IV, 4.18, N is Ff-prime over UI, and by IV, 4.10(2) Ff- constructible over I U {@}. Hence M is Ff-constructible over I U {a}, but as II U {@}I = A, IU {a} is F:-constructible over 9. Hence M is Fi- constructible over 9, hence M is F:-prime over $3.

DEFINITION 1.8: (1 ) I(A, F, T) is the number of F-saturated models of T of power A, up to isomorphism. If F = Ff we write I f ( A , T).

(2) IE(A, F, T) is the maximal number of pairwise non-elementarily embeddable F-saturated models of T of power A. If a maximum is not obtained, and the supremum is a limit cardinal p, we write the value as p-. If F = Ff we write IEt(A, T). We omit A if we do not restrict the cardinality (so the value may be m or 00-).

512 CLASSIFICATION FOR MODELS [CH. X, $2

X.2. The dimensional order property

(Hyp) In this section T will be stable and K be K,(T).

Remember that T is unstable iff it has the order property, and unstable theories are complicated in some respects, e.g. they have many non-isomorphic models. However, a stable T may have an order hidden in it. For example, consider for A > KO, A E A,, the theory T of the model (B, F,, F,) where

B = A U {(a,P, y> :a ,P , y < A , and ( a , p > ~ A * y < 4,

Clearly T is not only stable, but even KO-stable, and N,-categorical ; however, by cardinality quantifiers we can define an order (if A is an order).

We shall consider here a property, which clearly means there is a hidden order property. From later sections we can see that for T superstable and Fio-saturated models it is the only one. Note that here the order and independence properties coincide.

DEFINITION 2.1: T has the dimensional order property (dop for short) if there are models4 (1 = 0,1,2), each Ff-saturated, 4 E M , , 4, and {M,,@} is independent over 4, and the Ff-prime model over M, U@ is not F;-minimal over M, U 4 .

In this section we first develop a number of equivalent forms of the dimensional order property. These are summarized in Lemma 2.4. Condition 2.4(d),o is the form of the dimensional order property used in Theorem 2.5 to show that if a superstable T has dop, then Iio(T,A) = 2A (when T is stable in A).

LEMMA 2.2: Let && < M , , 4 , each 4 F;-saturated, A 2 ,u 2 K ,

{M,,&} independent over 4, M F;-atomic over M, U@ and M is F,”- saturated (and M, U M , c M ) . Then the following conditions are equivalent :

(a) M is not F,”-minimal over MI U 4. (b) There is an injinite indiscernible I E M over M, U&. (c) There is p € S m ( M ) orthogonal to M, and to 4, p not algebraic. (d) There is an inJinite I S M indiscernible over M, U@ such that

CH. x, $23 THE DIMENSIONAL ORDER PROPERTY 513

Av(I,M) i s orthogonal to Ml and 4. Notes by 1.6(5), tp,(I,M, U 4 ) i s almost orthogonal to M, and to 4. Proof. The equivalence of (a) and (b) is the content of IV, 4.21, and (d)*(b), (c) are trivial; we now prove (b)*(d), (c)=(d) . The "notes" in (d) can be proved by 1.6(5).

(b) * (d). We can assume 14 = No, and suppose I E M is. indiscernible over Ml U 4 but not orthogonal to M, (by symmetry). So by definition, Av(I,I) is not orthogonal to some ~ E S " ( M , ) .

< K , Av(I,I) does not fork over J, Av(I,J is stationary, and let {bn : n < w } be an independent set over (Ml U A& U I,Ml) of sequences realizing r (hence indiscernible over M, u A & U I ) . By V, 2.7 for some k, tp(bon ... bk,M,U4UJ, tp(aon ..."ak,M1 U& U J are not weakly orthogonal.

As M is F,-atomic over Ml U 4 , for some B, E M, U 4, 141 < p, tp,(I',B,) I- tp,(I', M, U A & ) where I' = J U {al: 1 5 k} (for p singular use 2.3(1)). Hence for some C EM, ~ 4 , ICI < KO, tp(bo^ ..." bk ,4 U C U J) and tp(a0 -. . . A 9, B, U C U J are not weakly orthogonal. Now tp(bo^ ..." b,,M, lJ4 U J does not fork overM, (by the choice of the 6 s ) and M, is F;-saturated and IB, U C U < p + KO + K = p , so some co " . . . ^ c k ~ M l realizes tp(bon ... "bk,B, u C uJ) (see III, 0.1). Sotp(aon ..." Gk,B,UCUJ), tp(co" ..."ck,B,UCUJarenotweakly orthogonal, hence

W.1.o.g. I = J U {an: n < w} ,

tp(a0 * " 974 u c u J) Ytp(a0 . . en 974 u c u J u 4 . . . - c k ) 5

hence (byAzV2forF8,) tp,(I',B, U C) y tp,(I',B, U C U (roe ... Q), hence by monotonicity tp,(I',B,) y tp(I',M, U 4 ) contradicting the choice of 4.

(c)*(d). So let rESm(M) be (not algebraic) and orthogonal to M l , 4 , It suffices to prove that r r (B U M,, U 4) is realized in M for every B c M , IBI < K . [This is because then we can choose I& E M , 141 < K , r does not fork over B, r rlj, stationary, and 5n E M realizing r r ( l j , U {bl:Z < n} UM, U 4 ) , and then I = {bn:n < w } is as required - indiscernible by 111, 1.10( l).]

We can, of course, increase B as long as IBI < K . So w.1.o.g. r does not fork over B, r rB is stationary, for 1 = 0,1,2, tp , (B,4) does not fork over B n 4 , and let F realize r , 4 = B n Ml and tp,(B,M, U A&) does not fork over B, U 4.

As IBI < K , M is Ff-saturated, i t suffices to prove

stp(c, B ) I- r r (B u M, u 4).

514 CLASSIFICATION FOR F g o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [ C H . X, 92

For this let F2 €4 (1 = 1,2) and it suffices to prove stp(c,B) I- r r (B u 6, u 6,).

We can find C E&,, ICI < K such that tp(b,,&, UB;) does not fork ove r l$UCfor l= 1,2.

Now tp,(B,&) does not fork over B nib$ = 4 and C E 4 so tp*(B,& U C) does not fork over &. By symmetry, tp*(C,& U B) = tp*(C,B) does not fork over 4. Extend tp*(C,B) to a type q over M which does not fork over 4. Then qrM, is orthogonal to r and parallel to stp,(C,B) so by V, 1.2(4), stp*(C,B) is orthogonal to r . Hence,

(1) Stp*(c,B) I- stp*(c,B u C ) . Now tp(6, U B , , 4 ) does not fork over 4 [as it is Gtp,(M1,4)]

hence stp,(6,,4 U B,) does not fork over &, U 4. Also tp(6,,&, u 4) does not fork over B, U C by the choice of C , hence by 111, 0.1(2), tp (6 , ,4 U B) does not fork over B, U C. Now B, U C EM,, B, U C E B U C E 4 U B, hence stp,(61, C U B) is parallel to some complete type over M,, hence is orthogonal to r, hence to stp*(c, C u B). So

(2) stp*@, c u B ) I- stp*(C, c u B u 6,). Now tp,(& U B,,M,) does not fork over &, [as it is E tp*(M,,Ml)],

hence tp(6,,M1 U B,) does not fork over 4 U B,. Also tp(&,,&, U B,) does not fork over B, U C (by C's choice), hence

tp(b,,M, U 8) does not fork over 4 U C. As B, U C E 4, and B, U C E B U C U 6, G Ml U B, clearly stp(6,, C U B U 6,) is parallel to some complete type over hence is orthogonal to r , hence to stp(c, C u B u 6,). so

(3) stp(c, c u B u 6,) t- stp(c, c u B u 6, u 6,). By ( I ) , (21, (3) clearly (4) stp(c,B) t- stp(€,B u 6, u 6,)

which, as mentioned above, is sufficient.

CLAIM 2.3 : Suppose A 2 K , 4 (1 = 0, 1,2) are Ff-saturated, & < M,, 4, and {M1,4} is independent over 4. Then

(1) Every Ff-isolated PES"(M, U 4 ) is Ff-isolated. (2) A model M is Ff-atomic over M, U 4 iff it is Ff-atomic over

M, U 4. Hence if K 4 p, x 5 A , and an F:-prime model M over M, U 4 is F,"-minimal, then M is F:-prime over M, U 4 (and F:-minimal).

Proof. (1) Suppose B E M, U 4 , IBI < A, F realizes p and stp(c,B) t- p = tp(F,M, U 4 ) . We can assume that tp,(B,&,) does not fork over B n&,.


Let C G B, ICI < K be such that p does not fork over C, tp,(C,&)

for any EM, U 4 , there is &EM, U 4 such that stp(a',B) is a stationarization of stp(it, C).

We can find A E & such that tp,(a U C,&) does not fork over A , its restriction to A is stationary, IAl < K , and let = a,- a,, EM, (1 = 1,2). Let A = {at: i < i(O)}, and we can find a;€&, such that stp,((a;:i < i(O)),Bn&) extends stp,((a,:i < i(O)),Cn&) and does not fork over C n &, hence over C. Now find €4 such that stp,((a,:i < i(O))-a;,B U a;-,) extends stp,((a,:i < i(0)) -at,C) and does not fork over C. It is easy to check that it' = a; is as required.

does not fork over C n&. Now by 111, 4.22 it suffices to prove:

(*I

(2) trivial from (1).

LEMMA 2.4 : The following properties of T are equivalent for A 2 x 2 K : (a) T has the dop ( = dimensional order property). (b)A,x There are F,"-saturated models & , M , , 4 , A&, <MI,&, {M1,4}

independent over 4, such that the Fi-prime model 4 over M, U 4 is not Fi-minimal.

( c ) ~ , ~ There are F,"-saturated models i & , M l , 4 , & < M,,&, {Ml,4} independent over 4, and there is an Fi-atomic non-Fi-minimal model ik& over M, U 4 .

(d), There are sets A,, A,, A, such that A, E A,,A,, [All < A , {A,, A,} is independent over A,; and there is an infinite I indiscernible over A , U A,, orthogonal to A , and to A,, and tp,(I, A, U A,) is almost orthogonal to A , and to A,. Moreover, i f it, (I = 0,1 ,2) are such that { A l , A,, a,} is independent over A,, and tp(al, A, U A , U A, U a, U a3-,) doesnotforkoverA,Ua, for1= 1,2, then

stp,(I, A , U A,) I- tp,(I, A, U a, U A , U a2).

We can replace a, by B,.

Remark. Note that (a) does not depend on the cardinals A,x .

Proof. As (a) is (b)K,K it suffices to prove: (i) (')A,* * ( d ) K ; (ii) (d)K * (d)x; (iii) (d)x * ( b ) A . x ; (iv) ( b ) A , x * (')A,,*

(i) (c),,, (d)K. So let &,M,,M&& exemplify ( c ) ~ , ~ . Now by 2.2(d) there is an infinite I indiscernible over M, U 4 such that Av(I,&) is orthogonal to M, and to 4. First assume K > KO, and w.l.0.g. 14 = KO.

516 CLASSIFICATION FOR F ; o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $2

By2.3(1), tp,(I,M, U&)isFz-isolated, andsoforsomeA E M , U&, IAI < K , with stp,(I,A)t- stp,(I,M, U&) . We can also assume that tp,(A,&) does not fork overA fl4 and tp,(A,A fl4) is stationary. Let A, = A n 4 . It is easy to check that {A,,A,} is independent over A, (as {Ml,&} is independent over 4, and use 111, 0.1). Clearly tp,(I,A, UA,) is almost orthogonal to A, and to A,, and even the stronger assertion in 2.4(d) holds.

If K = No, T is superstable and so there is a finite J G I such that Av(I, I) does not fork over u J and Av(I,J) is stationary, and continue as before with J instead of I, noticing that tp*(I,A, U A, UJ) does not fork over u J, is stationary, and is orthogonal to M, and

(ii) (d), * (d),. As K 5 2, if A, (2 = 1,2,3), I exemplify (d),, then

(iii) (d), =s (b)A,X. Let A,,A, ,A, , I exemplify (d),. Let 4 be an F;-saturated model of T , A , E 4. By using

automorphism of 6 we can assume tp,(Al,& UA,J does not fork over A,. Next choose an F;-saturated M,, such that U A, EM,, tp,(M,,& U A 2 ) does not fork over 4, and an F;-saturated 4, such t h a t 4 U A, E & and tp,(A&,M,) does not fork o v e r 4 (this is easy). So clearly {MI,&} is independent over 4, and by the latter part of

to 4.

they exemplify (d),.

stp,(I,A, u A, ) I- stp,(I,M, u 4). (4,

So tp,(l,M,U&) is F,"-isolated, so there is & F,"-prime over M, U&, such that I E&.

As M4 contains an infinite indiscernible set I over 4 U M,, it is not F,"-minimal (by IV, 4.21) so we have proved (b)A,,.

(iv) (b)A,, * (c)~,,. Trivial (for x singular, prove by induction on a that if (M, U 4 , ( a t : i < a ) ) is an F;-construction, then M, U& U {ai: i < a} is Fz-atomic over M, U 4 : for this use 2.3(1).

THEOREM 2.5 : Suppose T has the dop, T stable in A and K 5 p < A. Then T has 2A non-isomorphic F;-saturated models of cardinality A.

Proof. Let A, ,A, ,A, , I be as 2.2(d),, 14 = No and let J be any set of indices. W.1.o.g. we shall work in Eeq (see 111, Section 6).

Let A; = aclA,; and we can define, for 1 = 1 , 2 , s € J , an elementary mapping f f, such that : (a) Dom f: = A;, (p) f f, PA; = the identity,


( y ) tp,(f#i)), U{fF(A;):(t,Ic) # ( s ,Z) ,~EI ,~E{~,~}} doesnot fork over Ah.

Next, for every s , t ~ J (not necessarily distinct) we choose an elementary mapping f8,,, whose domain is A; U A; U I , and which extendsf,1,f:. (This is possible asff Uf ," is an elementary mapping, which is true because tp(Al,Ah) is stationary (by 111, 6.9(1)) and the independence of {A; ,A;} and of {f,1(Al)lf$42)} over Ah.) Let A: =f:(A,) (so I&I < K ) , 1 8 , t =fs,t(l). Now:

(st) stp*(I,,,,A,1 u A 3 I- StP, (&,,? u 4 u u 4 u u qu). VE I veI (V, U) # ( 8 , t )

This holds by 2.4(d) for

B, = UV;: ( v J ) z (s,1), (42)) u {~u,v:{u,4 n W I = fVl 4 =& UA,' u u { 1 8 , v : v # t , V E J ) ,

B,=&uUA,2UU(Iu, , :u#s,uEJ) .

Let ct be any set indiscernible over A,' UA:, extending power p+. Clearly (st) continues to hold for the IZtls.

Let R be any two place relation over J , and let C, = U,, ,A: U UR(8,t) I; , , and let MR be an F,"-prime model over C,.

It is easy to check that IJI = A implies llM,II = A (remember T is stable in A , obviously IC,( < A , hence llM,II < A ; on the other hand, clearly A , $ A,, hence IC,I 2 A , hence llM,II 2 A ) .

of

Now, using (st) we show: for s, t E J , there is inM, an I of power p+ realizing tp,(C,,,A,1 U A:) iff R(s, t ) .

If R(s, t ) holds, clearly there is in MR an I (of power p+) realizing tp,(c, ,Af u A t ) : c, itself. Suppose R(s, t ) fail, I E MR, 14 = p+ realizes tp,(C,t,A,' UA,2). We shall work in Eeq. Let I' E I , (1'1 = KO. If stp,(l', A,' U A:) -= stp*(l,,,,A: U A:) we get an easy contradiction : by IV, 4.9, dim(I', C, ,M) 5 p, and by (st), dim(I', C, ,M) = dim(I', A,' U A:,M) which should be 2 14 = p+ > p, a contradiction. How- ever, if stp,(I',A,1 u A:) # stp,(c,,A,' u A:) , as tp(l,A,1 u A:) = t p (c , ,A t U A:) , stillA,,A:,A:,Isatisfies 2.4(d),, so we can prove the assertion corresponding to (st).

So in M, the relation R on (A: U A: : t E J ) is defined ; and [Ail < K ,

A = A'" (as T is stable in A , K = ~~(5")). So as in VIII , 3.2, we can prove that there are 2A pairwise non-isomorphic Fz-saturated models

(st 1)

518 CLASSIFICATION FOR F $ o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $2

of T of cardinality A. The case which most interests us, p = KO, T superstable (i.e. K(T) = No), has the same proof as in VIII, Section 2 : we have just to choose the right @ (see there). For more, see (remembering A = A‘”!): [Sh 85e], 111, $3 and (for K = KO) [Sh 891, 111, $7 and [Sh 88aI.

Remarks. ( 1 ) For most h we can also get 2A such models no one ‘elementarily embeddable into the other, e.g. for h regular >(TI; more generally, see also [Sh 831, [Sh 891, new version, 111, $7.

(2) In 2.5, if T is not stable in h we can still get similar results.

For the reader unsatisfied with this, we give two cases in more detail.

FACT 2.5A. If T has the dop x 2 h > K,A > p 2 K , A regular, then there are sets A, (i < 2A), each of power x , such that, letting M, be F,”- prime over A,, the following hold:

(i) The 4 ’ s are pairwise not isomorphic. (ii) For i # j, 4 is not elementarily embeddable into 4. (iii) For i # j, there is no elementary mapping from A, into 4.

Proof. Let S* = (6 < A : cf6 = K}, and for each SES let qa be an increasing sequence of successor ordinals of length K converging to 6. For every S c S* let

Js = x , R, = {(qb( i ) , 6) : ~ E S , i < K}, A , = CRS, M s = MRs,

A$ = u A: u u &,a> : (4 a> E m , 2<2 i < A

(note that A$ E A,) . Now suppose S,, S, E S, S, -4 is stationary, f : A:, +Ms2 an

elementary mapping, and we shall eventually get a contradiction. This clearly suffices. By renaming, we can assume that f is into

B = 4 U {ac: i < A}, where 4 = A,, U U A:, ,<A 2<2

tp(ai,4) is F,”-isolated where 6: = 4 U {a, :j < i}. Let A* > 2X be regular, and choose 4 < @(A*), E), 114(( < A ,

4 n h = a,, (4:j 5 i ) ~ & + ~ , and f , A:,,B,&,(ac: i < h),R,, ( A : : 1 < 2 , i < x),(C,:,:s,tEJ,,) (m = 1,2) belongs to No. Choose CE&-&,

Next choose M < (H(h*), E), ~ E M , M n K an ordinal 5 < K, 6 = 6,.


(4 : i < A) EM, and all the elements which we demand to be in No will be in M too.

Now f(A: U A: U u I ~ v c ~ o , c ~ ) gives us the desired contradiction.

FACT 2.5B. There are @, c such that T c T,,IqI 6 A(T) and @ is proper for (0, Tl) (see Def. VII, 2.6 and Lemma VIII, 2.3) such that for every linear ordering I :

(1) EM(I, @) is Fio-saturated. (2) There is a formula

$ @ p q = ( 3 ~ o , . . * , ~ ~ , - . * ) i < ~ + A ~ ) c ( . . . , ~ t c ~ , z ) , . . . , g , ~ , a<p+

P)~ jirst order, g, z of length < K , such that, for s, t E I ,

EM(], 0) + $[6,, 41, ins < t ,

where

6, = ( fc(at): i < IA, U A,I < K ) , and if K = No, 6t = a,.

Proof. Let J be {i: i < 3,+,}, where 8, = &,([TI), ct has power A, lM0, be F;-prime over u, , ,A: U U { I z t : s < t } (so R is the natural ordering). Now expand Mo,-by A(T) functions and get M, such that:

(a) For every b €6, EM, stp(6, a) is realized in the closure of by the functions of N,.

(b) P = {it, : SEJ) where a8 E A,O U A:, and if K = No then equality holds (i.e. the range of a8 is A: U A:) and always A: U A: c { ft(a8) : i}.

(c) f( - , (d) M, has Skolem functions. (e) If s < t , I' G Skolem hull of as U at is countable, infinite and

U at the same type as It , 8 , then Av(I', Skolem hull of

Now apply the proof of VII, 5.4 to get EM'(w,@), a model of = Th(MR), which realizes only types Mk, realizes.

at) (s < t ) is a one-to-one function from M, into ct.

realized over a8 U a,) is omitted in M,.

2.5C Corollary of 2.5B: Iio(A, T ) = 2A for A 2 A(T)+N,.

Proof. From the @ of 2.5B we can derive @, proper for (w2w, c) such that for some (not first-order) L-formulas P)~, n < w , VE%, q ~ " w

EM(I, @) t=qfl(cT7, tiv), if and only if v = q rn

(just let I = " * o x (0, I} if 7 is a proper initial segment of v, I t


, ( q , O ) < ( ~ , O ) ~ ( v , l ) < ( q , l ) , a n d i f r l ~ ~ o , r n < k, then(qn<rn),l) < ( v - ( k ) , O ) and if q ~ ” w , (7,O) = ( q , l ) . EM’(””w,@,) will be EM1(I ,@) , choosing for ~ E ~ ” w , B7 = @,, = Now for A > ITII( + A(T)) the theorems of VIII, Section 2 work. For A = A(T) by the proof in VIII, Section 3.1, Case I for almost every A we get the result, and we can complete also the remaining case. Anyhow our main interest is

Remember that for T superstable, IT1 + K, 5 A 5 A(T), I (A, T ) = 2A was proved in IX, 1.20.

Now we shall mention a topic, not necessary for the rest of the book, but naturally connected to the dop. Just as we have looked at “hidden order” we can look for “hidden unstability”, like the one caused by K(T) > KO.

DEFINITION 2.6 : T has the discontinuity dimensional property (didip for short) in the cardinal p (p regular) if there are Ff-saturated models Ma (a < p ) such that a < p*x < M, and the Ff-prime model over u,,,& is not F,“-minimal over Ui<JQ

T) .

I

Now if T has the didip for p, then p < K(T) and there are 2A Fi- saturated non-isomorphic models of power A if A > x 2 K , T stable in A, at least when (VA, < A)A:, < A, cf A = A.

The proofs are parallel to the proofs in VIII on the number of Fi-saturated models of power A, when x < K(T) regular.

Also, the parallel of 2.3 holds, and if x > K is singular, T does not have the didip for cf x, then the Fi-prime model over any set A is minimal. Also, if each & is F,Q-saturated (i < a ) M < 4, {ly : i < a} is independent over M, and N is F,Q-prime over UtJ& and in addition T does not have the dop nor any case of didip, then N is F,Q-minimal over Utda&.

X 3. The decomposition lemma

(Hyp) In this section T is superstable without the dimensional order property.

The main result of this section, the decomposition lemma, states that (for superstable T without the dimensional order property) every Fg;saturated model is Fio-prime over a non-forking tree N,(qeI) (i.e., I E w’llMll is closed under initial segments, and

CH. X, $31 THE DECOMPOSITION LEMMA 52 1

t p ( 8 , u {q: 7 6 v}) does not fork over A$(L(7)-l), when l(7) > 0, and 7 Q v - q ~ q ) . This is a kind of structure theorem, so this division line (superstable + not dop) is significant. Though we shall eventually prove that some such theories (the deep ones, see Section 5 ) have many non-isomorphic models, all of them do not have a family of K Fio-saturated models no one elementarily embedded into another, with K of arbitrary cardinality (this is with the help of [Sh 82bl). Recall that for A c B E C, B < A C means that for each F E C, tp(F,B) is orthogonal to A .

THE ATOMIC DECOMPOSITION LEMMA 3.1 : Suppose Nl < M are Fgo-saturated models. Then there are elements EM (i < a ) and models N2,c, M , such that:

(a) Y a,, <M, < M , (b) 4, N2.t are Fin-saturated, (c) tp(a,,Nl) i s regular, and for i #j, tp(a,,N,), tp(a,,N,) are

(d) a,EN,,,, and N2,t is Fio-prime over Nl U {at}, (e) N,,, <N,ii& and 4 i s maximal with respect to this property (in

(0 M i s Fio-prime and F~o-minimal over uI<oik& (g) {4 : i < a} i s independent over Nl, and tp , (4 , Nl U {a,}) i s almost

orthogonal or equal,

fact, for no aEM-M, i s tp,(M, U {a } ,q , i ) orthogonal to Y),

orthogonal to Nl.

Proof. Let I = {a, : i < a} G M be a maximal set, independent over Nl of elements of M realizing over Nl regular types and by V, 1.12 w.1.o.g. satisfying (c).

Let N,,, S M be FJto-prime over Nl U{ai}. So (c) and (d) hold trivially. By V, 3.2, we have:

FACT A. tp(N,,,, u, <=,, +&,) does not fork over Y.

Now for each i < a, we define by induction on j < llMll+ an element ~ , , , E M - N , , , U {b,,,:y <j} such that tp(b,,,,N,,, U {b,,,:y <j}) is Fio-isolated or is orthogonal to Nl (we can assume that the second possibility occurs only if N2,$ U {bt,,: y < j} is the universe of an Ft0- saturated model). There is a first P ( i ) < llMll+ such that bt,8cr, is not defined. Obviously (by l .6(5)) :

FACT B. N,,, U { b t , B : P < P( i ) } i s the universe of an Fio-saturated model which we denote by 2y (clearly M , < M ) .

522 CLASSIFICATION FOR F g o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $3

Now :

FACT C. tp,(4, u,,,M;) does not fork over 4.

To show Fact C, we just prove by induction on 6 5 zi<u,!l(i) that if we let A{ = &,, U {bf ,B:&,/3( j )+/3 < 0, then for every i < a, tp(Af, U,+tA6) does not fork over &. The induction step is by V, 3.2, 111, 0.1 (when we add an element realizing an Fio-isolated type) and V, 1.2(3), 111, 0.1 (in the other case).

We prove by induction on /3 6 p(i) that:

(*I 4.i <N,&,L { b , , y : y < For /3 = 0 this is trivial, as well as for p limit. So let us prove it for

p+ 1. Let tp(c,&,,) be a type which does not fork over X Since 4 is F,"(,,-saturated, by V, 1.2(3) it suffices to prove that tp(c,&,J, tp,({b,,,: y 5 /3},N2,J are weakly orthogonal. This is equivalent to

tP(G&,,) I- t P ( m , , u { & , y : y s pH.

tP(c94.t) I- tP(c>N,,, u @t,+ y < PI).

By the induction hypothesis on /l

Hence tp(C,Nz,, U {b t , y : y < /3}) does not fork over N2,,, hence (by transitivity, see 111, 0.1) does not fork over N,, and it suffices to prove it is weakly orthogonal to tp(b,,,,N2,, U { b , , y : y < p}). If the latter is Fio-isolated this holds by V, 3.2 and if the latter is orthogonal to N,, this holds by V, 1.2(3).

FACT E. M is Fio-minimal over ui<uik$.

PROOF OF FACT E. Let M' < M be F,"-prime over u,,,4 (we know that there is such M'). Suppose M' # M which holds if M is not F,"- prime over U f < = 4 , and M' can be chosen so that it holds if M is not Fio-prime and Ffto-minimal over ut<u&; we shall eventually get a contradiction. Choose ~ E M - M ' with R[tp(b,M'), L, 001 minimal (it is < 00 as T is superstable).

By V, 3.5, tp(b,M') is regular. Let us first assume that tp(b,M') is not orthogonal to N,, then by 1.4 there is a regular type p E S"(N,) not

CH. X, $31 THE DECOMPOSITION LEMMA 523

orthogonal to it, hence by V, 1.12 some b ' d 4 - M ' realizes the stationarization of p over M'. But this contradicts the maximality of I = (a, : i < a} as b' can serve as a,.

So we can assume tp(b,M') is orthogonal toNl. Choose a setB EM', IBI < K(T) = No such that tp(b,M') does not fork over B. SinceM' is Fgo-saturated we can also assume tp(b,B) is stationary. Also there is a finite S E 01 such that tp,(B, Utes&&) is Fgo-isolated [as tp,(B, U i K a q ) is Fio-isolated by IV, 4.3 asB -C M', andM' is FgO-prime over u$<a&]. So there isM* E M' Fgo-prime over uies&& and B E M so that tp(b,M*) is a regular type orthogonal to Nl. We prove that this is impossible by induction on 181. If IS1 = 0 then tp(b,M*) is not orthogonal to Nl as M* = N,, a contradiction. If IS1 = 1,s = {i} we get a contradiction to the definition of &&. Suppose IS( = n + 1 and assume by induction that for any U S a with =< n, if N* < M is Fk-prime over uiEu&& and rES(N*) is a regular type which is realized in M , then r is not orthogonal to N,. Let ~ E S and choose M+ EM*, Fgo-prime over U,,,,,,,q such that M* is Fio-prime over M+ U && (for some fixed ~ E S ) . By V, 3.2 (Ill+,&} is independent over Nl. Since T does not have dop, by 2.3, tp(b,M*) is not orthogonal to one of&& or M+. Let N denote the model tp(b,M*) is not orthogonal to. By Lemma 1.4 there is a regular ~ES"(N) such that q is not orthogonal to tp(b,M*). But tp(b,M*) is orthogonal to every (regular) complete type over Nl. Since non-orthogonality is transitive on regular types (V, 1.13), it follows that q is orthogonal to every regular type in S(Nl), i.e. by 1.4, q is orthogonal to Nl. But q is not orthogonal to tp(b,M*) and b is in the Fio-saturated model M , so by V, 1.12, the stationarization of q on M* and hence q is realized in M . But then q and N contradict the hypothesis of induction.

So we prove Fact E and we can check that the only part of 3.1 to be proved is the last phrase of (e) which we leave to the reader. For (g) use l.6(5), 1.6(1).

THE DECOMPOSITION LEMMA 3.2 : For any Fg0-saturated model M we can f ind a set I S u'llMll (Jinite sequences of ordinals < IlMll) closed under initial segments, and q , a , for ~ E I and p , ( q ~ I - { ( )}) such that :

(1 ) N, < M is Fgo-saturated. (2) No is F&-prime (over 9). (3) pl,yr> = tp(a,-(i),N,) is regular, and for qn (j) €1, p,-(i),

p,-(,> are orthogonal or equal.


(4) N,-(,) is Fgo-prime over N, U {a,yl)}. (5) tp*(N,-(i), U{N,: V E I but not 7 -(i) 4 v} does not fork over N,

(6) M is Fgo-prime, Fgo-minimaZ over u,,,N,. (7) tp*(U{N,:q 4 v ~ o , N , ) is orthogonal to qrn when Z(7) = n+ 1.

( for 7 -(i) €1).

Proof. We define by induction on n, a set L, of sequences of ordinals of length n, models N,, M;, and elements a, for each EL, such that :

(1) r l € l , , m < n i r n p l i e s r ] r m E ~ , and&={( )}. (2) N, is F;to-saturated. (3) N< ) is Fgo-prime (over $3) M( ) = M . (4) is Fgo-prime over N, U {ayTt)}. ( 5 ) Let = t~(a ,~,) ,N,) , then it 1s regular, and paYr) ,py(5) are

equal or are orthogonal. (6) M,, is Fgo-saturated, N, G q E M . (7) 7 4 v impl i e sN,zN,EM,Gq . (8) M;, is Fg;prime over U,M;,y2).

(10) { ~ y c ) : 7 - (i) €0 is independent over N,. (11) A , = {a,y2): 7 n(i) € I ) is a maximal subset of&(, (or even set

of sequences from 3) independent over 8. (12) A , is a maximal subset of M which is independent over N7 and

every element of it realizes over N, a type orthogonal to Ub<l(,)N,rk if

The definition is easy: for n = 0 trivial, for n+ 1, for each ~ E I , we apply 3.1 with N , , q standing for N,,M and get U ~ , N ~ , ~ , & and let a,-(r) = a,, N,,(,) = &,c, M,,?$) = & (so L,+, is the set of v's of length n + 1 for which N, is defined).

Let I = urnIn. Now all the conditions of the lemma are obvious except "M is F,"-prime and F,"-minimal over UVEIN," and ( l l ) ,

Let us first prove ( l l ) , (12). If (11) or (12) fails for 7, let a exemplify i t ; i.e. a€&&, tp(&,N,) is orthogonal to N,r(k-l) if Ic = Z(7) > 0, and &#N, and tp(a,N, U {a,yi): 7 -(i)~o) does not fork over N,; w.1.o.g. tp(G,N,) is regular.

(9) N,-( t ) <N,M, , IY , ) '

7 # < >.

(12).

As in the proof of 3.1,

CH. X, $31 THE DECOMPOSITION LEMMA 525

and as M,, is Fie-prime over U {hi!&,, : r ] ^ (i) €4, tp(a, U,M,,?,)) does not fork over N,; by V, 3.2,

This proves (1 1) (i.e. when GEM,,) ; for (12) note that by the above t p ( a , q ) does not fork over N,, hence is the stationarization of tp(&,N,) which is orthogonal to N,r(l(,)-l) when it is defined. Let k = l(r]), and we now prove by induction on 1 k: that tp(n ,~ , , ,_ , , ) does not fork over N, .

We have proved for 1 = 0 and for I+ 1 notice that as t ~ ( a , q ~ ( ~ - ~ , ) does not fork over N,, i t is parallel to tp(a,N,), hence orthogonal to N,r (k- i ) , hence to qr(k-l-l). SO for j # q(k - 1 - 1), t~(a,M,,~(~-,)) is orthogonal to tp* (%r(k-l-i)yj), qr(k-l-l)).

As {qr(k-,-l)-(j) : j < a) is independent o ~ e r N , ~ ( ~ - , - ~ ) , tp(a,%,,,_,,) is t~*(Uj#~(k-l-l)M,,r(k-l-l)-(j),~r(k-l-l)), and

So as M,,r(k-,-l) is Fio-prime over Ujqr(k-r-l,-(j), by IV, 4.10(2),

so also the latter does not fork over N,. For 1 = k we get a contradiction to EM = M( ), as tp(n, M( )) does not fork over N,, and a#N,.

Now we shall prove that M is Fie-prime, Fie-minimal over U,,,N,. So suppose M' G M, M' # M, M' is F:-prime over U,,,N,, and choose EM-M', R[tp(b,M'),L, a ] minimal, hence tp(b,M') is regular. Now tp(b,M') is orthogonal to each N,. If not, choose r] with minimal length, then by 1.4 for some regular q€Sm(N,), tp(b,M'), q are not orthogonal and q = p,?,) for some i, or q is orthogonal to every pVy,). Let qf €Sm(M') be the stationarization of q over M', so by V, 1.12 there is & E M which realizes qf, so tp(af,M') does not fork over N,. By choice of T,I (being of minimal length) (if r ] # ( )) tp(b,M') is orthogonal to Uk,l(,,N,rk, hence by V, 1.13 also tp(ar,M') is, so we get a contradiction to (12). We conclude tp(b,M') is really orthogonal to each N,.

Choose a finite B EM' over which tp(b,M') does not fork, and a finite I* G I closed under initial segments such that tp,(B, U,,,N,) does not fork over U ,,,. N,. As M' is Fio-prime over U,,,N,, tp,(B, U,,,N,) is Fie-isolated, so tp,(B, U ,,,. N , ) is Fio-isolated (see IV, 4.3).

526 CLASSIFICATION FOR F;,-SATURATED MODELS [CH. X, $3

Let N < M ' be F;,-prime over u{q: r]~l*} and B C N. We have found an Fg0-prime model N over uVEI*q, so that there is a regular type in Sm(N), orthogonal to each N, (T , IEI*) , realized in M, and I* is finite, closed under initial segments. We get a contradiction to the statement in the last sentence by induction on II*I. If I* = { r ] r l : 1 < lg(r])} this is trivial, and if not choose distinct r ] , u in some I* n L, with n minimal. Let No = qr(a-l), N' < N be FJto-prime over u{N,: r ] Q

CT E I*} a n d P c N be Fio-prime over u {N, : not r] 4 a, but r E I*} such that N is Ffto-prime over iV' U P. Now by 2.3 (and as T does not have the dop) for some 1 E { 1,2}, there is a regular complete type over N z orthogonal to No, realized in M, and we get a contradiction to the induction hypothesis on II*I.

In fact our proofs also prove:

LEMMA 3.3: Suppose I E "'A is closed under initial segments, {NV:r]eO is a non-forking tree of F,"-saturated models [i.e. r ] Q ~ - N , < N , , T , I E I - { ( )}*tp(N,,U{N,:uEI, not r ] 4 v}) does not fork over N,r(l(,)-l), and each N, is Fio-saturated].

If T does not have the dop, M F,"-prime over U I E I q , then M is F,"- minimal over U,,,N, and every qESm(M) is not orthogonal to some N,.

Proof. First note

FACT 3.3A. If S;, S, are non-empty subsets of I which are downward closed, then {uvEs,N,, uVEs2Ny} is independent over UYESln

Proof. By the local properties of forking, it suffices to restrict ourselves first to the case 4-4 finite, then to the case 8-4, 4-4 finite, and finally &,A!$ finite (using 111, 0.1).

Now we prove the statement by induction on IS; U & I ; w.1.o.g.

So choose 7 ~ 4 - 4 of maximal length, and by the induction hypothesis and transitivity of forking (see 111 ,O . l ) it is enough that tp,(N,, u{N,: V E S ; U S,, u # r ] } ) does not fork over N,r(z(,).-l), but this follows from the hypothesis. Now return to 3.3.

If one of the conclusions fails we can reduce it to the case I = I* is finite; then, as in 2.2, the two conclusions are equivalent. If both fail, there is M' i M, M Fi0-prime over U T E I . q , bEM-M', tp(b,M') orthogonal to every N,; and continue as in the last part of the proof of 3.2: from the choice of B.

S; z S; n 4 , S, z S, n 4 , clearly < )ES; n 8 .

CH. X, $41 DEEPNESS 527

X.4. Deepness

(Hyp) In this section T is superstable without the dop.

As we know the number of non-isomorphic Fi@-saturated models in every N, 1 IT1 + N, for unsuperstable T and for T with the dop, we concentrate on the case in the hypothesis. By the decomposition Lemma 3.2 we know every Fio-saturated model is Fi0-prime over a non-forking tree of F:-prime models. Clearly, if there are few trees then there are few models, but the converse is less clear. Anyhow, clearly the most important distinction is whether the tree ( I in 3.2) is always well-founded. If it is always well-founded, naturally some rank is defined (called here the depth), and if the rank is small the number of models is small.

In this section we introduce the basic relevant notions and the simple facts about them. Notice that we could have chosen some other variants of the notions, but by later sections we can show that they would be equivalent.

We define the depth so that the results on the number of models can be stated smoothly (this is why 4.l(iii), 4.2(1) are such that the depth is not a limit ordinal).

Let us make some more specific remarks. Note that if the tree is not well-founded there are 4 < q+14+l Fio-prime over 4 U {a,}, tp(al,&) regular orthogonal to &-l. So the rank is defined as an attempt to build such a sequence (i.e. the rank of (N,N',a) is 00 iff there is such a sequence, N = No," = Nl, a = ao).

In Definitions 4.1 and 4.3 we give some variants of this, in 4.2 we define the relevant property of a theory (deepness), in 4.4 we prove various facts, and in 4.5 the essential equivalence of some variants is given. Now 4.6 says that, looking for high depth, it is enough to look at types not orthogonal to $3. For "canonical" examples see 4.9, 4.10.

DEFINITION 4.1 : Let K' = { (N,N' , a) : tp(&,N) is regular, and N' is Fio-prime over NU {a} and N is Fio-saturated}.

For every member of K' we define its depth, an ordinal (zero or successor but not limit) or infinity 00, by:

(i) DP(N,N',~) 2 o iff (N,N',~%)EK';

528 CLASSIFICATION FOR F E ; o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $4

(ii) Dp(N,N’,a) 2 a+ 1 (a zero or successor) iff for some N”,c’ :

(iii) Dp(N,N’, a) 2 S+ 1 (6 limit) iff Dp(N,N’, a) 2 /3 for ,8 < 6; (iv) Dp(N,N’,a) = co iff for every ordinal /3 Dp(N,N’,a) 2 /3,

Dp(N,N,E) = a iff D~(N,N’,E) 2 a but not Dp(N,N’,a) 2 a+ 1.

(N’,N”,a’)€K‘, N’ < N N ” and Dp(N’,N”,a’) 2 a;

DEFINITION 4.2: (1) The depth of the theory Dp(T) is U{Dp(N,N’, a ) : (N,N’, a ) EX} when this is finite and U{Dp(N,N’, a ) : (N,N’, a ) EK‘}+ 1 when this is an infinite ordinal.

(2) The theory T is deep if its depth is 0 0 ; otherwise it is shallow.

DEFINITION 4.3: (1) KA = { (N,N’ ,a) :N,N’ are F,”-saturated, EN', a#N,N’ F,”-atomic over NU {a}}. Dp((N,N’,a),K) is defined as in 4.1 for any set K of triples. If (N,N’, a) #K we interpret Dp((N,N’, a), K) as Dp(N,N’, a), K U { (N, N’, a)} (not closing under isomorphism) and if K = K’ we omit it.

Let K” = {(N,N’,&)EK’: tp(a,N) is regular}. (2) For a tree I, we define DpA(7,I) (A a cardinal, ~ € 1 ) : DpA(7,

I) 2 a+ 1 iff for A v’s, 7 Q v, Dp(v,I) 2 a, and for a = 0 or limit DpA(7,1) 2aiffDp,(y,I) 2/3forevery/3<a.SoDpA(r],I) = a i f i t i s 2 a but not 2 a + l . Let Dp,(I) = sup{Dp,(q,I):q~I). For A = 1 we omit A. (3) Dp(T, K) is defined as in 4.2.

LEMMA 4.4: ( I ) If (N,N’,a)€K, (N’,N”,a’)€K,N’ <,N”, then Dp((N,N’, a), K) 2 Dp((N’,N”, a’),K) (the inequality is strict except when both are co) [this holds for any class K of triples]. (2) If a<Dp( (N,N’ , a ) ,K)<m, (N,N’ ,u)EK, a not limit, then

some (No,No,a0)€K has depth a (in K). (3) Dp(N,N’,n) = 00 iff there are&,al (1 < w ) , & + ~ < N l & + 2 , & = N , 4 = N’, a0 = a, (&+1,&+2, at) EX. (Similarly for any “reasonable”

(4) If a = Dp(T) or a = Dp(N,N’, a), a < 00, then a < (21’1)’ and in fact a < &(IT[) (look at VII, $5 on the definition of S ( l f l ) and information on i t) .

K. 1

(5) The depth is preserved by automorphisms of 6. (6) If (&,q,&,)~K;l~ (1 = 0,l) and tp(al,q) are parallel or not

orthogonal, thenDp(No,N’o,ao) = Dp(N,,Nl, 4). Ifonly (Nl,Nl, al) cKNO, tp(al,Y) regular, still Dp(&,N’, ao) 5 Dp(&,x, at).

CH. x, $41 DEEPNESS 529

DEFINITION 4.4A : We let, for regular p, Dp(p) be the depth of (N,N’, a) EK&, when tp(a,N), p are parallel.

PROOF OF LEMMA 4.4. Easy. Note for ( 3 ) we need (6). (1) Trivial. ( 2 ) We prove it by induction on /? = Dp(N,N’, a); for /3 = a.there

is nothing to prove; for p > a, use 4.l(ii) applied to Dp(N,N’, a) 2 /?+ 1, to get N”, a’ with y = Dp(N’,N“, a’) 2 a, but by 4.4(1), y < /?, and use the induction hypothesis.

(4) By (6), Dp(N,N’, @) depends only on tp(@,N) up to parallelism, and by (5) it is preserved by automorphisms of a, hence there are 5 2ITI possible depths. But by ( 2 ) the ordinals which are the depth of some triple form an initial segment of the set of the non-limit ordinals, hence

a = Dp(N,N’, a) < co -a < (21T9+.

The first phrase has been proved above. We work as in VII, Section 5. For the second phrase, let 23 = (H(h) , E, T, S(lTI)), H(h)- the family of sets of hereditary cardinality < A , and w.1.o.g. T c ITI. Now consider 23’ elementarily equivalent to 23, with “ T and IT1 ” standard : but non-well-ordered “ordinals” < S(T). (23’ is known to exist.)

In this model we can consider various notions and check whether they are absolute, i.e. whether if %’ says something holds it really holds. Now this holds for (a) being a model of T, (b) R ( p , d , h ) = n ( A finite, h 5 No), (c) being orthogonal types, (d) non-orthogonal types, (e) tp,(A,B) does not fork over C , and (f) A c B, p €Sm(B) ; p has a unique extension in Sm(N U B ) when

tp,(N,B) does not fork over A . We can conclude by 4.4(1) that also if %’I= “Dp(p) 2 a*, a* an

ordinal”, then p has depth 2 the order type of ( ~ ~ 2 3 ” : a < a*}. If a* 2 S(lT1) this is not well-founded so Dp(p) = co.

( 5 ) Trivial. (6) We prove by induction on y that [Dp(No,No,@o) -5 y or

Dp(N,,T1, ao) 5 y] implies the equality. We can choose F:-saturated N,, No UN, c N,, and by 4.4(5) w.1.o.g. tp(q,N,) does not fork over & (for 1 = 0 , l ) . By V, 3.2 , No is F:-constructible over N, U ao, and let

5 30 CLASSIFICATION FOR F g o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $4

N2 be Fz-primary over N2 UW, = N2 UN, U a,. In N2 tp(al,N2) is realized, so w.1.o.g. al cY2, Nl G N2, Y2 F,“-prime over N, U al and over N2 U X1 (and over N2 U a. and over N2 U No).

By symmetry it is enough to prove Dp(N,No, a,) = Dp(N2,r2, go), because checking the definition we can observe that Dp(N2,X2, ao) = Dp(N,,W2,al). Now the inequality 5 is trivial (with the induction hypothesis for y ) and for the inequality 2 we have to replace parameters with others of the same type. For proving the second phrase act as before, observing that by the first phrase Dp(Nl,Nl,

(3) First suppose that there are &, @,, and let a, = Dp(&,&+,, al). By 4.4(1), a, 2 al+l, and if a, # 00, a, > alf l . So if a, is # co, then a, ( 1 < o) is a strictly decreasing sequence of ordinals, a contradiction; so a, = 00, but a, = Dp(No,Nl,a,) = Dp(N,N’,a), so we have proved the “if” part of 4.4(3).

Now suppose Dp(N,N’, a) = 00. We define by induction on 1 , &+l, @, such that is Fz-saturated, Dp(&,&+,,a,) = 00, 4 <Nl-l&+l

(when 1 > 0 ) . So let No = N , Nl = N’, a, = a. So the induction hypothesis holds. For l + 1 as D P ( & , & + ~ , ~ , ) = 00, there are &+2,

a,+, = (a ,+2) , such that (4+1,&+2, at+2)~K’ , D P ( ~ + , , & + ~ , 2 ( 2 9 + , hence by 4.4(4), D ~ ( & + ~ , q + ~ , a , + ~ ) = 00. As we can carry the induction we have proved the “only if” part of 4.4(3).

The following lemma shows that there is no real difference between the various Dp( -, K)’s, in particular, whether we use a or @.

a11 = DP(N,,W2,ao).

LEMMA 4.5: (1) For any (N,N’,a)€KA

Dp((N,N, a),&) = Dp(N,N’, a).

(2) If K’, = { (N,N, a ) EK’&:N is Fio-atomic over NU {a}} , then on K’,, Dp(-,K‘), Dp(-,Kl) are equal .

( 3 ) For any complete type p , Dp(p) = sup{Dp(r):r a complete regular type not orthogonal to p } .

(4) Suppose N<,N”, & E N , @“EN”, tp(a‘,N) tp(a”,N’) are regular, tp(N’, N U a’) is almost orthqonal to N and tp(N”,N’ u a”) is almost orthogonal to N’, Then Dp(N,N’,a“) > D(N,N”,a“) (or both are 00).

Proof. ( I ) Remember that by 111, 4.22, if M is F:-saturated, M’ is F:-prime over MU a, K < A , then M is Fgo-atomic over MU a. 80

CH. X, $41 DEEPNESS 53 1

easily by the definitions and 4.4(6) Dp((N,N', a),&) 2 Dp(N,N',@). So it suffices to prove by induction on a that:

(*) Dp((N,N', a),&) 2 a Dp(N,N', a) 2 a for (N,N', a) EK,.

For a = 0, a limit and successor of limit this is trivial; for a = /?+ 1, /? not limit, there is (N' ,N",~' )EK, , N' <,N", and Dp((N',N", @'),K,) 2 /?, hence by the induction hypothesis Dp(N,N", a') 2 @. Apply Lemma 3.1 forN,N" standing forN,,M and get ai,N2,,,& (i < i(0)). By V, Def. 3.2, Th. 3.2, w.1.o.g. N2,, = N and i(0) is finite (= w(@',N)), and let y = Max,,,(,,Dp(N',N,,,,a,) (so y is not limit); clearly it suffices to prove y z/?. Otherwise, as y is not limit there is (N",N*, a*) E K I , N" <,N*, Dp(N",N*, a*) 2 y . As tp(&*,N") is regular, orthogonal to N', clearly as in the proof of 3.1 (see Fact E) for some i < i(0) and regular q ~ h " ( N ~ , ~ ) , tp(@*,N") and q are not orthogonal. Let c realize q, N'2,i be Fio-prime over N2,, U {c}. So by 4.4(6)

(a) DP(N",N*,@*) = Dp(N,,i)&,C). Now as q is not orthogonal to tp(a*,N"), it is orthogonal to any

regular complete type over N' (as non-orthogonality is an equivalence relation among regular types, see V, 1.13) hence q is orthogonal to N'. Hence <N,N2,1, hence (by 4.4(1))

(b) DI>(N',N,,,,a,) > DP(N,,*>N'2,i,q. As Dp(N",N*, @*) 2 y by (a), (b), Dp(N,N,,i,ai) > y , contradicting

y's definition. (31, (2), (4) Easy.

LEMMA 4.6: If (No,T0, a0) €KIXo, tp(ao,No) is orthogonal to 9 and has depth <a, then there is ( N l , N ' l , @ l ) ~ K x o such that tp(iil,Nl) is not orthogonal to $3 and

Dp(N,,N'o, a01 -= Dp(N,,N;,Q

Proof. By 4.4(6), w.1.o.g. No and No are Fio-prime over 9. Let B C No be finite, such that tp(ao,No) does not fork over B. There is B' realizing the stationarization of stp,(B,pl) over No, and let M' be FEo-prime over No U B'. By 1.7, M' is Fio-prime over 9. Hence by IV, 4.18, No,M' are Fio-prime over B , B , respectively, hence there is an isomorphism from M' onto No taking B' to B. So there is a model N < No such that tp,(B,N) does not fork over 9 and No is Fio-prime


over NUB. Hence by V, 3.9 there is a finite set J = {b, : i < n} C _ No independent over N, of elements realizing regular types, such that No is FP, -prime over N U J. By V, 3.2 there are N z < No F;lo-prime over NU{b, ] such that N , is Fio-prime over ug,,w. As tp(ao,N,) is orthogonal to 9, is parallel to stp(ao,B), and tp,(B,N) does not fork over 9, by 1.1 tp(ao,No) is orthogonal to N. On the other hand, by 3.3 tp(ao,N) is not orthogonal to s o m e v , hence by 1.4 some regular q e S m ( x ) is not orthogonal to tp(60,No), so by V, 1.13 it is orthogonal to N.

Clearly Dp(N,w,b,) > Dp(q) = Dp(&,&,ao) (see 4.4(1), 4.4(6), respectively) and tp(b,,N) is not orthogonal to $3 as it is not orthogonal to tp,(B,N) (because bgENo,No F" - rime over NUB, tp(B,N) does not fork over 9). Now put ( f l N ; , it,) = ( N , v , b l ) .

'' 'deI

THEOREM 4.7 : T) (the number of non-isomorphic F;,- saturated models o power N,) is at most ~,,(T,(la1~'~') for shallow T, so it is <Sa(lTi,(lalZ' I) = Sa(lTl,(l~l) < 3(Z~~~)+(IaJ) and if T is countable < ~ J I 4 ) .

if

Proof. Immediate by 3.2, and the bounds on S(lT1) (see e.g. VII, 5.5 and 5.5(2)).

LEMMA 4.8: If T is superstable without the dop, then R(x = x , L , 00)

2 Min{w, Dp(T)}.

Proof. Suppose k < o, k < Dp(T). Then we can find N 7 , ~ as in 3.2's notation for qek"A (any A) . Now we can prove by induction on k-Z(q) that for every bEq\N7

k-l(q)+R(tp(b,N,),L, a) < R(x = x , L , 00).

Example 4.9. A very natural example of a superstable T without the dop which is deep, is the following T: !!&, = Th("'w,f) when f ( q ) is q if r] = ( ), and q r n if qen+lo.

Notice that a model of Tdp consist of trees, exactly one with a root (i.e. f ( x ) = x ) , in which every element has infinitely many immediate predecessors (i.e. y's such thatf(y) = 2) . A similar example is TZp = Th(O>w,. . . ,P,, f,,, . . .) where P, = ,w, f,, is a partial function: f rP,.

Both theories are N,-stable, and by expanding a little we can get elimination of quantifiers.

CH. X, $51 DEEP THEORIES HAVE NON-ISOMORPHIC MODELS 533

Examples 4.9A. Examples of shallow theories can be obtained similarly to 4.9; we prefer to use the T"* from 11: the language consists just of the two-place relations E, (i < 01). The axioms of T state: each E, is an equivalence relation, for i <j, E, refines E,, moreover each E,-equivalence class is the union of infinitely many distinct E,-equivalence classes. Also E, has infinitely many equivalence classes and each E,-equivalence class is infinite. It is not hard to check that Dp(N,N', a) = i iff i = y when y < u, i = y+ 1 otherwise, where y = min{j : there is b E N , bE, a} (and y = u if there is no such j).

CONCLUSION 4.10 : For every ordinal /3, which is a natural number or a successor ordinal, for some T = T;, IT1 = 1/31 +KO, and ' io (Na , TI = ',(la1 + '0).

Proof. For p < w or /? = u + 2 we use the previous example ; for /? = S+ 1, S limit, take the sum of models of TI' (i < 8) with disjoint languages. The computation is easy, but i t is a worthwhile exercise for the reader as we are proving in the chapter that every shallow theory T is in some sense similar to

X.5. Deep theories have many non-isomorphic models

(Hyp) In this section, T is superstable without the dop.

Clearly, if T is deep, we can construct trees like the one we get in 3.2, and try to prove that we get many models. The freedom we have is to determine various dimensions. So when u = N, this is easy. Generally notice that we have much less freedom than, e.g., in 2.5 (when proving that the dop implies there are many models).

This section is dedicated to the proof of:

THEOREM 5.1 : If T is deep, A(T) 5 K, , N, < K,, then Igp(Ka, T ) = 2n=, i .e. T has 2 N ~ non-isomorphic FiB-aaturated models of cardinality Ka.

Remark. But the lemmas will be used also for shallow theories. We shall concentrate on the case N, > A(T), where A(T) is the first cardinal in which T is stable.


DEFINITION 5.2 : We call (4, a,: r ] E I , v E I+) a representation i f : (1) I is a tree with root ( ) of height su, and we let 7- be the

unique predecessor of r ] for r ] E I+ = I - { ( )} and let I- = (7- : r ] E I+} . (2) N< ) is Fgo-prime over 9. (3) If r] = v; = v;, then pUl = tp(a,,,N,,) is regular, and pul,pUp are

equal or are orthogonal. If all p, with v- = r ] are equal, q,, will denote their common value.

def def

(4) For ~ ] E I + , N,, is Fio-prime over ( 5 ) For r ] E I - , {a, : v- = r ] } is independent over Nq. (6) If T E I , ~ - - is defined, then tp(a,,,N,-) is orthogonal to N,,--.

U {a,,}.

Convention 5.2A : When we say ( ' p is orthogonal to N,,- but r] = ( ) ", we interpret this as being always true.

Remark. We shall write in short <N7, a,, : r] E I), though we do not need a< ), so a( ) is any element of N<) or undefined.

DEFINITION 5.3 : We say (N,,, a,, : r ] E I) is an F-representation of M if it is a representation and M is F-primary over u,,,,N,. If F = Fg0, we omit it.

LEMMA 5.4 : Let (Nq, a,, : r ] E I ) be a representation, then : (1) tp,( U {q: r] Q Y}, {q: not 7 4 v}) does not fork over qq-) (for

(2) For Y E P , tp,(U,,,,N,,N,) is orthogonal to q-. (3) For r ] , V E I , p,,, p , are orthogonal or equal (and then 7- = v-). (4) Each N,, is Fgo-prime over 9. (5) If M is Fgo-prime over UQEIN,, ~ E S ~ ( M ) is regular, then p is not

(6) (q, a,, : r ] E I ) represent some model of power A(I) + III. (7) Dp(N,,-,N,,a,,) is at least Dp(r],I) (see Def. 4.3(2)) even at least

r ] E I + ) .

orthogonal to some p' E Sm'(N,) for some r ] . I

( - I ) + Dp(r],4 + 1.

Proof. As in the proof of 3.2, or easy using 3.3, 1.7. (1.7 is needed for (4b)

LEMMA 5.5: (1) Every F;to-saturated model hus a representation. (2) If (@,,, a:: ~ E I , ) Fio-represents 4, 1 = 0, 1, F :&+Il an iso-

morphism (for partially ordered sets so it preserves the level), F : u,,,,o q + U , , s r l q is an elementary mupping, it maps onto wF(,,]


(equivalently, for each r ] €4, F‘ DomF’ = U l e 1 , ~ ) , then 4, MI are isomorphic.

embedded into MI. (We may demand F’ maps q onto q into PF(,) and tp,(F’(q), WF(,)-) does not fork over F ( q - ) .

i s a n elementary mapping ontoPF(,,,

(3 ) If in (2) F,F are not necessarily onto, t h e n 4 can be elementarily or at least maps

Proof. (1) This is by 3.2 (and 5.4). (2), (3) Trivial.

LEMMA 5.6 : For p > 0, (3, av : r ] E I ) Fgo-represents an F&-saturated model iff for every r ] E I , and regular p E Sm(N,) orthogonal to N,- for at least NBv’s, p , i s not orthogonal to p .

Prooj. Easy by 5.4(5), and usual arguments. Let M be F&-prime over u,N,, and suppose A C M , IAl < N,,

P E S ( A ) is omitted. Let tp(a,M) be a stationarization of p over M; clearly w.1.o.g. p is stationary. By V, 3.9 there are {a,: i < n}, independent over M , realizing over it regular types such that tp(a, M U {ac : i < n}) is Fio-isolated : w.1.o.g. tp(a, A U {ar : i < n} is Fie- isolated, and tp(a,, M ) does not fork over A and tp(a,, A ) is stationary. We now try to define by induction on i , b,EM realizing stp(a,,A U { b , : j < i }) . We have to fail for some i , so w.1.o.g. p is regular.

By 5.4(4) p is not orthogonal to some regular qES(Nv), which is not orthogonal to some P,-(~,, so we know p , ~ , ~ , , , are not orthogonal. By V, 2.3, 2.4, dim(p,M) = dim(p,-(,,,M), which is ZN, by hypothesis, so we finish.

LEMMA 5.7 : If T i s deep, then for every tree I with root ( ) and height I - w , there i s a representation (3, a, : 7 E I ) (in fact q, i s well deJined for every ~ E I - , i.e. pvYI> = q,).

Proof. Easy. By 4.4(3) there are Fio-saturated y,y+l Fio-prime over & U {a,},

tp(al ,y) regular, 4+1 < N,y+z. Complete the partial ordering of I to a well-ordering and let {yi: i < i*} be a list of the members of I in increasing order. Let n(i) = l(r]J and m(i) be maximal such that qr rm(i) ~ { r ] , : j 0). Now we define by induction on i an elementary mapping Fi. F, is the identity on No, Fr is an elementary mapping with domain N&), extending F,,,, such

NVi = Fc(Nvg), a,,$ = Fi(an(,)), and the checking is easy. that tp*(F,(N,,,,), q<cq(N,@))) does not fork over F,($)(N(iJ. Let


DEFINITION 5.8 : For a representation (N,, a,, : 7 E I) we let :

defined by : (1) E (E of the representation) is the equivalence relation on I+

yEv iff p,, = p , (iff p,, p , are not orthogonal).

We write P , , , ~ instead of p , and NTIE instead of N,,-. (2) The representation is standard if q / E is uncountable for every

(3) We say J is a A-large subtree of I (relative to E) if for q E J + there are s;lE G q / E of cardinality < A such that J = { ~ E I : for no

7 E I .

1 < ~ ( 7 ) does 7 r ( I+ 1) E S ; r ( n + l ) I E } .

(4) We define similarly “J is a (<A)-large subtree of I”. (5) J is a A-big subtree ofZ if (J E I, [v < 7 in I & Y E J =+ V E J ] and)

If we omit A we mean KO. for 7~ J I{a: ~ ” ( ~ ) E P \ J ) J G A. Similarly for (< A)-big.

CLAIM 5.9 : Suppose (3, a, : 7 E I ) i s a (standard) representation of M. Then for any stationary type q over a finite subset of M :

(a) ifq is not orthogonal to p,, then dim(q,M) = 17/EI, (b) if q is orthogonal to every p,, then dim(q,M) = No.

Proof. Easy.

CLAIM 5.10: Suppose <q,Z,: ~ E I ) is a representation of M , u E I finite and downward closed, N c M i s Fio-primary over UvEuN,,, ~ E I - u , P - E U , a ” 6 ^ a ~ M , tp(a,N) regular, {a,aJ depends on N, tp(6”E,NU a) i s Fgo-isolated, r = stp(c,6) i s regular orthogonal to N . Then r i s orthogonal to p , when p & V E I .

Proof. Because we can find a representation (Wq,ai:q~IJ of M with :

(i) q > = N , (ii) if ~ E I ; , ~ ( 0 ) > 0, q is an Fio-prime model over N, U N for

= a,), and for every such v there

-6°F.

some V E I - u , v-EU, v # p (and is such 7,

(iii) qo, is F;Io-prime over N u (iv) if ~ E I ~ , V E I are as in (ii) then for any sequence

7 , q A 7 ~ I l ~ v A 7 E I , and if ~ - F ( ~ ) E I ~ , thenIV&,) is Fio-prime over Nv-T-(t) U N&r and {N,-.T-(t), qer} is independent over N,,,,.


MAIN LEMMA 5.11 : Suppose for 1 = 1,2 , (fl,, i$; 7 € 4 ) is a standard representation of i&, and F is an elementary embedding of Ml into 4. Let 4 be as in 5.8. Then there is a function h from 4 into 4 such that :

(i) h is one-to-one h(( )) = ( ) and: rE,p * h(7)E2 h(p) ; ~ f i , ~ , is a countable subset of VIEl, s : / ~ , a countable subset of h(q) /E, ,

(ii) for ~ E I ; , F(&),p&) are not orthogonal,

(iv) i f yO€Il, ~] ,Ev, c q0/-&, vo uncountable disjoint to sioIEI, for ~ E V ~ we have Z,,-C,E~, tp(d,“C,âi,?-) is the same (for all ~ E V ~ )

and for some a,[7 h(u,) €Il A pin(.,) is parallel tp(d,, 4)] then for all but finitely many ~ E V ~ , h(7) 4 h(7 -(a,)).

(v) If F is onto 4, h is onto 4, and (iv) holds for h-’,4, Il too.

(iii) DP(Pi/El) = DP(P~(q)/E2)~

Proof. Note that by 5.9, V, 1.16 (and as the representations are standard) :

(1) dim(pk, = 17/41 for 7 €4. First note that for any ~ E I : , dim(pi,Ml) 2 N, (as the repre-

sentation is standard). Hence dim(F(p:),&&) 2 dim(F(p:), F(Ml)) = dim(p:,Ml) 2 dim(p:,Ml) 2 N,.

Hence by 5.9 for some v = v,~Il,F(pi),p.:~ are not orthogonal. Clearly, v, is not unique, but v,/& is unique and depends on 7/E1 only, and

(2) 17/41 = dim(pi,M,) G dim(F(P:),m = dim(p,p&) = Iv,/E,I. We shall now define hvlE, = h r ( v / E l ) : q/El +. v,/El for each El-

equivalence class separately.

Class I. If Dp(pvIE,) = 0, hTIEl is any one-to-one function from VIE, into v,/E,.

Class 11. Dp(p:/,,) > 0.

By 7.2, - piE, is trivial, and also P:,,/,~ is trivial by 7.3(4). Let b, = bilEl G Ni- be such that prlIE, does not fork over 6ilEl,

pilEl r 6ilE, is stationary. Let 6tlE, E N :,lE2 be such that pf7/B, does not fork over 6:lE2, pf71E, r 6;lE, is stationary.

Now F(&~)E&, 4 is Fin-atomic over UvEI2x. So there is a finite u = u,,~, E 4, V - E U , u, closed under initial segments and tp,(F(6i), dVEu x ) I- tp(l”;6:), byE12x). Hence tp(F(bi), UvEI2N~) is Fgo-isolated ovel*uvEu,q. So there is N, Fin-primary over u,,, x, N G 4,4 Fin-

538

primary over uvEIzx U N and F(6;) E N . Let q = q/El be a maximal set of sequences from A$ realizing

By V, 1.16 it is countable. Easily q U {a; : p E r]/El} is a maximal set of sequences from Ml realizing pilE, r 6: independent over 6;. As dim(F(p,’),F(6;),N) = KO, there is a countable subset s ; / ~ , E q/&, s.t. {F(a:) : p E q/E, - s;} is independent over (N, F(6$?,)). Also {q : v E v,/ E,, v 4 u, hence tp(a,,N) does not fork over (8-)} is a maximal subset of 4 of sequences realizing independknt over (N,N,-). Also s,blEz = { v : v E v,/E,, av EN) is countable. Now as p i , p,2 are ‘ trivial, the relation “ {F(@;), @} depends on 5; U 6; ” for GE r]/El - v E vt/E2 - ~ , b / ~ , , defines a one-to-one function from {p : p E r]/El - s;} into {v: v E v , / E ~ -s:}. Choose s ~ / E , ~ , 2 / , ~ s.t. s;/E, 5 s ; ! ~ , 5 7/E2, s ; / ~ , E s % / ~ , E v,/&, Js;/~,I = (s;/~,I = do, and the function maps r]/El-s: onto v,/E,-s;. We define h, on r]/El-s: according to the function mentioned above and h, rs: as any one-to-one function from s; into st. If F is onto A& there is no problem making be onto v,/E,.

Now let us check that the demands on h are satisfied. Demand (i). Holds by its definition (and h/El : I i /El +I: /& is one-to- one as the types ptlE, ( ~ ] / E , E I ~ / E , ) are pairwise orthogonal).

Demand (ii). Trivially, by the choice of h.

Demand (iii). As p~IEI,p&,)IEp are regular, not orthogonal, by 4.4(6).

Demand (iv). So suppose r ] , ~ I i , r ] , ~ v , G qo/E,,‘vo n s ; / ~ ~ = 9, vo uncountable, for r]~w~,a,, c d,,- E , E ~ , tp(a,nd,AC,,q-) = q (i.e. does not depend on r ] E wo), tp(d,, ct) is stationary regular and parallel to some pi-(=,), by renaming w.1.o.g. to

We have to show that for all but finitely many q € v o , h(r]) 4 h(r]^(O)), i.e. (see 5.10) F ( P & ( ~ ) ) is orthogonal to p i when

Now we proceed as in the definition of h. We choose 6 * ~ q - such

CLASSIFICATION FOR F&-SATURATED MODELS [CH. X, $5

r4, independent over 6:.

Let v1 = h”(vo).

h(r]) R P.

that for r] E vo,

tp(G,nd,-c,,6* u a,) I- tp(@,-d,-E,,N,- u a,) (possible as q exists).

Now we would like to have F ( ~ * ) E N , this can be achieved by increasing uBoIEI, to uiolEl preserving its finiteness. This, naturally, costs us the omission of a finite number of members of wo,vl:


IUC~/E , -U~~/E , I by 5.10 we finish (note that tp(Z7,c7) (7Evo) are pairwise orthogonal).

Demand (v). The same reasons for the existence (and uniqueness) of h/E, : +A/&, show that it is onto.

In order to get demand (iv) for h-’, note that if we allow KO exceptions, the proof of (iv) dualizes. For the finite version note that the proof of demand (iv) for h-’ needs only that s:,st are large enough in order to guarantee the existence of the “dual” of NVlE,.

COROLLARY 5.12 : Under the assumption of 5.11, the function h there satisfies :

(1) For some large (see Definition 5.8(3)) subset J1 of 4, h r J1 satisJies: it is (a one-to-one function from J1 into 4) preserving (*) for 7 , P E J l , 7 4 P e h ( 7 ) 4 h(P).

(2) If in addition F i s onto 4, for some large subsets J1, J, of 11, 4, respectively, h r J1 is a one-to-one function from J1 onto J, satisfying (*) (and even h-’ satisJies the parallel demand).

p} into I:’= {vEA:Iv/&;I 2 p}. If in addition x is regular >KO, (<A(T), naturally) and for every 7 E If”, I(7 <a)/E, : 7 ^<a> E If”}/ < x, then for some ( < X)-large subset If*X of I f P , h r I’;.X satisjies (*).

(4) If F is onto 4, h maps If” onto I:”, and for x as above w.1.o.g. maps I’;*X onto a (<x)-large subset of I:”.

(3) For any given cardinal p, h maps If” = { ~ E I ’ : Ir]/E,I

5.13 Proof of 5.1. So K, 2 A(T), 8, > N,. By 5.7 (and 4.5) there is a representation (q, a:: 7 EIO) where

I’ = ,’Ka,q,, is well defined. By 4.6, w.1.o.g. qo does not fork over 9.

As we want F;;saturated models, by our system of notation (which is not convenient in this case) we have to increase I0 to take care of this. So we can find J,I c J, and N7,a, for r ] ~ J--Io such that:

(1) (A$ a,, : 7 E J) is a representation, (2) for each 7 E J, q/E, -I has cardinality K, exactly, (3) each regular ~ E S ” ’ ( N , , ) , ~ E J, is not orthogonal to some pry,>.

(a) c 1, (b) if v 4 - { ( 6 ) , ( >>, then 7 1 4 c 4,

For each 6 < N, choose 4 such that

540 CLASSIFICATION FOR F&-SATURATED MODELS [CH. X, $5

(c) DP,$5),4) = 5, (d) ~€1: implies 7 r 1 = (5). (e) if ~ € 1 ; then I{a: q1(a)~15}1 = N,.

For any set S E N, let I ,= u4,

5 E S

J, = { ~ E J : ~ E I , or for some m, q rmEI,, q r (m+ 1) e J - 4 ,

M, be a model represented by (q, a, : a E J,).

By (l), (2), (3) above, and 5.6, M, is F;;saturated and of cardinality N,. Suppose S( l ) , S(2) E N,, P is an isomorphism from M,,,, onto M,,,,. Let h : J,(,) +- J,(,) be as in 5.1 I .

Clearly any regular type over M,,,, of dimension >N, is not orthogonal to some p,, q EI,(,). So necessarily h maps I& into I,(,). Similarly, h-l maps I,(,, into I,(,,. Hence w.1.o.g. it maps { ( f ) : k ~ S ( l ) } onto { (5) :5~S(2)} (as q<> does not fork over 9). Now use 5.12.

THEOREM 5.14 : If T is deep and Ha is 2 N,, + A( T ) but smaller than the Jirst beautiful cardinal, then

IE" N,, T ) = 2'~.

Proof. We use the theory of K-bqo as developed in [Sh 821. Let 9 - , ( w , =) be the class of ( I , f ) , I a well-founded tree (with root, no w-branch and a t most w-levels,f:I+-w) ordered by (Il, fl) < (I,, f,) i f and only i f the first is embeddable into the second, i.e. there is a function G : A -+h, I, I= s < t implies: I , != G(s) < G(t) and fl(s) =

By [Sh 821 there are 2'. pairs ( I , f ) each I of power N,, no pair embeddable into another and w.1.o.g. each r"'(N,) (use [Sh82] 4.10, and translate i t into trees as in the proof of [Sh 82, 5.71 for example). W.1.o.g. for each such I there are sequences in I of arbitrarily large (finite) length.

So let {(I: , f :) : i < 2'~} be this family, so I: E "'(N,). Let g : Ha+- N, be such that (Vi < N,)l{j < N,:g(j) = i} = N,, I: = { q ~ " > ( N , ) : g ( v ) gf (g(v(Z)) : 1 < Z(q)) €I : } , f : (q) = f:(g(v)). We leave the case a = 1 to the reader, so by obvious monotonicity w.1.o.g. B 2 1. Let for i < 2Na, 4 = { q : v ~ I : , or for some m, v r m E I : , N , < q(m) < N,+K,, and for every 1 > m if (31c)[Z = L2+(fi(qrm)+m+2)2+m < Z(q)], then q(Z) < N,, otherwise q(Z) < NB}.

5(

f,(G(s)).

CH. x, $51 DEEP THEORIES HAVE NON-ISOMORPHIC MODELS 541

Let I = "'(N,+N,), and by 5.7 there is a representation (N,,,a,: ~ € 1 ) such that q,, is well defined. We choose J , I E J , IJI = K , and N,,,a,, for V E J - I such that

( 1 ) (N,, a, : 7 E J ) is a representation, (2) for each V E J - I , q/EJ has cardinality K, exactly, (3) for each regular ~ E S ' " ( N , , ) , . ~ ) E J , is not orthogonal to some

Let for i < 2 x ~ , J b = { r l ~ J : rl~l,orforsomeZ,rrZ~4,r]r(Z+l)41). Let 2y be represented by ( N , , a , : y ~ J b ) . Clearly each 4 is FiB- saturated and has cardinality Nu. So suppose i #j < 2'~, F is an elementary embedding of 4 into 4. We shall use 5.1 1, so we have a function h as described there. Now applying (iv) on hr4 we can conclude that for some &-large subset If of 4, h r I ; preserves Q . By the choice of 4 (and 4-I,), necessarily h maps If into I,.

8 for some increasing function k : w + w , for q ~ I ; , l ( h ( v ) ) =

Now by the way we have defined 4,J (and as I : , I i are well founded) (*I0 k is the identity.

m < w } , then max{m:q rmEl : } = max{m: v r m ~ l ; } (by the choice of 1-1;). Hence for ~ € 1 ; II I:, Z(h(7)) = Z(7). But with an assumption on the ( I , f)'s we finish.]

Now as for every 7 E I: - { ( )}, I{ v €1: : v- = q-}l = N, and similarly for I:, clearly by the definition of 4, I, : (*)I

PrYa>.

We shall show later that w.1.o.g.

k(l(r l ) ) .

[Why? For any ~,vE~(N,+N,) s.t. A ~ < ~ r l r z € ~ f , h ( r l r z ) ~ { v r m :

h maps I;l n I: into I;, and (*I2 for r l E C mf:(rl) =fi(h(rl)).

Now as 1; n I: is an No-large subtree of I : , clearly there is a function g* from I! into I;l n I: preserving level and 4 and = , such that for q E I y , g ( g * ( q ) ) = 7. Now the composition g+ Ef (ghg") is a function from into I; preserving Q (but is not necessarily one-to-

fi(g(g*(a))) = fi(q), contradicting the choice of the c's. one) for v 4 : f , ( g + ( r l ) ) = ( f ,g)h(g*(rl)) =f:(h(s*(rl))) = f i l ( g * ( r l ) ) =

We still have to prove that w.1.o.g. 8 holds. We shall prove later:

CLAIM 5.15: If T is deep, then we can find (4, a,: 1 < w ) as in 4.4(3) such that:

for every 1 > 0 for some q, and 6, €4, tcp,[a,, 6,] and for every


m > I , tp(a,,q) is orthogonal to every type to which cpl(x, 6,) belong,

(*), tp(ao, No) is not orthogonal to $3, (if we waive regularity, it does not fork over 9).

Completion of the proof of 5.14. I n the proof of 5.7, we can use the (4, a, : l < w ) from 5.15 (instead of 4.4(3)). Also in our present proof we could have used (N,, : r ] E "'(N3, + N,)) derived in this way. So there are formulas cp,(x, Cl) ( I < w ) 6, EN( ) such that for each i < 2u~, r ] ~ I l ,

cpl(q,(", h,,J €!I,,,

l(r]) < m < w * q,, is orthogonal to every type to which cp,(,,,(x, 6!(,,J belongs. W.1.o.g. u,b, C No.

Now look at the proof of 5.11 (applies to B':N+q) . So for some countable u1 c 4, u2 E 4, closed under initial segments,

and N c 9, Fjo-primary over U,,EuSq the following hold : (4 F(U,<,b,) c N , (b) tp,((6,: 1 < u) ,F(&)) does not fork over F ( A ) c N c 4, and

A G%,U, 6J c A, A countable. As q< ) does not fork over $3, h maps { r ] € 4 : Z(q) = l} into { r ] E 4 : Z(r])

= l}. So w.1.o.g. [ r ] €1: * h(r]) r 1 #u2 and r ] r 1 4u1]. Let us prove that hrI: preserves equality of the level. Let ~ € 1 ; . If l(r]) = 1, we have already noted this.

Now :

Observation. If q1, q2 €I: , Z(sl) = l (q2) > 1, then for some automorphism H of 6 over N( ) , H ( ~ , , , ) , P , ~ are not orthogonal.

Hence,

Observation. If rl, r ] , €I: , Z(q1) = l (q2) > 1, then

orthogonal,

H(F(pVp)) are not orthogonal.

(1) for some automorphism H of 6 over A,H(pql ) ,p l l l are not

(2) for some automorphism H of 6 over F(A) U u,<w6t ,H(F(p , , l ) ) ,

However :

observation. If V~,!~EJ, l (v l ) # l (v2) > 1, then for no automorphism H of 6 over Ul<"bt , are H(p,,),pUB not orthogonal.

As non-orthogonality is an equivalence relation, we finish.


Proof of 5.15. Choose ~ ( x , 6) (&,a, : 1 < w ) such that ( 1 ) (&,u,:Z < w ) is as in 4.4(3). (2) !=v[a,,6] for 1 < w , and 6 ~ 4 . (3) Under (1) + (2), u* = R[q(z, 6),),L, 001 is minimal. By 4.4(3) we can find such ~)(z,&),&,a,. We can prove that N, is

Fjto-primary over No U a, (see Exercises 5.1, 5.2), hence for CEN,, tp(c,2\’o U a,) forks over 4, hence t ( c ,N , ) forks over 4. So for 1 > 0, R[tp(a,,N,), 1, 001 < u*, so for some 6, €4, and q, : != rp,[a,, 6J and u* > R[v,(z, 6,), L, a]. Let A!$ = { m : 1 < m < w , and tp(a,,Nm) is not orthogonal to some (regular) type to whichT,(s, 6,) belongs}. Suppose A!$ is infinite. Let k j = {n(Z, i) : i < w } , n(Z, i) < n(1, if 1). Now let P, = N n ( 2 . m), and for each rn, as Pm is Fio-saturated, there is a regular r, ES(N:) to which cp,(x, 6,) belongs and rm is not orthogonal to tp(a,,,,,,,P,). So r , is realized by some U ~ E N , ( , , ~ ) + ~ . Now pi(”, El) , (Pm, a: : m < w ) easily satisfies (1) + (2) (use Exercise 5.1, Claim V, 3.3), a contradiction to the choice of u*.

So each A!$ is finite. So we can find {Z(k):k < w},O < Z(k) < l ( k + 1) < w , A ! $ ( ~ ) E {i:i < l ( k + 1)). Let and (x,a:: k < w ) , ~ ) , ( ~ ) ( z , 6z(k)) are almost as required. For the requirement “tp(ao,No) does not fork over x”, prove as in 4.6 (and then use Exercise 5.1).

EXERCISE 5.1: If No is Fio-prime over 9, &+, FiO-prime over 4 U 4, for 1 = 0, 1, and tp(Gl,N1) is orthogonal to No, then N, is F;fio-prime over No U a,.

= &(k),a,* =

EXERCISE 5.2: If 4 is F;fi.-prime over A for i < i(*),& is increasing and i(*) < then u,y is Fi.-prime over A . (Hint: Use the characterization theorem IV, 4.21.)

EXERCISE 5.3 : If No is F;fi.-prime over A, (N, : a < a( *)) increasing continuous, Fi7-pfime over Nu U a,, and for each a, (8: tp(aa,Na) not orthogonal to tp(a,,N,)} has cardinality GN,,, then Nu(,) is Fi7- prime over A . (Hint : see [Sh 88aI.)

EXERCISE 5.3A : Show that in Lemma 5.5(1) we can add:

(Hint: Use Exercise 5.3.) (8) The representation is standard.

THEOREM 5.16: (1) If T is shallow, then IE: (Nu, T ) < 3a(lT,, and even IEiAT) < i.e. sup{lKl: K a family of #:;saturated models, no one elementarily embeddable into another} < aaclTI).

544 CLASSIFICATION FOR F~;SATURATED MODELS [CH. X, $5

(2) If T is deep, K~ the$rst beautiful cardinal > ITI, then IEibN,, T )

(3) If T is deep, and IT1 smaller than the$& beautiful cardinal K ~ ,

< K; and even IEio(T) < K;.

N, < K,,, then for /3 < a, IEijN,, T ) = Min{K;, 2w-}

IEiAT) = K;.

Proof. By 5.5(3) we can translate the problem of proving the upper bound to problems on embeddability of labelled trees. That is, if (q, a,, : 7 E I) represents M, we can define model q such that

(a) 3 is a model of T with universe c {u: a < 21Tl} x (1 +1(7)), (b) there are isomorphisms I,, from q onto q, f V r r E f , , for

So the function g, g ( 7 ) +q (7 E I ) has a range of cardinality 22'T' (really, 2ITI suffices). [Now if for 1 = 1,2, Ml has representation (fl,,, a,,: 7 € 4 ) and above we have chosen Mi, (7 EZ), gr as above, then

(*) if there is a function h from 4 into 4, one-to-one, preserving Q , b , (and the level) g2(h(7)) = gl(r]), then Ml can be elementarily embedded into 4.1

For 5.16(1), see below (i.e. 5.16C implies the result by 5.5(3), 4.4(4)). For 5.16(2), (3) use what we have proved and 5.14 (for N, 2 K ~ , note that the proof of 5.14 still works).

1 < w.

Now the following problem on A-well-ordering should have been dealt with in [Sh 821, but we forgot. We give below just what is needed here.

DEFINITION 5.16A: (1) A quasi-order Q (i.e. a pair (IQI, <), < a transitive, reflexive relation (but maybe x < y < 5, x # y ) is A-well- ordered if for any qf E Q, ( i < A) for some i < j, qi < q,.

(2) Q is A-narrow if for any qf E Q (i < A) for some i # j , q1 < q,. (3) 9(Q) is the family of subsets of Q. (4) For a given Q, Go is the following quasi-order on 9 ( Q ) :

A GOB if and only if for some function f from A into B, (VqEA) q

( 5 ) For a given (2, is the following quasi-order on 9(Q) :A GOB if and only if for Borne one-to-one function f from A

<f(q).

into B, ( V q E A ) q < f ( q ) .

DEFINITION 5.16B: Let .&,(Q) be the class of pairs (I, f ) , I a tree


(with root and ,<u levels) and depth Dp,(I) < a, f a function from I into Q, quasi-ordered by embeddability, where F is an embedding of (Il,fl) into (&f2) if F is a one-to-one function from I, into 4, preserving level, S Z , 4 , depth and f,(r) ,< f 2 ( F ( q ) ) for 7~1,.

THEOREM 5.16C: If 1 1 Q 1 1 = ZA,u > 0 an ordinal, then Y<a(Q) i s a,(A)+-well-ordered.

We shall prove 5.16C and 5.16D later.

LEMMA 5.16D :For any cardinal K , and ordinal 5, letting 8 = 2", there is a function F from P("() to '5 such that for A , B E P ( " ~ ) , A < l B i f F ( A ) < F(B).

Remark 5.16E. '5, "5 are ordered coordinatewise.

DEFINITION 5.16F: We say A E "5 is of kind (w,p,p) if: (i) 7 is a sequence of ordinals ,<[ of length K ,

(ii) w is a subset of K,

(iii) for every a= (ui:i < K ) E A , [i$w A i < K * a t = yc][iEw=.

(iv) if y: < yi for i ~ w , then { a ~ A : y : < a r < y i for ~ E W ) has 016 < Ycl,

cardinality IAI, (v) IAl = P*

DEFINITION 5.16G : We call A E "5 simple if it is of some kind, we call it ,<A-simple if it is the union of ,<A pairwise disjoint simple subsets.

Observation 5.16H. If A , , A , E P ( " [ ) , A , of kind (wz,yz,p,) and w1 = w2,pl ,< pz,yl ,< y2, then A , < , A 2 .

CLAIM 5.161: Any A E "E i s (,<Z")-simple.

Proof. Let x be regular large enough, and for 5 < K+,Y be an elementary submodel of ( H ( x ) , E) such that { E : ,< 2") E Nc, llNsll =2", AEN~, SEN,, KEN, and [ a sNnla l ,<~*aEN, ] , and

6 < 5 4 €4 and let N = N,+ Ef u, 3. Define an equivalence relation E, on A

iiE$if and only if A (VEENg)(ai < E = pi < E A a( = E = pa = 6). t < K

546 CLASSIFICATION FOR F ; o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $ 5

Clearly the number of Ec-equivalence classes is < 11N11" = 2", so for some &*€A for no [ < K+ is h*/Ec simple (otherwise {a/Ec:a€A, &/Ec is simple but for no E < [ is &/E( simple} a partition of A to < 2" sets, each of them simple).

Let yr = Min{EEN:a: < 5> y = ( y t : i < K ) ,

w = {i < K:a:$2\3.

Let for @< y , ~ ~ , ~ , ~ = {aEa*/Ec:@rw < & r w < yrw). We shall prove that for some [ a*/E,, is of kind (w, 7, l&*/EcI).

The only problem is (iv), so let y?eKe, r w < yr w and suppose B,c,7. = { h ~ & * / E ~ : y ~ r w < Erw} has cardinality <lh*/EcI.

By the choice of N, as {yt: i < K } E N , also for some [ ( O ) < K+,

~ E Y ( ~ , and A EY(~, , hence {E*/Ec(,,+g,A, y} E q(o)+c+l E N. So there is y* with the above properties in N , hence in NScl, for some

if we have chosen [ < K+ with Ia*/Ecl minimal, also (iv) holds. C(1), Y(0) +c+ 1 < [(I) < K+ and Ia*/Eg(l)l < IBl.c,7) < Ia*/Eg(o)+gl, 90

Proof of Lemma 5.16D. Straightforward.

Proof of Theorem 5.16C. By induction on a for a non-zero, we prove that there is a function F : 9.J (2) + A(u)[ (for some ordinal 5) where h(a) = &+=(A) such that

(U) 6 (4,fz) ifW1,fl) < F(4,A). For a = 1, since in '2 there are 2A pairwise incomparable elements,

this is easy. For a = /3+ 1, /3 non-zero use the induction hypothesis and Lemma 5.16D (the reduction to (9(YGp), < I ) is easy). For a = /?, /? limit, note that to deal with (9(u,,,YG,), < I ) , we can reduce it to problems on (P(YG,,), < I ) for y < /3.

EXERCISE 5.4 : (1) Suppose T is countable, superstable without the dop. Prove w2 < Dp(T) < GO implies IEibT) = 3Dp(T).

(2) The natural upper bounds when T is not necessarily countable and/or Dp(T) < w2.

EXERCISE 5.5: Show that for a < w 2 , there is a countable superstable T without the dop of depth w + a + 1, and IEi$ T) = 3u+1.

CH. x, $51 DEEP THEORIES HAVE NON-ISOMORPHIC MODELS 547

5.17. Discussion on invariants. So the decomposition (h$ al : ! € I ) totally describes a model M which it represents ; however, it IS not read easily nor uniquely from (the isomorphism type of) M.

We can try to correct this somewhat. A technical point.

RE-DEFINITION : In the definition of a representation we replace (8, (4) by:

(2)' IIyII G W), (4)' tp(N,,,N,,- U a,,) almost orthogonal to Zt-.

DEFINITION 5.17A : (1) A representation (q, at : 7 E I) is derived from the representation ( q , a i : V E J), if J s I, at = a: for 7~ J, there are for 7 ~ J , s , E { v : q q v~l>,[vn(a)~s, A ~ " ( a ) YES,,], Isl] <A(T), ( ) E J , T ) is Fgo-prime over U p E 6 c 2 N ; for y e J - { ( )}, let v be 7 r J for a maximal 1,, < l (q) , and IVf is Fgo-prime over ~ u { ~ : p ~ s , , } , and l = u { s , : ~ ~ J ) . (We could rename the elements of J so that it is closed under initial segments. Of course, r ] ~ J * s , , n J = (71.)

(2) Two representations (fl,,, a: : 7 € 4 ) for 1 = 1,2 are isomorphic if there is an isomorphism F from ll onto 4 and an isomorphism F' (for 7 E 11) from onto A'&), such that Il I= Y <

DEFINITION 5.17B : Two representations will be called equivalent if they have isomorphic-derived representations.

* F,, s Fv.

We can prove

LEMMA 5.17C : Two representations represent isomrphic models if and only if they are equivalent.

However, is the equivalence class of representation of M a reasonable invariant ?

We may try to replace p , , ( ~ ~ I , l ( q ) = 1) which w.1.o.g. does not fork over 9, by cP@,,)-simple types over 8. Now for types of depth zero we need just their dimension, types of depth>O are trivial, so we can represent a,, by an equivalence class for the natural dependency relation on the simple type (this is not so far from IX, 2.3's proof). However, the equivalence relation is not necessarily first order, so we do not know to continue this. A primi, we can use CeQ with a non-first-order equivalence relation, but we do not know enough corresponding theory to make this work, and get, as desired,

548 CLASSIFICATION FOR F$-SATURATED MODELS [CH. X, $6

instead (3, a, : 7 € I ) , a non-forking tree of imaginary elements such that the model is prime over it and is quite unique.

Note that in the representation (N, ,u , , :TEI) not all types, e.g. p(8,r), are necessarily there in the tree. We could have replaced NO by N&, and we could have replaced q,,,) by any Fio-saturated submodel including U E(8,c) . Moreover, if we try to continue to shrink N(,,,) the process is not necessarily well founded (when Dp(T) 2 w ) . We may be tempted to take an inverse limit of the possible representation of a model, but it is not clear we get anywhere.

We can use standard representations (see Exercise 5.3A) and represent p,’s by a canonical base, but we still have non-orthogonal, non-parallel p,’s.

For a solution which is satisfying for me, see XIII, Section 1. By the way, 5.17C is true also for ( G;,+, q)-decompositions.

X.6. Infinite depth

THEOREM 6.1: If K, 2 A(T)+N,+,, Dp(T) 2 w , then Ii$Ka,T) 2 &qNT)(l4 +No)-

Prmf. Compared with the proofs of 5.1 and 5.14 we have one advantage : the clause in 5.1 1 saying DP(JI;,~,) = Dp(p:(,),E,) becomes meaningful. For finite depth in say 5.12, it would be many times enough to show that h usually preserves the level (i.e. when the level in the tree determines the depth). For infinite depth this is not true; however, n+ Dp(T) = Dp(T), so we can “dedicate” some low depth to “mark” the level.

Let x = IDp(T)I + No. First assume a 2 w . We can find a representation (q, U: : q c I * ) , s.t.

(i) tp(a&,Nf)) is not orthogonal to 9 and has depth i, (ii) tp(a:-(,),N,) has depth i, (iii) if p is a regular type of depth i not orthogonal to 9, i > 1, then

(iv) if qcI* -{( )}, and there is a regular type over N, orthogonal

(v) if Dp(p:) = 2, then q - ( i ) ~ I * - i = 1. (vi) qn (0) $I* and vn( 1) € I * implies Dp(p:) = 2.

( i ) E I * ,

to If,- of depth i, i 2 2, then y ^( i ) e I* ,

CH. X, $61 INFINITE DEPTH 549

Clearly there is such a representation, II*l = JDp(T)J < h(T) < N, (or both are finite), and (by 4.6) Dp(T) = u { i : ( i ) ~ I * } + l .

Now we define by induction on i, for q E ( I*) - , a set H ( q ) .

Case I : q(Z(q)- 1 ) = 1, H ( q ) = K>.

Case 11: Not Case I and q(Z(7) - 1) = 2, H ( q ) = { h ; h a function with domain H(qn(l)) = {N,}, and

h ( R ) = {Nl(7)+l>.

Case 111: Not Cases I, 11: ~ ( Z ( q ) - l ) 2 3,

and range E { A : No < h d N,} U { O } , and I{ Y : g( Y ) # O>l d N,}. H ( 7 ) = {g : g is a function with domain u {H(q (i)) : q A (i) €I*},

An exercise in cardinal arithmetic is to prove IH(( ))I = 3Dp(T)(lal+N0) (using Dp(T) 2 o) (check by cases). Now for every Y E H ( ( )), we choose I,, (K,a,: U E I ~ ) , ~ Y ( U E I ~ ) and functions g : , g y , s.t. :

(a) (q, a,: ~ € 1 ~ ) is a representation, (b) gY is a function from Iy onto I*, preserving level (equivalently

length), Q , mapping Iy-I ; , I ; , into I* - ( I* ) - , (I*)-, respectively, (c) g: is a function with domain IF , g:(r]) E H ( g Y ( q ) ) , (d) if 7 ( u ) E I ; , then g$(q ^(a)) is in the domain of gt(r), (e) if ~ E I F - , 2 is in the domain of g y ( q ) , then ( { q m ( a ) ~ I y :

( a ) ) = Z>l = (g:(q)) (2) ; if 7 ~ 1 y - I ; - (so q(Z(q) - 1) = 1) then

(f) fY js an isomorphism from q onto qycy,, extending f& for

There is no problem, and IIyl = N,+h(T). We now can find Jy,N~,a,(uEJy--Iy) s.t. (g) (q, a, : u E Jy) represent an F&-saturated model My of cardi-

(h) Iy E Jy and for q E Jy of length m, if q #I, then

g:(q I{q^(a) : q - w EIyIl = N,,

z < Z(v).

nality N, (see 5.6),

I{v E J, : tp(a,, N,) = tp(a,, Nv-)>l d K,, (i) if /3 = 0, Jy = Iy. For notational simplicity :

So it is enough to prove: (j) for Y # 2, Jy n J, = {( >>.

(*I if Y # 2 are from H ( ( )), thenMy,M, are not isomorphic.

550 CLASSIFICATION FOR F g O - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $ 7

So suppose F is an isomorphism from My onto M,. We use the representations (x,uv: U E J,), ( e , a v : U E J,), and apply 5.11, obtaining a function h from J, onto J,. So there are #,-large J& G J,, Ji C Jz such that h r J k is onto Ji and preserves 4 (by 5.12).

A s q ~ J h A I { u : q Q VEJ:}~ = N,*I{v:h(q) 4 VEJ,}~ = N,clearlyh maps J& n (&-I;) onto Ji n (Iz-I;) (remember Dp(p,Y) = Dp(pi(,,)). Similarly (looking for types of depth 2 ) , J$ n &-I;-) is mapped onto Ji n (1; -I;-) . However, looking at the definition of H ( q ) when q(Z(q)- 1) = 2, we see that on J& n (I;-I;-) h preserves the level. From this we can easily deduce that h J& n I, preserves the level. As it also preserves the depth, easily (by induction on Z(u)) :

for V E J : nI , ,B , (V) = g,(h(v)).

The rest is straightforward too. If 01 < w use Exercise 6.2.

EXERCISE 6.1 .- If T is multidimensional then I ; (N,, T) 2 )a + wJ for N, 2 A ( T ) + N ~ + ~ ; also I (N , , T) 2 la+wl if N, 2 I ~ + H , .

EXERCISE 6.2: Suppose A,(1< n+ 1) is increasing, al€Al(O < I < n), tp(a,+2,Al+l) is orthogonal to A,, tpI(A1+l,Al U al+l) is almost orthogonal to A,. Suppose also that there are p pairwise orthogonal regular types, each orthogonal to A,,, but not to A,+1, and lAll = N8. Prove that n < Dp(T) and for K, > A(T)+#,, a-p 2 p

G & K u , T) 2 an(la-BIP)* And even (if n > 0) 2 2,,-1 [Ia+wl""-B+'b)].

EXERCISE 6.3: Suppose that for y there are no A,,a,(I < n+ 1) as above for n = Dp(5"). Prove I{o(K,, 2') < =I,(lal<P) for a large enough, and always I;(K,, 5") < 3 , ( l a l < P + 2IT9 for every a. Phrase and prove the parallel for I;p(Nu, T) (Hint: Repeat the proof of 4.7.)

X.7. Trivial types

(Hyp) T stable, K = K,(T).

DEFINITION 7.1 : We call p~h'"(B) trivial if it is regular stationary and the following holds:

CH. X, $71 TRIVIAL TYPES 55 1

If I_ is an independent set of sequences realizing p , 6 realizes p but tp(b, B U u I ) forks over B , then for some F E I , tp(6, B U C') forks over B.

LEMMA 7.2: Suppose Dp(N,N',@) > O,P,ES"(B) stationary and parallel to tp(a,N) (which is regular). Then p is trivial provided T does not have the dimensional order property.

Proof. Assume T is superstable. W.1.o.g. B = m,NFio-saturated, I is finite and let I = {az: 1 < n}. Suppose there is no such C, i.e. {6, it,} is independent over N for each 1. Let M be Fio-prime over NU I U 6. As I U 6 is finite by Definition V, 3.2, Theorem V, 3.6(2), V, 3.9( 1) there is a finite set J G M independent over N, of sequences realizing regular types, such that M is Fio-prime over NU J.

FACT. There is b* E I U {6} such that for no J E J, tp(6*, N U J ) forks over N .

Otherwise there are E J, ba E J, such that each of the pairs {a:, az} ( 1 < n ) and {ba, 6} is not independent over N .

So for each 1 , tp(az,NU {a;: 1 < n}) forks over N and tp(b,NU {az: 1 < n}) forks over N , so by V, 1.14 tp(6,NU {a:: 1 < n}) forks over N . By the choice of ba, tp(bo,N U 6) forks over N, hence by V, 1.14 again tp(Lo,N U {a: : 1 < n}) forks over N . As /is independent ba E {&p : 1 < n} , so let ba = a;. So, by our choice, tp(6,NU a:) forks over N, and tp(@;,NU al) forks overN, hence by V, 1.14 tp(6,NU aZ) forks overN; a contradiction to the assumption " I , 6 is a counterexample" to 5.10. So we have proved the fact.

Let J = { Z l : 1 < m,}. As M is prime over NU J, there are 4 Fie- prime over NU d z , such that M is so-prime over u,,,& (see V, 3.2). Let N* < M be Fzo-prime over N U b*. As 6* realizes p (as all members of I U {b} do), by 4.4(6) Dp(N,N*, b*) > 0, hence there is a regular q* E Sm(N*) orthogonal to N, and let q be its stationarization over M. For each 1 < m,, {b*, dz} is independent over N (by the fact), hence by V, 3.2 tp,(&,N*) does not fork over N. Hence by 1.1 q* is orthogonal to &. Hence q is orthogonal to each &. But as T lacks the dimensional order property, q is not orthogonal to some 4. This contradiction proves the lemma.

What if T is not superstable? W.1.o.g. B = m, N is (TI+-saturated, let ]be a maximal subset of I independent over N , so tp(6,N U uJ forks over M, hence is orthogonal over N , hence FP,,+-isolated, so there

552 CLASSIFICATION FOR F E ; O - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $ 7

is M Fh,+-primary over N U UJ to which 6 belongs. Continue as above.

LEMMA 7.3: Let T be stable, K = K,(T). ( 1 ) Suppose r, E S ~ ( ~ ) ( A J for 1 = 0, 1 , r, parallel to r,, r, is stationary,

regular and trivial, then so is rl . ( 2 ) Suppose r izSm(A) is a (stationary) regular trivial type and {a, 6}

is independent over A. Then for any F realizing r , tp@, A U iz U 6) forks over A iff tp@, A U a) forks over A or tp@, A U 6) forks over A.

(3) Suppose r€S"(A) is stationary regular and trivial. For any a there are do, . . . , dn-, E r ( 6 ) such that : {do, . . . , d,,-,} is independent over A, tp(d,, A U a) forks over A for 1 < n , and n = low,(a, A ) . So for any J ~ r ( 6 ) , tp(d, A U a) forks over A iff tp(& A U dz) forks over A for some l < n. Also tp(a,A U u,,,~,) is orthogonal to r.

(4) Suppose ro,rl are stationary regular and not orthogonal. Then r, is trivial if r, is trivial.

( 5 ) Suppose A c B, tp(a,B) does not fork over A, rgSrn(B) is stationary regular and trivial, not orthognal to stp(a, A). Then for some eEacl(,4 U a), e$aclA, stp(e, A ) is c13(r)-simp1e of weight 1. In fact there are e,, . . . , en-, as above, tp(a, A U {el : 1 < n}) orthogonal to r. If stp(ti, A ) is semi-regular, stp(e, A ) is regular.

(6) If for 1 = 0,1, p, = stp(@,, A) is not orthogonal to r , r a stationary regular trivial type, then p,, p , are not weakly orthogonal; in fact p , has an extension over A U a, which forks over A.

Proof. ( i ) By the definition of parallel, r, is stationary and by V, 1.8( l ) , r, is regular. Let A, U A, E M, M F:-saturated, h > (A, U A,) and r2 be the stationarization of ro (and r,) over M. Clearly r2 is regular and stationary.

Suppose r, is not trivial, then there are b, a,, . . . , an-, realizing r l , tp(6, A, U a, U . . . U an-,) forks over A,, but tp(6, A, U am) does not forkoverA,form < n. W.1.o.g. tp(6fiaon...nan-,,M) doesnotfork over A,, and then clearly 6, ti,, . . . , an-, exemplify r2 is not trivial (use 111, 0.1). So tp(6,M U a. U . . . U an-,) forks over M, hence over Ao,-and 6, go, . . . , an-l realizes r2 and r, G r2. So by V , 1.11, clearly tp(b, A, U I) forks over A, where I = r,(M) U {a,,. . . , an-,}. Obviously for every FEM, tp(6, A, U F) does not fork over A,.

So 6, I exemplify ro is not trivial (except that I is not independent, but this can be discarded by re's regularity) ; a contradiction, hence r, is trivial as required.

CH. X, $71 TRIVIAL TYPES 553

(2) The implication -= is trivial. So suppose cis a counterexample to the other direction. Let M be an Ff-saturated model, A EM, tp,(M,A U a U 6 U a does not fork over A. By some application of 111, 0.1 clearly {a,6} is independent over M , tp(c,MU a) and tp(c, M U 6) do not fork over M, tp(c,M U E U 6) forks over M, and tp(c,M) is a stationarization of r , hence by (1) is a stationary regular trivial type. So w.1.o.g. M = B. Note: nz may be infinite below.

Let {dm:m < no} be a maximal set of sequences realizing tp(c,M) independent over M such that tp(dm,MUa) forks over M, and similarly let {Ern: m < n’} be a maximal set of sequences realizing tp(c,M), independent over M, such that tp(cm,M U 6) forks over M . By V, 3.16A, no = low,(&,M), n’ = low,(6,M) and tp(d,^ .. . -d3-’,M U a) and tp(Eon.. . nEnl-l,M U 6) are Ff-isolated. So clearly {do, . . . , d,,o-l, e,, . . . , E , , ] - ~ } is independent over M and

M U E U 6) is Ff-isolated (see V, 3.2). tp(d, -... dno-’ e, ... en]-’, So let N be Ff-prime over M U a U b, {do, . . . , d,,o-l, go, . . . , a,-,} E N .

By V, 3.16(8), (10) low,(an6,M) = lowr(a,M)+low,(6,M), and { d o , . . . ,d,o-l, e,, ...,En-,} is a maximal subset of r(N) independent over M.

Now what about c? F realizes r € S m ( M ) , F depends on &^6 (i.e., tp(C,M U a U 6) forks over M), hence by V, - 1.16(3), (1) tp(c,,M U r(N)) forks over M, hence tp(c,M U {Jo U . . . U d,o-l U co U . . . U forks over M. By the definition of triviality, tp(c,M U J l ) forks over M or tp(C,,M U Ez) forks over M for some 1. By symmetry, suppose the former. Clearly by V, 1.9(2) {&,c} is not independent over M , a contradiction.

(3) Assume, for simplicity, T is superstable (otherwise use V, 3.16, 3.16A). Let A E M,M Ff-saturated, tp,(M, A U 8) does not fork over A. By V, 3.9 there are n, do, . . . , dn-l such that : tp(Jon . . . - Z,,-’,M u a) is Ff-isolated, al realizes the stationarization of r over M, n = low,(&, M) = low,(a,A), and {do , . . . ,dn-l} is independent over (M,A) . By the first part, qz = tp(dz,M U a) forks over M, hence q1 forks over A , hence for some 6,, tp(d,,A U

The only property missing is tp(dz,A U a) forks over A. If this fails we get a contradiction to part (2) (with A, a, 6,, dt here standing for 2, a, 6, C there). The second sentence is by V, 3.12(a) and triviality.

We will still have to prove “tp(@,A U ul, ,dz) is orthogonal - to r ” . Suppose not. Let N be F:-saturated A > IAI + K , A U UZ<,,dl EN, tp(a,N) does not fork over A U Ul5,d,.

As the conclusion fails, there is d, realizing the stationarization of

-

- -

A - - - - A n-

- - -

- - -

6,) forks over A , ~ , E M .

554 CLASSIFICATION FOR F E ; o - ~ ~ ~ ~ ~ ~ ~ ~ ~ MODELS [CH. X, $7

r over N , tp(d*,N U a) forks over N , hence over A. As above we get tp(@, A U d,,) forks over A. But this contradicts the second sentence in (3).

(4) Left to the reader (see V, 1.14). (5) Let {do, . . . , be as in part (3) of the lemma (replacing A by

B) , I, = { d ~ r ( C ) : tp(d,B U d,) forks over B},

E A U u I,, and and Q),, 6, be such that 6, E B , Q ) , [ ~ , , 8, and Q),(E, a, 6,) forks over B. By 11, 2.2(8) there are such that for every d~ I,, ~)[d,@,6,] iff @,[d,cf,6,], and w.1.o.g. (by increasing 4) is a concatanation of sequences from I,. Let e,ECeQ be defined by q/&, where

By the second sentence in part (3), for every automorphism F of 6- which is the identity over B U a, F maps Il into some I,,, and if m = I, then F(e) = e. Hence el can have at most n possible images (varying P). Hence el is algebraic over B U a.

Let, for i < [TI', f, be an elementary mapping with domain C = B U a U u,(d, U {e ,} U Q,f, r ( A u a) = the identity, stp,(f,(C), u,,,f, (C)) does not fork over A U C, and extend stp,(C, A U a), and fo is the identity.

SUBFACT : For i # j, {f,(dm),f,(d,,,)} is not independent over U,f,(B).

For every i > 0, tp,(f,(B), u,,,f,(B) U U,,, d, U C) does not fork over A U a, and tp,(f,(B), A U a) does not fork over A [as tp,(B, A U C ) does not fork over A by symmetry], hence by transitivity tp,(f,(B), uj,,f,(B) U Ul<,dl U a) does not over A. So by 111, 0.1 tp,(U,,, f , (B) ,B U U,,,d,) does not fork over A, hence over B, hence tp,(U,,, z,, U,f,(B)) does not fork over B. So {d,: 1 < n} is independent over (U,f,(B),B). Clearly ( f , ( C ) : i < ITI') is indiscernible over A U a, hence for every i {f t (dl) : I < n} is independent over (u,&(B),f,(B)). We can also conclude that tp(a, U,f,(B)) does not fork over A. Now we know (for i > 0) that tp(a,f,(B) uf,(d,,,)) forks over f,(B), hence tp(C, u,f,(B) Uf,(d,,,)) forks over f ,(B), so by the previous sentence tp(C,Uj&(B) U&(dm)) forks over u,f,(B). So by V, 3.12(a), {f,(dm): i < 12'1') cannot be independent over u,f,(B). Hence, for some i, tP(.fi(Jm), uj&(B) U Uj+,fj(dm)) forks over f ,(B), and by the indiscernibility and the finite character of forking, for some n

CH. X, $71 TRIVIAL TYPES 555

tp(f,(dm), u,&(B)u~-l&(dm)) forks over U,f,(B). We can choose a minimal n, so {f,(dm) :j = 1, n} is independent over U,f,(B). Applying 7.3(2) several times we find that for some Z,1 < Z < n, tp(f,(dm), u,f,(B) Ufz(dm)) forks over u,f,(B). But quite clearly

tp(U if,@) :j # O,z},fo(B) Ufo(Jm) uft(B) Uf i (dm)) does not fork over A, hence {fo(dm),j,(dm)} is not independent over f ,(B) Uf,(B). By the indiscernibility we can replace 0, Z by any i # j, thus proving the subfact.

As r is regular, for I > 0, tp(d,,B U um,,Im) does not fork over B and easily tp(d,,B U um,,(dmA cmn(em))) does not fork over B, hence tp(dl,B U Um,,(dmn Emn (em)) U Ui,,f,(B U &-con(eo>)) does not fork over B.

We can find an elementary mapping g :

Dom(g) = auU,f, B U U d m n ~ m n em) , ( m 0

g r U,f,(B U don C, (e,)) is the identity and stp,(range(g), U,f,(B U aonC0 n(eo)) extends stp,(Dorng, u,f,(B U do n ~ o n(eo))) and does not fork over U,f@ U don qn (e,)). Now {dm: m < n} U {g(dm):O < m < n} is independent over u,fi(B) by the previous paragraph, hence it is independent over (U,f,(B), B).

As we have noted above, e, is algebraic over B U a, hence e, = g(e,) is algebraic over g(B U a) = B U g(a).

Checking closely, we see d a t for any automorphism F of CCeq

which is the identity over B U aU g(a),F(e,) = e, (as F(e,)E{e,:Z < n} andF(e,) = F(g(e,)) E{g(e,) :I < n}). So e, is definable overB U a U g(a), say by +(x, a, g(a), b*) (b* E B ) . Let

n - ‘‘1 w o n 5, Yl z1) = “ (W [+@A yo, ql,f,(K*N = +(x, y1, TJ,6*))1

holds for infinitely many i’s”.

By 111, 2.5, 1.7, E is a formula which is almost over A, and let e* = a ”g(a)/E. Now exactly as in the proof of V, 4.11, e* #A, tp(e*,A) is c13(r)-semi-simple not orthogonal to r and e* E acl(U,fi(B U {e,})) but here, by the subfact, any two of {fr(e,) : i} depend on U,f,(B), hence low,(e*,A) = 1.

The only point left is “e* E acl(A U a) ”, but for any i we know that (k large enough) e*Eacl(Uf:f&(B U {e}) 5 acl(A U aU u::f&(B)) (because e,Eacl(B U a); see above). As this is true for every j and


{ f f (B) :j} is independent over A U a, clearly e* E acl(A U a). The rest is obvious.

(6) Let A E 4 for i < 17J+, Nl < ITI, {q: i < 111'1 independent over A, and r has a stationarization T,ES"(&). Let u,q c M , M F:- saturated, and w.1.o.g. tp(aona1,M) does not fork over A, and w.1.o.g. r€Sm(M) . By the assumption there is J realizing T , a; realizing tp(at,M) such that tp(ai,M U 2) forks over M. For each i there is Ji realizing the stationarization of ri over M, tp(& M u d) forks over M . By V1.9(2), tp(& M U q) forks over M (for i < JTI, I < 2). As r, does not fork over 4 by 7.3(2) for each 1, i, tp(&;,Y U J ) forks over Y. Again by 7.2(2), {a",, $} independent over iVf is impossible. As this holds for each i as A E Y,{q: i < ITI'} independent over A, easily tp(a;,A U &) forks over A.

CHAPTER XI

THE DECOMPOSITION THEOREM

XI.0. Introduction

(Hyp) T is stable.

This chapter is similar in spirit to Chapter IV, i.e. we give here an axiomatic treatment, not claiming for it any inherent meaning, just that it is a good way to organize the material. The notion we try to axiomatize is “a decomposition”. The most elegant case was presented in Chapter X, Section 3, but in order to get the “main gap’’ for other cases we need other such theorems. In Section 2 we present the axioms, notions and prove the existence of decomposition (and the possibility to extend a partial one). T is a set of “small” types, E * a notion of “strong submodel ”. So we try to get decomposition theorems for models M realizing every p E T over M, by G *-submodela. In Section 3 we define various T’s and E *’s and prove that the axioms are satisfied for various pairs.

In Section 1 we prove some more facts on “ p stationary inside A ”, (weak) orthogonality and conclusions of “ p eSrn(M) is not regular ” or “ P , Q E S ~ ( M ) are not orthogonal” (for M not necessarily Ffr(T)- saturated).

XI.l. Stationarization

DEFINITION 1.1 : (1) We say that p €Srn(B) is stationary inside A if B E A and there is a unique extension of p in Sm(A) which does not fork over B.

(2) We say that p is stationary inside ( B , A ) when p is an m-type which does not fork over B , B E A , and for every qESrn(B), if p U q does not fork over B, then q is stationary inside A.

557

558 THE DECOMPOSITION THEOREM [CH. XI, $1

FACT 1.2: If B G , A (see Dejnitions 3.1(1) and 3.1(4)) and p ~ s " ( C ) does not fork over B , then p is stationary inside (B, A ) .

Proof. Suppose not. By Definition 1.1 w.1.o.g. p € S r n ( B ) , so as p is a counterexample, for some ~ E A and p p U {p(~,a)} and p U { -p (~ , a)} does not fork over B . By 1.4 below, for some 6~ B and 6, I= $[a, 61 and every a' satisfying l=6[a', 61 satisfies this. As B = , A there is such a' E B , easy contradiction.

FACT 1.3: If p ' ~ S " ( A ) , p = q , q forks over A , then for some Q, = p(Z,6) , q l-p(Z, 6), and p U {p(x,6')} forks over A for every 6'.

Proof. It is easy from the next subfact.

SUBFACT 1.4: For every ~ E S " ( A ) , and + = + ( ~ , g ) there is #@,a),

for any 6 , p U {+(Z, 6)} does not fork over A ifs != 6[6, a]. A such that:

Proof. The set of qEST(l6l) such that p U q does not fork over A is finite, in fact the number is <Mlt(p, +, H,,).

Let {ql , . . . , qn} be an enumeration of those q's. We know (II,2.2( l ) , (a), and 11, 2.13(1), (3)) that for some Bl(g,al),

+(z, 6) E ql if and only if I= el[6, all. Let 6(g,a) = V:..,6,(S,al). So we can conclude (by 111, 0.1, the extension property) that :

p U {+@, 6)) does not fork over A iff for some 1, +(z, 6) Eql iff for some I I= 6,[b, at] iff I= 6[6, a].

The only point left is ~ E A . However, easily every automorphism of 6 which is the identity on A maps to an equivalent formula. By 111, 2.3(1) this implies that #@,a) is equivalent to a formula with parameters from A .

Remark. Alternatively, for 1.4 we compute R [ - , + , k ] for large enough k depending on p .

FACT 1.5: Suppose {atn&: i < w } is indiscernible over C, tp(&, CU U,<.a, U U,,,b,) does not fork over F U a,, tp(&FU u,<wai) does not fork

CH. XI, $13 STATIONARIZATION 559

overcand tp(6,,FU U5<,E), tp(d,aU u,,,a5) arenot - weaklyorthogonal. Then for some n, tp(bfl, dt,,a,) and tp(6,”. . . bfl-l, UiSfl a -t ) are not weakly orthogonal.

Proof. Clearly, tp(bt,aU u5,,a5 U uj,i6,) does not fork over F U ai (otherwise there is a formula witnessing it, and as {a,n 6,::j < w } is indiscernible over F, we may assume w.1.o.g. that the formula is over FU uj,,a5 U u5<,6,, a contradiction to one of the hypotheses). Also {a, : i < w } is indiscernible over and (as {ai : i < w } is indiscernible over F, tp(d, F U u,,,a,) does not fork over aand by 111, 1.6(1) strong splitting implies forking).

As tp(6,,FU u,<,@,), tp(d,FU u,<,a5) are not weakly orthogonal there are 6; and p such that % realizes tp(60, FU u,,,it,) and l=p[ao, b,, al ,at , . . . ,ak-l,d, F ] , l=1p[aE,&,al, . . . , ak-l,d, El. By the indiscernibility of {at: i < w } over aU d, there are 6, b;(i < w ) such that G, b;( realizes tp(6*,cU u5<wa,) and

p[at9 6, at+l, * * * 1 @t+k-l> d, b7 p[@i, b;(, ai+l, * * * 9 al+k-l - 3 d, F] .

Now the sequence (EO”&,al” K,tizn &,,a3CG,...,a2tAb;;l, azi+l n6&+l,. . .) cannot be an indiscernible set (as the formula p(, . . . , d, C) contradicts 11, 2.20). So it cannot realize the same type as (gon 6,, alA61,. . .). The conclusion is immediate.

Remark 1.5A. ( 1 ) This is essentially like IV, 4.12, p. 190. (2) Note that we can replace at by a: = ai -c; (3) We can let the sequences be infinite.

FACT 1.6: Suppose in 1.5 that in addition there is 2t’ realizing stp(d,FU ui<,at) such that tp(d’,au u,,,a5 U 6,)fork ower F U 4. Then there are n < w and & realizing stp(6,,FU Ut<,at) such that tp(&,, U ~ - , ~ , U F U U , , , Z , ) ~ ~ ~ ~ ower F U ~ ~ .

Proof. Work in Ceq and replace at, F by an enumeration of acl(F -at), acl(C), respectively. Now apply the previous fact.

FACT 1 .7 : Suppose tp(6,M U a) is not almost orthogonal to M , l= @(6,a). Then there are al,6,(l-= 0 , l ) such that:

(i) tp(aon60,M) = tp(a^b,M), tp(al,M) = tp(a,M), l=@[61,al], (ii) the type tp(a1,M U ito U 60) does not fork ower M ,

(iii) l=p[ao, 6,, iz,, 61, F] and {p(a0, Z, a:, 6:, @} U tp(6,, a. U @ fork ower a, U M , for some p and FEM, for every a:, 6:.

560 THE DECOMPOSITION THEOREM [CH. XI, $ 1

Proof. As tp(6,M U a) is not almost orthogonal to M , we know that for some 8, tp(d,M U a) does not fork overM but tp(&M U U 6) forks over M. We can choose by induction on n < w , sequences %,,,6,, such that anh6, realizes stp(&^6,M), and tp(an”6,,MU dn6^CU Ul<,,a2^6J does not fork over M . By 1.6 for some n < o there is i!$, realizing stp(b0,MU lJl<,,,al) such that tp(&,CU U,,,al u U,“,,6J forks over M U 4; w.1.o.g. n is minimal. By 1.3 there is a formula p and CEM such that

- (i) bp[ao, i!$,, ai,&, . . ,a,, b,, a n + l , an+2, - . . ,4,

(ii) for every a:, . . . , e, 6:, . . . ,5:, {p(- - -* 5* a* 6* -* 6* a”* a o , ~ , a i , 1 , 2 , 2 , . . . ,an, ,+1,...,a:,@}U tp(b0,M U ao) forks over M U a0.

h -

As {ao 6, a1, a2^ b,, . . . , a, 6,, . . . , ah} is independent over M (by the minimality of n) , tp(a2 n62 -. . .n a,, ak, M U a. âl) is finitely satisfiable in M, hence we can find in M a sequencep = a; “& a; & . . . a; b:, A a;+, A. . . a;such that l=(39) [p(ao, i!$,, a,, 8, P, 6) A +(g, a,)] (remember I== @[6, a]).

6, an+,^ . . .

Let & satisfy this formula, i.e.

k p[Eo, i!$, ; Gl, 4, @, C] A +[&, all. Clearly, ao, g, a,, 6 are as required.

FACT 1.8 : Suppose p €Sm(M) is not regular, @(2, &) ~ p . Then there are c , c ~ , c ~ such that

(i) C, Co realizes p , and I= @[c,, a], (ii) {C, Co} is independent over M ,

(iii) tp(C,M U C, U c(,) and tp(C0,M U c,) fork over M .

Proof. A s p is not regular, by Theorem V, 1.9( l ) , (3) there are cl(l < n) , C and iP realizing p such that { E l : 1 < n} U {c} is independent over M, tp(@,M U UzGnC2) forks over M and tp(C,M U UZGn cl U @) forks over M. W.1.o.g. n is minimal. Still n 2 0. If for every I < n, tp(q,M U u;,,+, Em U F) does not fork over M , then tp(Co n. . .n C,,, MU @) does not fork over M, a contradiction. By renaming and monotonicity of forking, tp(Co,M U uu:,,~~ U P) forks over M. By 1.3 there are formulas pl,qz and EM such that letting d = C l h . . . n c n :

(4 l=Vl[@, CO, d, S I ? (b) != V2P, e, 4, d, %I,

CH. XI, 5 21 THE AXIOMATIC TREATMENT 56 1

(c) for every i?*, d‘, {pl(i?*, Z, d, al)} U tp(co,M) forks over M, (d) for every P*, d, G,{v2(Z, P*, ci, d , al)} U tp(c,M) forks over M. So Wy)[vl@,cO,&al) A v 2 ( ~ , ~ , co,&al) A @ ( ~ , a ) ] (use @ as a

witness) and {d,c0,c} is independent over M (as {q: 1 < n} U {F} is independent over M). Hence for some d’ EM

b(3Y) [Q)i(Y, co, $ 9 a1) A v2(G 8,4,$, a1) A @(Y, a)].

So let F+ satisfy

l=vl[a+, co,d’, all A v2[c,c+, cO,d’,al] A @[c+, a].

Clearly, F,Fo, F+ are as required (for F, c0, el) in the fact.

CLAIM 1.9 : Suppose p E Sn(N) is not orthogonal to some type q to which @(~,6) belongs, where b€N,l(Z) = m. Then

(1) for some r , @(z, 6) ETES~(N) , r is not weakly orthogonal to p . (2) If N c N* and N* realizes p , then $(N*, 6) # @(N*, 6).

Proof (1) Let N E M,M F&,-saturated, and tp(a,M), tp(c,M), are complete types over M which are non-orthogonal stationarization of p , q , respectively. By V, 1.2(3) tp(&,M), tp(c,M) are not weakly orthogonal, so w.1.o.g. tp(a,M U C) forks over M, hence over N. So for some JEM, EEN, btp[a,~,d, E] and for every c‘ ,d, { ~ ( Z , E ’ , d’,i?)} Up forks over N. Now as t=(3Z)[v(@,Z,Z,?3 A @(~,6) ] , {a ,d} independent over N for some d” E N, k ( 3 ~ ) [ ~ ( a , Z, d‘, i?) A @(z, b)] ; hence for some F’, I=cp[a ,F’ ,d’ ,F] A @[F’,b]. Clearly, r = tp(F’,N) is as required.

(2) Follows by the proof of (1): if N EW,CZEW realizes p , obviously k(3Z) [v(a,Z,d’,i?) A @(Z,b)] and we can choose c ’ ~ @ ( W , b ) , such that kg~[a, c’, d‘, C] A @[?, 61. Now c‘#N as tp(a,N U c‘) forks over N.

XI.2. The axiomatic treatment

We try here to axiomatize the various variants of the decomposition theorem. Notice that we have such theorems for Fi0-saturation, < ( T superstable without the dop, in X, Section 3) for FE, -saturation, < (T totally transcendental without the dop, in [bh 82bl) for FL0-saturated, E, (in [Sh 86, Section 2]), T superstable satisfying


the (< co, 2)-existence property; defined here in XII, 4.2) and we shall consider a few others,

We could have omitted the regularity in our treatment, but we can have it in all our applications. The most pleasant case is T = Tko (see Def. 3.1) E * = Sio (see Def. 3.1), and a reader who does not like our generality can concentrate on it. The proof that our axioms are satisfied by some pairs is postponed to the next section.

CONTEXT 2.1: We shall consider a pair (T, E *) where T is a set of types (i.e. m-types for m < w ) and E * is a two-place relation on the family of sets A E 6. If T = Ti, we omit it (see Def. 3.2) and if E * is Ek0 (see Def. 3.1) we omit it. Let T(A) = ( ~ E T ; p a type over A) .

Ax(Ai) : C * is preserved by automorphisms of 6, A E * B implies A c B , i f A $ B l e t A s * B m e a n s A c * B U A , a n d a s s u m e A s * A .

Ax(A2):A E * C a n d A EBECimpl iesA E*B.

Ax(A3): C* is transitive.

Ax(A4): If A c *$, for i < a,$ increasing, then A E * u,,,$. Ax(A4;,) : If A C B and A C * C for every C E B of power <A, then A E * B (we write h instead <A+).

Ax(A5) : If A, C * B for i < a, A, increasing, then U,<,A, C *B.

Ax(A6) : If {All: V E ~ > is a non-forking tree, Arrk c *Al , then A , ) E * U,,IA,.

DEFINITION 2.2: (1) A set is T-saturated if every type PET@) is realized by some ~ E A .

(2) F(T) = { ( p , B ) : p ~ 8 ~ ( A ) for some m < w,B E A, and there is

(3) A is T-atomic over B if A is F(T)-atomic over B. (4) A is T-primitive over B, if B E A and for every T-saturated

A* extending B there is an elementary mapping F , F ( A ) E A * , F I B = the identity.

q c p r B , q E T , q w

CH. XI, $21 THE AXIOMATIC TREATMENT 563

(5 ) A is T-prime over B if A is T-saturated and T-primitive over

(6) A is T-minimal over B if A is T-saturated, B S A, and every B.

T-saturated A*,B E A* c A, is equal to A.

Remark. Usually any T-saturated set is a model (follows by Ax(B2)). So usually we denote such sets by M,N even if this is not necessarily assumed.

Note that T-primitive is not necessarily equivalent to F(T)- primitive.

Ax(B1) : T is preserved by automorphisms of a.

Ax(B2): Every finite type is in T

Ax(B3): I f M C A , p E tp(n,A),pET, tp(&,A) does not fork overM, and M is T-saturated, then p is realized in M.

Ax(B3-) : Like Ax(B3) but M 5 *A.

Ax(B3--) : Like Ax(B3) but M E *A U ii.

Ax(B4) : If B is F(T)-constructible over A, then B is T-atomic over A.

Ax(B4+): If B is T-atomic over A , A G B , and tp(6,B) is T-atomic, then tp(6,A) is T-atomic.

Ax(B5) : If p ( Z , Q) is a type, p ( Z , 6) E T and p ( Z , 6) U {cp(%, 6)) is consistent, then for some q(y) ET which brealizes, Domq s Domp(Z, y) and, for every 6' realizing q ( p ) , p ( ~ , 6') ET and p ( ~ , 6') U { c p ( ~ , 6')) is consistent.

Ax(B6) : If p(Z , g) ET, then p(Z,6) ET.

Ax(B7): If 6 is a limit ordinal, i <j < 6*M, C M , , each Mi is T-saturated, then ui<6 Mi is T-saturated.

Ax(C1) : If N S Nl E C , N E *A, {A,Nl} is independent over N , C is T- atomic over Nl U A, and Nl,N are T-saturated, then Nl c * C .

564 THE DECOMPOSITION THEOREM [CH. XI, $ 2

Ax(Cl-) : Like Ax(Cl), but A is T-saturated too and N C *Nl.

Ax(Cl--) : Like Ax(Cl-), but C C acl(Nl U A ) .

Ax(Cl+) : Like Ax(Cl), but N not necessarily T-saturated.

Ax(C2) : Suppose M C_ *P E N , all T-saturated and for some FEN, C # P tp(E,W) is not orthogonal to M . Then for some F E N , a#IP and tp(F,P) does not fork over M .

Ax(C3) : Suppose M C_ * A , A C N , A # N and M , N are T-saturated. If for every EN, &#A, tp(5,A) is not almost orthogonal to M , then for some EN, 5$A, tp(5,A) does not fork over M .

Ax(C3+) : Like Ax(C3), but we strengthen the conclusion to : tp(6, A ) does not fork over M and is regular.

Ax(C3-) : Like Ax(C3), but we assume : M c *N.

Ax(C3*) : Like Ax(C3+), but we assume : M S * N .

Ax(C4) : If M G * N , M # N , then for some FEN, F#M and tp(i7,M) is regular.

Ax(D1) : Every PET@) can be extended to a T-isolated complete type over A .

Ax(D2) : SupposeM E * A , A C N , ~ E T ( A ) andM,Nare T-saturated, then there is FENrealizing p such that tp(c,A) is almost orthogonal to M .

Ax(D2--) : Like Ax(D2) but M C * N .

Ax(D2-) : Like Ax(D2), but suppose also & E Ml c . . . c M, = M , each Ml is T-saturated and C _ * N and for each I < n(VB) [Ml+, c B G N A tp*(B,4+l) is almost orthogonal to 4 C *B] and tp,(A,i&+,) is almost orthogonal to 4.

Ax(E1) : Suppose & S *Ml(Z = 1, 2){M1,4} is independent over ik& and each &(Z < 3) is T-saturated. Then there is a T-prime T-atomic model M over Ml U 4.


Ax(El+): Like Ax(El), but we omit the hypothesis && E*& (but require that they are T-saturated).

Ax(E2): For Ml as in Ax(El), every ~ E S " ( M , U 4 ) which is T- isolated, is almost orthogonal to MI and to 4.

Ax(E2+): Like Ax(E2), for&& c Ml T-saturated, {&Il,@.} independent over &&.

DEFINITION 2.3: ( 1 ) Let A( E *) be the first A such that for every A EB, there isA' ,AEA'E*B,JA'J<A+lAl .

(2) Let A(T) be the first A such that the following is impossible: p 2 A, A, increasing continuous for i < p+, lA,l < p, and for every i some p€T(A,) is realized in A,+1, but not in A,.

if A,(i < A+) is increasing continuous, lAJ < A, then for some i :A , s * U,<h+Aj and every p€T(A,) realized in U,A, is realized in A,.

(4) A,(T, E *) is the first A 2 IT1 such that: if A,(i < A+) is increasing continuous, lAtl < A, then for some closed unbounded subset S of A+, for every i€S of cofinality cf(A),A, E*U,A, and every p€T(A) realized in U,A, is realized in A,.

(3) A(T, E *) is the first A 2 12'1 such that: (*I

DEFINITION 2.4: (1) We say that ( N , , @ , , : r ] ~ 1 ) is a (T, c*)- decomposition if:

(a) I is a set of finite sequences closed under initial segments (but a( is immaterial),

(b) each y is T-saturated, llyll < A(T, c*), (c) q r r G*q, (d) tp(@?,q-) is orthogonal to N,-- (when Qr]) > l) , (e) t p * ( 3 , q r k U E,rcr+l,) is almost orthogonal to qrk (where k <

(f) : r ] (i) E I ) is independent over N,, (g) tp(@,,N,-) is regular, (2) we define a (T, E *, A)-decomposition similarly, replacing

w, r]€I ) ,

A(T, E*) by A.

Remark 2.4A. (1) Remember T,- = r] r (Z(r]) - 1).

A)-decomposition if A satisfies 2.3(3) (*). (2) All we shall prove for (T, E *)-decomposition, hold for (T, E *,


DEFINITION 2.5: (1) We say that ( q , a , : q ~ I ) is a (T , c*)- decomposition inside M if (in addition to (a)-(g)) :

(h) N, c M for every t E I, (i) if N, E A E M , tp,(A, N,- U a,) is almost orthogonal to N,-, then

(j) M is T-saturated, (k) N<) E *M. (2) We say that (N!, a,, : 7 E I) is a (T , c *)-decomposition of M if it

is a maximal (T , E *)-decomposition inside M (maximal - under the natural ordering).

CLAIM 2.6: (1) If (N,,@,: ~ E I ) is a (T, c *)-decomposition, then (N,: ~ E I ) is a non-forking tree.

(2) A n y (T, c*)-decomposition inside M can be extended to a (T , = *)-decomposition of M (there is always a (T, E *)-decomposition inside M: the one with I empty).

Proof. (1) Use (e) and (f) of Definition 2.4.

Nn =*A,

(2) Easy.

CLAIM 2.7 : (1) [Ax(B2)] Every T-saturated set is a model.

Ax ( B 3 7 . (2) Ax(B3) implies Ax(B3-) and (B3-3; and Ax(A2), (B33 imply

(3) Ax(Cl+) implies Ax(C1) implies Ax(Cl-) implies Ax(Cl--). (4) Ax(C3+) implies Ax(C3) and Ax(C3*), each of which implies

(5 ) Ax(D2) implies Ax(D2-) implies Ax(D2--). (6) Ax(Dl), (B4) imply Ax(EI+) implies Ax(El) , also Ax(E2+)

implies Ax(E2). (7) Suppose that whenever &(I < 3) are T-saturated {M,,k&} is

independent over 4, and 4 c *Ml, then acl(M, Uk&) is T-saturated. Then Ax(B2), (Cl--) imply Ax(Cl-).

Ax(C3-). Also, Ax(C3') implies Ax(C4).

(8) Ax(Cl-), (D2-), (El) imply Ax(E2). (9) Ax(B3) and Ax(B5) imply Ax(E2); also Ax(B3-), Ax(B5) and

(10) Ax(B3-), Ax(B5) and Ax(D1) imply Ax(D2). (11) For A < p, Ax(A4;,) implies Ax(A4;J. If Ax(A2), then

(12) Suppose T satisJies Ax(B3). If M is T-saturated, p ( z , Q) E T ( M ) , A b realizes p and tp( 6, M U a) is not almost orthogonul to M , then there

are al -b1 ( I = 0 , l ) satisfying (i), (ii), (iii) of 1.7 and aln 6, realizes p .

Ax(Cl--) imply Ax(E2).

Ax(A4) ifl AX(A~;,~).


(13) if T is superstable, Ax(B3) implies Ax(B7). (14) Ax(B4+) implies Ax(B4).

Proof. (8) Let M be T-prime and T-atomic over M, Ui& (exists by Ax(E1)). Let us prove that tp,(M,M, U 4 ) is almost orthogonal to M,. Let C be a maximal subset of M such that tp,(C,M, U 4 ) is almost orthogonal to MI. Clearly, M, U 4 E C. By Ax(Cl-) M, E *M, and M, C* C hence by Ax(D2-) (even (D2--)) C is T-saturated. As M is T-prime over M, U 4 , M is isomorphic over M, Uik& to some M-,M, U 4 c M- c C. Now tp,(C,M, U 4 ) is almost orthogonal to M,, hence tp,(M-,M,U%) is almost orthogonal to M,, hence tp,(M, M, U 4) is almost orthogonal to M,.

(9), (10) Use 1.3. (12) Repeat the proof of 1.7.

THEOREM 2.8: [Ax(Al), (A2), (A3), (B2), (B3), (B4+), (B5), (B6), (B7), (Cl-), (C2), (C4), (D2-), (El), (E2) and: (A4) or (A6), (Cl)]. Suppose T does not have the dop. If (A$a,,:qeI) is a (T, c *)- decomposition of M, then M is T-prime T-atomic and T-minimal over U,,,N,.

We break the proof to a series of claims.

CLAIM 2.9: (1) [Ax(A3), Ax(Cl-)]. Let &,M(l< 3) be as in Ax(E1). Then 4 E *M for 1 < 3.

ik& E M,*, {M,,M,*} is independent over 4 and each model is T-saturated. Then every T-isolated p E Sm(Ml U 4) has a unique extension in Sm(Ml U M ; ) .

(2) [Ax(B2), 0331, (B5), (B6)I. Suppose 4 E *Mi94 E *4,

Proof. (1) Trivial. (2) Easy too. Let p be a counterexample. By Ax(B2), (B3) (or use

XII, 2.3) M, U 4 EtMl UM; (see Definition 3.1) hence by 1.2 p is stationary inside MI U Mz. So (as p is a counterexample) some extension of p in Sm(Ml U M,*) forks over M, U 4. Let a realize such extension, so a realizes p and for some 6eM,*, FEM,, and cp t=cp[a, 6, F] and cp(a,6, Q fork over M, U 4 . By Fact 1.3 w.1.o.g. for every 6, F’, p U cp@, b’, c’) forks over M, U 4 .

As p is T-isolated there is p, E p, p, I-p and pOeT and w.1.o.g. F E Domp,. Clearly, p, U {cp(Z, 6, q} is consistent, hence by Ax(B5)


there is q(q, Z) ET such that q(q, Z) E tp(6-E, Domp,), and for every b' c realizing q(g,q, p , U{cp(Z,6',c')} is consistent. Now tp(6, Ml U 4) does not fork over 4, E E M ~ and q(g, C) ET (by Ax(B6)), and q(g,C) E tp(6,M1 U&). So by Ax(B3) some 6 ' ~ & realizes q(q,C), hence 6' F realizes q(q, Z), hence p , U {cp(~,6', C)} is consistent. As p , I-p, also p U {cp(x, 6', C)} is consistent, by the choice of cp it forks overM, U 4 . As~ES"(M, U & ) , E , ~ ' E M ~ U&, we get a contradiction.

- A -I

CLAIM 2.10: [Ax(Al), (AZ), (A3), (BZ), (B3), (B4+), (B5), (B6), (B7), (Cl-), (El ) , (E2), and : (A4) or (A6), (Cl)]. Suppose I Gu' a is closed under initial segments, ( N , : q € I ) is a non-forking tree, each N, is T - saturated and Nqrr E *%. Then there is a T-prime T-atomic model M over UqEIq, and 4 E * M for every ~ E I and tp,(M, U,,,N,) is almost orthogonal to each 4.

Proof. Let I = {vr : i < /3} be such that qr = 7, r k - i < j. We define by induction on i a model 4 such that :

(a) u,<,+rq, c 4 9

(b) 4 is T-prime and T-atomic over U,,,+,N, U q for each y < i,

For i = 6, limit. We let 4 = u j , d M j .

Now (a), (d), (e) are trivial, (c) follows by Ax(A4) (or use Ax(A6), ((31): by Ax(A6) q, s * U { N , : a < l+i,vj, 8 q,} and then by downward induction on k < l(v,), using Ax(Cl), 3, E *u{q=:a < 1 +i, q, r k < va}, so for k = 0, qj E * Ua<,+&,=; now use Ax(C1) with q,, q,, uu..l+,qm,l& standing for N,N,,A, C, respectively, using Md is T- atomic over U,<l+r N,,, proved below). Now (g) follows from (g) for j < i, by the finite character and monotonicity of forking. As for (f), let Ed!& then for some a < 6, a€&, hence tp(q U 1 < l + a ~ , ) is almost orthogonal to each q,j < l+a by (g), tp(E,U,,,q,) does not fork over u,<,+&,, so easily it is almost orthogonal to each N,, proving (f). Also, tp(c;u,,,+,q,) is T-isolated, hence for some p E

is almost orthogonal to each q,(j < 1 +a) and as (q : q E I) is a non- '~('7 U,<,+aq,) P I - tp(c, Uj<l+uNq,) and P E T . ~ u t as t p ( ~ , Uj<l+aq,)


forking tree, clearly (by XII, 2.3) tp(c, u,,l+,q,) I- tp@, U,,dA?. We can conclude that p I- tp(c, u,,,q,), hence tp(c, u,,,N,) IS T- isolated.

The same argument proved half of (b); M, is T-atomic over u,,,qj U My. For the other half (T-prime) let M+ be T-saturated, u,<l+,l'(j EM+, and we shall define an elementary embedding of M into M+ over uj,I+ui'$. For this we define by induction on i < 6 an elementary embedding 4 of 4 into M+, Fi rN = the identity for j < 1 +a and 4 E I$ for j < 1 +a. F, is the ident!iy ; for limit we take union, for successor we use (b) from the induction hypothesis. So Fb is as required.

Really this proves (b) for y = 0; the general case is proved similarly. Mi is T-saturated by Ax (B7).

For i a successor, let 1 + i = j+ 1, where j > 0 and let l(7,) = k,, 7, r (k,- 1) = 7,. By (g) for i- 1, {N,,,4-l} is independent over q,. Those three models are T-saturated (M+l by (b), the others by Def. 2.4), by (c) qa G*JI - , and by Def. 2.4. qm ~ * l $ .

So Ax(E1) applies and a T-prime T-atomic model 4 over NVj U M ) exists. By Ax(E2) tp*(4,NTj UM)) is almost orthogonal to N, and to 4, and by 2.9(1) q, G * ~ , M ; ~ * 4 . Now (a), (d), (e) hold trivially, (c) holds (by Ax(A3)). We can get (f) by 2.9(2), and it implies (g). Now (b) can be proved using (B4+).

CLAIM 2.11 : [Ax(Al), (A2), (All), (B2), (B3), (B4+), (B5), (B6), (B7), (Cl-), (El) , (E2) and: (A4) or (A6), (Cl)]. Suppose ( q : r ] ~ I ) is as in the previous claim, and M is T-prime over u,,,y. Then tp*(M, u,,,N,) is almost orthogonal to N , and N , G *M for each V E I , and M as T-atomic over u,,,N,. Proof. By 2.10 there is M+ T-prime over u,,,N, which satisfies the conclusion of 2.11. As M is T-prime over UTEII$ there is an elementary embedding ofM intoM+ over U,,,q, so w.1.o.g. M c M+. Easily M inherits the required properties from M+.

CLAIM 2.12: [Ax(Al), (A2), (A3), (B2), (B3), (B5), (B6), (Cl-), (C2), (El), (E2) and: (A4) or (A6), (Cl)]. Suppose T does not have the dop. Suppose (3, a,, : 7 E I) is a (T, E *)-decomposition u,N, G M, M # M+, u,,,N, EM+ c M,M+ is T-prime T-atomic over U V E I q , for each V E I , N, G *M+ and tp,(M+, U,,,,N,) is almost orthogonal to 4. Then for some FEM, F$M+ and ~ € 1 tp(c,M+) does not fork over N, and is orthogonal toN,- if 7 # < ). [Actually we use only (a), (b), (c) ofDejnition 2.4(1).]


Proof. Let h = ~ ~ M ~ ~ + + ~ T ~ + , and the following fact is true (under 2.12’s hypothesis).

FACT 2.13 : (A) We can find Ff-saturated q ( g E I ) such that N,, E q, qrk 5 q, tp , (q ,M U u {v : g # v and not g 4 v}) does not fork over

(B) For such q, (q : g E I ) is a non-forking tree and there is M*, Ff-prime over UIeIT,M+ C M*. (We assume M+ is T-atomic over

Y.

U,,& and Ax(B2), @3), (B5), (BW)

Proof of 2.13. Only the second phrase of (B) is problematic, and for this it suffices to prove that tp,(M+, U,,,N,) I- tp,(M+, U,,,q). This property of M+ is inherited by any submodel, hence it suffices to prove that for some M-, T-prime over U,,,,I$ tp,(M-, u,,,,N,) I- tp*(M-,u,,eIq). Let (qr: i < /3) be as in the proof of 2.10 and we define by induction on i </3 & as there and M: such that in addition

(h) 4 E M:,M: is increasing (with i), (i) M: is Ff-prime over U l < l + r q (hence tp,(M:, U,<,+,q) I-

In the successor case (1 +i = j+ 1 > 1) we first define 4, then prove tp*(N,q, U4-,) I- tp,(&,q, UM:,) (by 2.9(2)), then prove tp*(&,N, U M:-l) I- tp,(&N*, U M:-,) (by 2.9(2)), and at last defining Mi*.

End of proof of 2.12. So for any FEM,F#M+, tp(F,M*) is not orthogonal to some v (as T does not have the dop and tp(F,M*) is parallel to some complete type overM*, and use X 2.2). Choose such F, v with minimal l ( v ) . But {T ,M+ U C) is independent over N,, hence tp(C,M+) is not orthogonal to 8. Now apply Ax(C2).

tP*(M:, U,,&)).

CLAIM 2.14: [Ax(A2), (C4), (D2-)]. (1) Suppose (N,,a,,:qeI) is a (T, E *)-decomposition inside M, V E I , tp@, u,,,,N,) does not fork over 8 and is regular and orthogonal to 4- if v # ( ), and p = v -(a) 4 I . Then we can deJine ap = F, and N p such that (N,,, a,, : g E I U @}) is a decomposition inside M.

(2) If tp(F,U,,,N,) is not necessarily regular we can still deJine ap # u,, I N, 9 3. Proof. We define by induction on i < A+ = h(T, E *)+ a set C,,N, U FC C, C M, tp,(C,,N, U @ is almost orthogonal to N,, C, is increasing

CH. XI, $23 THE AXIOMATIC TREATMENT 57 1

continuous, and for each i , l [C, E * C,,,] if possible and otherwise there is PET(C,) realized in C,+, but not in C, if possible. By the definition of h(T, E *) for some i , every P E T ( G ) realized in C = u {C,: i < h(T, E *)+} is realized in C, and C, E * C. Clearly, by Ax(A2) C, E * C,+,, hence

(*) if C, E B E M , IBI < h(T, E *), tp,(B,N, U onal to y, then C: c *B.

is almost orthog-

Also there is no p€T(C, ) not realized in C,, but realized by some EM such that tp,(Z U C,,N, U C) is almost orthogonal to 8.

Note thatN,E*C, if v = ( ) asN,E*M, and if Y # ( ) by (i) of Def. 2.5( 1).

We want to let &-(,)I = C,,:ly,> = F, but why is C, T-saturated? Ax(D2-) is tailor-made for this (with N,, C,,M, Nlro, NVrl, . . . ,y here standing for M, A, N , q , M,, . . . ,M, there).

(2) Proved as above. However, tp(c,N,) is not necessarily regular. But by Ax(C4) there is c ’ E N , ~ , > , F ’ $ N , , such that tp(t?,N,) is regular. Now tp(F’,U,,,N,) does not fork over N,, hence F’ satisfies the assumptions on F and we can repeat the proof with it.

Proof of 2.8. Easy.

CONCLUSION 2.15: [Ax(Al), (A2), (A3), (B2), (B3), (B4+), (B5), (Be), (B7L P1-L (C2), (C4), (D2-1, (El), (E2) anti: (A4) or (A6), (Cl)]. Suppose T does not have the dop. If (Nl,@,,:q~I) is a ( T , E *)- decomposition of M , then N , E * M , and tp,(M, U,,,,N,) is almost orthogonal to N , for each V E I .

Proof. By 2.11 and 2.8.

THEOREM 2.16: [Ax(Al), (A2), (A3), Ax(4;(,, E.)), (B2), (B3), (B4+), (B5), (B6), (B7), (C1-1, (CB), (C4), (D2-7, (El ) , (E2) and: (A4), or (A6), ((31); we can omit (C1-) if we add (El+), (E2+)]. Suppose M E*N,M,N are T-saturated, EN, Dp(a,M) = 1, and tp,(N,MU a) is almost orthogonal to M . Then we can Jind a and N4,qi,, ai for i < a such that :

(a) N+ E *N, tp(N4,M) does not fork over N+ n M , N+ n M E *M, and

(b) N4 U at E Yo, tp(ai,N4), is regular, (c) {a{t,: i < a} is independent over (M Uq,NJ,

h€N4,


(d) tp,(q*,, N$ U a(*)) is almost orthogonal to N$, hence tp*(q,,,N, U

(e) N is T-prime and T-atomic over M U u,qi,,

(g) If{*), N$ are T-saturated.

qi)) I- tP*(qf),M ulv, u (f) lIqi)lL IIlv,II are G U T , E*).

Proof. First define N+ to satisfy (a) and (f) (see Def. 2.3(4)), then a,(i < a) [just as a maximal family satisfying (c) and EN] then qi) [as a maximal subset of N satisfying (d)]. Now -Nrt) is T-saturated by Ax(D2)-. We define by induction on i < aM, T-prime T-atomic over M u u, < iN{,) increasing continuous in i.

THEOREM 2.17 : [Ax(Al), (A2), (A3), (B2), (B3), (B4+), (B5), (B6), (B7), (Cl-), (C2), (C4), (D2--), (El ) , (E2) and: (A4) or (A6), (Cl)]. Suppose T does not have the dop and is superstable. Then every T- saturated model has a (T, E *)-decomposition (Nq, a,, : q E I) such that M is T-prime T-minimal T-atomic over U q E I q , and N, G *M, tp,(M, uqEIN,) is almost orthogonal to each 4, for V E I .

Remark. What is the difference between 2.8 and 2.17 ? The first has an extra hypothesis, Ax(D2-), and stronger conclusion : every (T, c *)- decomposition will work, not just some.

Prooj. Like the proof in X, Section 3 but using the claims above and their proofs. DEFINITION 2.18: I f N E *M, then we define an equivalence relation E = E(N,,,,) ; its domain is { F E M : tp(c,,N) is regular and trivial}, aEb iff there is an automorphism F of M over N such that {F(a) , 6} is not independent over N (equivalently: for some & E M , {a’, b} is not independent over N and there are (No, a*), (Nl, a’) E CON(N,M) (see Def. XIII, 3.1), No& isomorphic over N.

XI.3. Specifying the axiomatic treatment

DEFINITION 3.1 : (1) Let A c : B means that A G B, and for every set of <p formulas in <p variables, all whose parameters are from A, if p is realized in B, then p is realized in A.

(2) Let A E : B means that A E B and for every a,EA (i < a < p) and ~ , E B (j < /3 < p ) there are b;EA (j < /3 < p ) such that

tp,((b,:j < /3),{ac:i < a}) = tp,((b;:j < /3),{at:i < a}).

CH. XI, $31 SPECIFYING THE AXIOMATIC TREATMENT 573

(3) Let A =,”B means that A E B and for every a,EA (i < a ,{a,:i < a}) = stp,((b;:j < /3),{a,:i < a}).

(4) If p = No we omit it, but we may write E, instead of s2. (5 ) We say (A ,B) satisfies the Tarski-Vaught condition if

A E ~ A u B.

Observation 3 . U .- (5) For x = t , s, a, if tp(@,A) is Ft-isolated A E t B , then tp(a,B) is F:-isolated, moreover tp(@, A ) I- tp(a, B).

Proof. Easy.

DEFINITION 3.2: Ti is the set of F,”-types (see Def. IV, 2.1) for x E {t , s , a}.

CLAIM 3.3: (1) A is TL-saturated iff A = m, N is p-compact. (2) For x = s , t , a , A is Ti-saturated iff A = m, N is Fi-compact (see

Def. IV, 2.1(3)) ; remember F;-compactness is the usual saturation, which, for p > No, is equivalent to F,”-compactness and to F;-saturation and that FL-saturation may be a very weak demand if there are few Fi-isolated complete types.

(3) A is Ti-saturated iff A S i B for every B extending A (for x = t , s, a ) . Similarly for Ti,,, ‘;I*. ,. (See Dejhition 3.4 below.)

(4) A is Tk-saturated if A = m, N model.

DEFINITION 3.4: (1) For x = s, t , a let A C;,,B means that for ever reduct 6- of 6 with < K relations and functions, in 6- A G i B .

(2) For x = s , t ,a let Ti,, means the set of types p which are in Ti for some reduct 6- of 6 with < K relations and functions.

Remark. Note that we shall use really only E ~ ~ , ~ , , the other notions being defined for aesthetic reasons.

DEFINITION 3.5: (1) Let A E ~ B (for A 2 2) means that A E B and for every a < A , sequences ~ E A , ~ E B (of length <A+No, so possibly infinite) and formulas c p { @ , ~ ) , yFL-,(r,,y,3T) for i < a (where 1Cy) = l ( 6 ) < A + N , , l ( f ) =l ( iZ) <A+N,anda<A,butcp,,yF,arefirstorder) there is ~ E A such that

(a) I=cpt[6,~] implies I=cpJb’,~r],


(b) R[@-,(zi,6,a),L, coo3 = 6 2 0 impliesR[@-,(~,b',a),L, co] = 6 2 0. (2) Let A G ~ B (for A 2 2) means that A c B and for a < A and

for every ~ E A and ~ E B , of length <(A+Ko),cp,@,i),@,(Z-,,9,i) (for i < a) there is 6 E A such that

(a) I= p1[6, ir] implies I= cp,[6, a], (b) if @,[%, 6, a] forks over A, then @&, 6, ir) fork over a. (3) Let A c i f B means A G ~ B and A s{B. If we omit A we mean

A = 2, and write E,, G~ instead c@, ~ f .

FACT 3.6: (1) We could waive (a) of 3.5(1) (hence the c p - , ' ~ ) . ( 2 ) If x ~ { f , e } and A G ~ C , ~ ~ E A , ~ E C , and tp(e,C) does not fork

over A and cp&, i, u), +,(q,ji, %) are formulae (i < a < A ) , then there is 6 E A satisfying

(a) if l=cp-,[6, a, c], then !==cp,[6, a, (b) if x = f, then R[@&., 6 , a ) , L , co] = R[@,(Z,, b, a) ,L , co] for

(c) suppose No+A 2 K,(T). If x = e, i < a and @, (%,., 6,ir) fork over

(3) Suppose A+No >K,(T). If A G:B ( A 2 2 ) A C C , {B,c)

for i < 01,

i < a,

a, then @&, 6, a) fork over a.

independent over A , then C G i B U C .

Proof. (1) Replace @&,9,5) by @-,(z,,y,S) A cp-,@,E). So if [6, a] A cp,[6, a], then R[@,(Z,, 6 , a),L, co] = - 1, contradicting 3.5( l ) (b) .

(2) Let cpf = p f @ , q ) be such that q is a finite sequence from A,i=rp:[b,~~]andforeveryb'~A,Ccp,(6,~,~)iff kcpf(6,q) (seeIII, 0.1 ; note: A E,B). Let d be a sequence of members of A of length < A + KO such that ir G d ,q E d for i < A. So there is rp: = q$@, d ) = cp:@, C-,). If x = f apply 3.5(1) to d , 6,q$, @-,(i < a) and get the desired conclusion. If x = e , as A K,(T) [w.l.o.g. tp(c, A ) does not fork over d ] , and apply 3.5(2) to d , 6,cp:, @-,(i < a).

(3) So suppose a < A, ~ E B , ~ E A , CEC are sequences of length <A+No, cpf@,i,u), @-,(i < a) are formulas, and it suffices to find 6 E A such that

(a) Ccp,,[6, a, C] implies t=cpt[6, ir, E] , (b) if @-,(Zi,6,a,c) forks over C , then @&6,a,c) forks over

au c. W.1.o.g. for each i, @&, 6, a, c) forks over C (otherwise replace it

by zo # zo) and trp,[6, a, c], and tp(6 nc, A) does not fork over a. Now


(*I {$@,,6,@,Q)} U {8(u,6,n’):tZ’~A, t=8[i?,6,a’]}forkoverA

Otherwise anE1 realizing it where a,c1 correspond to %,is such that tp (a^P,A U 6) does not fork over A. But clearly (by 1.2 as A E,B and tp,(B,A U C) does not fork over A), tp(i?,A U 6) = tp(P,AU6), hence w.1.o.g. P = i?; so C$-,[a,6,is,i?] but tp(a,A U 6~ c3 does not fork over A U i? (by 111,0.1(3) as tp(ani?, A U 6) does not fork over A), a contradiction.

So there is 8*(iZ, 6, at) E tp(E, A U 6) such that $&, 6, a, is) A 8,(@, 6, ii;) forks over A. W.1.o.g. ii; c ti and even 6, is $,(is, 6, a).

Now we apply 3.6(2) for x = e to A G:B, a,a,6, i? and the formulas pi@, 5, a) = pi@, 5, Q) A 8 , (U ,Y , if),

(zi his the variables).

$i(Ztn@,Y,5) = $Jzi,y,z,@) A ai(@,Y,z) (with Ztn@ standing for qnis, the infiniteness of the sequence does not bother us by the finite character of forking).

So we get P E A such that (a)’ k=p,[6, a;i?] A 8 , [ ~ , 6‘, a], (b)’ $-,(a, 6, 8, is) A 8JU, 6‘, u ) forks over a.

Now (a) follows from (a)’. Why does $&, 6 ,a , i?) fork over a U i?? Otherwise for some E,

t=$,[E, @,a, i?], and tp(a ,6 U i E U c3 does not fork over i r u E, As tp(i?,6‘ U a) C tp(i?, A ) does not fork over 8, by 111,O.l tp(f i? , ii U r) does not fork over a. However, $&, 6, a, is) A Qr(is, ti, a) forks over

(by (b)’) and t= $@, 0, a, E] (by the choice of E ) and C 8,[c, 6 , a] (by (a)’) hence

+&, 6, a, i;i) A #(is, 6, a)€ tp(Bni?, @ 6‘).

(When a,is correuponds to ~ , i ? . ) So tp(ani?,un6‘) forks over a, a contradiction.

CLAIM 3.7: (1) A c:B implies A r j ; B and A C:,KB. (2) A c j ;B implies A c f B implying A c ~ B for x = e , f. ( 3 ) A Gf implies A G ~ , ~ B .

Remark. Also, there are obvious monotonicity properties for the cardinals.

CLAIM 3.8: (1) The axioms (A1)-(A3) are satis$ed for E* being c:, ~ f , ~ (x = t ,s ,a, and A,K cardinals) and Ci, Gi.


(2) When A = No also Ax(A4), (A5) are satisfied and when p 2 A,

(3) If E * is sg0, =i0,., (z = t , a ) or E*, also Ax(A6) is satisfied. (4) If T is T,”,T,”,.(x = t , s ,a ) , then Ax(Bl), (B2) are satisfied. (5) If T is Ti or Ti,<, or T: ( A 2 K,(T)) or T,”,. (where A 2 K,(T) or

A 2 K + N, or A = K = No and every reduct 6- of 6 with Jinitely many relations and functions is superstable), then Ax(B3) holds.

(6) Ax(B4+) holds for T being Ti,T; when A is regular or cf

(7 ) Ax(B5), (B6) hold for T being T,”, T,”,.(x = t , s, a) . (8) If T is superstable, T is T,” (x = t ,a or x = a, A > No) then

Ax(4;,) holds.

2 K ( T ) .

Ax(B7) holds.

Proof. Easy, e.g. for (3), C, use 3.6(3).

CLAIM 3.9: (1) suppose K,(T) < A = cf A or K = A = cf A > No or K = A = ~ , ( T ) a n d ( T , E * ) i s o n e o f (T,”,E,”), (T,”,,,E,”,~), f o r x = t , s , a . Then Ax(Cl+) holds.

(2) The pair (Tko, G,) satisJies Ax(Cl--) for T superstable.

Proof. (1 ) Immediate by 3.3(3). (2) By 3.6(3) N, E *Nl U A, and we know C C acl(N, U A). As E,

satisfies Ax(A2) (see 3.8(1)) and Ax(B2) is satisfied it suffices to prove :

FACT 3.9A : I f N 4 6 , N E ,B, then N E , acl B.

Proof of the Fact. Suppose a ~ N , b ~ a c l B , and w.1.o.g. l=p[6,a], and $(z, 6, a) fork over a. We know that tp(6,B) is isolated, so there is T E B and formula 6 such that I= 6[6, c’], 8 ( g , q I- tp(6,N) and S(g , C) is algebraic. So let {bo, . . . , 6k-l} be the set of sequences realizing S(g, C) (note k > 0 ) w.1.o.g. 6 = KO. So for each 1, @(Z, cl, a) forks over a, hence vl<k@(~,61,@) forks over a. But this formula is equivalent to (38) [$(z, 8, a) A 6(g, E ) ] , hence this formula too forks over A. Apply the definition of N E ,A , with

fi,C,Z,a, (3!’g)6(g,a) A (Vg)[6(g,a)+p(g,Z)],

t ( Z , 4, Z) 2‘ (3g) [@(Z, 8, Z) A 6(g, a)] standing for a, 6 , ~ , g,p(g, z), $(z, g , ~ ) .


So for some E’EN, k(3!*g) S(g , C’) A (Vg) [a@, a’) +rp(g, a)] and @’(z, F’, a) fork over a. So { g : 8 ( g , ~ ’ ) } has exactly k members; call them &, . . . , bL-,. As N < 6, clearly &, . . . , K-l EN. Clearly, @’(z, F’, a) is equivalent to vz<c @(z, b;, a) but the former forks over a, hence for some 1, @@, b;, a) forks over a. Lastly, as I=(@) [S(g, F’) +rp(g, a)] (by the choice of F’), clearly l=rp[b;,@]. So b; is as required in Definition 3.5(2), so we finish the proof of the fact.

-

CLAIM 3.10: Suppose T is superstable, and (T , E*) is one of (T,”, G;), G,”) (z = t , s , a ; y = s ,a) (Ti, r’,), (Ti, ri). Then Ax(C3), (C3*) hold, hence Ax(C4) (by 2.7(4)).

Proof. By the implications among the various (T, G*) it suffices to deal with (Ti =,),LTio, re). Let z be f or e, respectively. Among the sequences&N:b#A}, choose one with minimal R[tp(6,M),L, co] and let

EM, S(z, 8) E tp(b,M) be such that u zf R[8(z , d ) ,L , co] = R[tp(6,M), L , 001.

Among the EN: 6#A, C 8[6, 8]} choose one with minimal R[tp(6, A ) , L , coo3 and let a ~ A , @ ( ~ , a ) ~ t p ( 6 , A ) be such that /3gfR[@(z,a),

W.1.o.g. tp(6,A), tp(a”6,M) does not fork over a, 8, respectively, L, 003 = R[tp(b,A),L, 001.

and @(z, a) I- S(Z, 8) and EE c a.

FACT 3.10A: 01 = /?.

Suppose not. So 01 > /?, and {@(z,a)} U tp(6,J) forks over 8, hence w.1.o.g. @@,a) forks over 8, and so it forks over M.

As tp(6,A) is not almost orthogonal to M , w.1.o.g. tp(6,M U a) is not almost orthogonal t oM; hence, we can apply Fact 1.7 so there are cp,aZ,Kz ( I = 0 , l ) and F E M as required there.

So clearly (6, witnesses this)

k(3g) [(p(ao, 60, a,, Y, q A @@, a,)].

W.1.o.g. tp(@o^60 aI”bl,A) does not fork over M , hence

tp(a,M u a, To) = tp(a,,M u a0 ”6,). so

[rp(@,, a,, 6, !A F ) A @(!I, a)] A (vz3 (@@, a) + 8)).


By 3.6(2) there is @ E M such that: (a) b(3y3 [rp(Eo, b,, a*, 9, @ A @@,a*)] and @(!A a*) F S(g, a), (b) i fx=f , thenR[@(g,&*),L,m] =R[@(g,&) ,L ,a] ,andi fx= e,

But if M $,A, hence M s ,A, then as @(g, a*) forks over d and then @@,a*) forks over d.

@(s, a*) I- 8(g, d) , clearly

R[@(Q, a*),L, 00) < R[8(8, d) ,L , 001 = a.

Also, if M s , A , then

R [ @ ( Y , W , L , 001 = R[(@W,@,L, 001 = B < a. So in any case

(b)’ R[@(Q,&*),L, 001 -c a. By (a), as tp(a-6,M) = tp(@, ”6, ,M) , clearly

b(3Y) [P(G 6, @, 8, F ) A @@,a*)]. Choose b* E N such that

k q[&, 6, &*, 6*, C] A @[6*, &*I. Now as k $[6*, a*]

R[tp(b*,M),L, 001 < B[@(g, @) ,L 001 -c a.

On the other hand, as I=p[a,6,8+,6*,~] and t p ( a ” 6 V , @ ) = tp(@, A 6on C, $) and the choice of rp, necessarily tp(6, CU iz U &* U 6*) forks over CU&. But as tp(6,A) does not fork over a, this implies tp(6, A U b*) forks over A, hence 6*#A. So 6* contradicts the choice of a.

So we have finished the proof of a = /3. We can conclude that tp(6,M) does not fork over M, so w.1.o.g. &EM.

This proves Ax(C3). Now we continue for Ax(C3f) by 3.10A w.i.o.g. + = e, = d.

FACT 3.10B: tp(6,M) is regular.

Suppose not, then apply Fact 1.8 to p = tp(b,M), @(z, a) ~ p , and get C,CO,Cl satisfying 1.8(i), (ii), (iii). By the symmetry of non-forking (111, 0.1) tp(cl,M u c0) forks over M.

So there are formulas p,(Z = 1,2) such that (after possibly increasing but still zt = J E M ) :

(a) ~ ~ 1 [ ~ , ~ 4 , ~ 0 , @ 1 ,

CH. XI, $31 SPECIFYINQ THE AXIOMATIC TREATMENT 579

(b) +v,[q, 4,a1, (c) v l ( ~ , ~ i , ~ ; , a ) forks over (4 i fx =f, R[v,(Z,ql,a),L, 031 <R[$( z ,a ) ,L , a ] , (e) if z = e , v,(Z, g, a) t- $(Z, a) A $(g, a), and p2(Z, 4, a) forks

We can assume w.1.o.g. F = 6. Clearly (c1 is a witness)

+w [Pl(F, 8, coy a) A Q)*(g, GJ, 4 $(g,a)l.

for any i ? ; ,F; ,

over a.

Remember that Fo realizes p , hence M G Z M U C0 (we assume M G z N as we are proving Ax(C3*)). As {c,c0} is independent over M , by 3.6(2) there is ELEM such that

C(38) [q1(c, g, c;, a) A vz(g, c;, a) A $(g, a)],

I= w;, al, and R[q2(g, F;, a),L, co] < a (a proof by the two cases, z = f, z = e , as in the previous Fact). Then we an find C;EN such that

t=pl[F, F;, F;, a] A y,[F;, F;, a] A $[I?;, a].

By (c) above, tp(C,MUF;) forks over a, hence over M, hence (remember 3.10A) tp(F,A U C;) forks over A, so C; # A . On the other hand, as F = 6: R[tp(F;,A),L, a] < R[tp(F;,aU F;),L, co] < R[v,(g,a;, iq,L, coo3 < a.

So i?; contradicts the definition of a. We have finished the proof of 3.10B too, and hence of 3.10.

CLAIM 3.11 : Suppose T is superstable. If (T, G *) is one of (TZ, G:), (Tf,K, GI) (z = t , s , a ; y = s ,a) or (Ti, G ; ) , (Ti, G;), then Ax(C2) holds; moreover we can get tp(C,N') regular.

Remark. For a simpler proof for (ti,, G ; , , ~ , ) , see [Sh 861.

Proof. The beginning is like that of 3.10. Again it is enough to deal with (Ti,, C ~ ) and (T:,, c,). Let z be e or f, respectively. We let M , P , N be as in Ax(C2), and 6,8(%, d) , a, a, $(a, a), B be chosen as in the proof of 3.10, restricting ourselves to 6's such that tp(6,W) is not orthogonal to M (with N' here standing for A there).

FACT 3.11A: 01 = p.


We can find an F&-saturated M*, for I = 0, 1, C1 EM* and b,, such that tp(alAbl,M) = tp(a-6,M),N1 E M*, {ao, al} is independent over M , tp(&,M*) does not fork over al, and {bo ,b l } is independent over M*. By V, 3.9 (and see Def. V, 3.2) there are Eo7 . . . , E,-l such that tp(Em,M*) is regular, {cm:m < k} is independent over M*, tp(6(), M* U um<k~m) and tp(Um,,Em,M* U KO) are F;-isolated, hence k =

Suppose tp(Em,M*) is orthogonal to M iff m 2 k(0). Now k(0) > 0 as tp(6, a) is not orthogonal to M , hence tp(&, at) is not orthogonal to M . Choose E, a0 c E c M*, such that:

e,-,,M*) does not fork over E, and its restriction to E is stationary,

- -

WO).

(A) tp(60nEonEl

(B) stp(Eon ...n~,,-1,~u60)o)stp(Eo . . .CEk-1 ,M*U60) ,

Now choose iP(n < w ) such that: (D) {En : n < w } C M* is an indiscernible set over M U a. U al based

n

(c) stp(60, E U U m < , ~ m ) k Stp(bO,M* u U m < k c m ) *

on ao, each E" realizing stp(c,ao). Then choose E:, . . . , E Z - , , such that [letting cn, = E:?. . --n e,,,,-,,

A -11 - - en, 2 = ~ C O ) * * * ek-1, en, 0-z En, 1" En, 21 :

(E) stp(Efl, A cn, a0 U b,) = stp(Eo rr . . . E , - ~ E, a. U a,), (F) tp(En,,,M* U 6, U UmZn Em,()) does not fork over E" U 6(). So { E ~ , , ^ P : n < w } is an indiscernible set over a. U go based on

a. z-a0 U KO. As tp(6, a) is not orthogonal to M, by V, 1.1, the types stp(b,, go), stp(61, a,) are not orthogonal, hence tp(6(),M*), tp(6,,M*) are not orthogonal, hence (V, 1.2(3)) not weakly orthogonal. So for some 62 realizing tp(&,M*), tp(b,,M* U 6,) forks overM*, hence over ao. So by 1.4 for some FEM*, and p - -

(a) k 9 ) [ b 0 , b2 , c, a,, a()] and w*l*O'g' p(g07 Y Z , ' 9 '19 xO) @(YO, ' 0 ) A @(&, '11,

(b) for any &, F', a:, {p(~, &, i?, a;, a())} U tp(6(), 4) forks over ao, and ah, if tp(&,, 4) = tp(ao, #), then {p@, b;, moreover for any 6, F,

F', a;, a;)} U {#(a, ah) : C W(6, a)} forks over a; ; Now we shall prove (0) if & realizes tp(6,,M*) and l=p[b0, &, F, a,, aO] and n < w , then

tp(Efl,,,M* U &) forks over e". If not, we shall prove by induction on m, k(0) < m < L, that

tp(qn.. .FE&-,,M* U &) does not fork over En. For m = k(0) we have just assumed this.

Assume we have proved for m < k and we shall prove for m + 1. By

CH. XI, $31 SPECIFYINU THE AXIOMATIC TREATMENT 581

(B) and (E) tp(a,"^. . .^c;-,,M* U 6,) does not fork over 8" U go, 8" E M* and

n n- c-- stp(a,"" . . . - a;-,- an, a. u 6,) = stp(ao . * . ek-1 e , a, u 60),

tp(Ft^ ... -C,"-,,M*) = F[tp(Con .. . e,-,,M*)l.

hence clearly for some automorphism F of M* over M U a, n-

Hence (as {E,, . , . , gk-,} independent over N*) tp(a;,M* U u,,, 8;) does not fork over M* (see (A)), hence over B", so it is parallel to tp(8;, a"), hence (as 8" a;, a-8, realize the same type over M U a,) it is orthogonal to M (but not to MU a,), hence (by (D) and X, Section 1) is orthogonal to MU a,, hence to stp(6,,a1), hence to stp(&,M* U U l < , a:) (by the induction hypothesis). So tp(a;,M* U & U u,,, a:) does not fork overM*. For m = k we get a contradiction by V, 3.2.

Next we shall prove (d) if & realizes stp(&, a. U a, U @ and I=cp[6,, &, F, a,, a,}, then for

every n large enough tp(cn,,,a0 U a, U F U & U a") forks over an. Suppose not. We know by III,4.19(2), III,4.21(11), III,2.10 that

for every n large enough, stp(&, a, U a, U F U 8" U an,,) = stp(&, a, U a, U F U P U En,l). Choose such n for which the conclusion of (d) fails.

6,,M*) does not fork over an. So we can find & such that stp(an,,n60n&,ao u a, u F U a") = stp(an,,^6,-&,a0 u a, U FU P), stp(cfl,,^bon&,M*) does not fork over a, U a, U F U 8".

So tp(&,M*) does not fork over a, U a, U F U a", and also tp(&, a0 U a, U aU a") does not fork over @, E a, U a, U F U En and stp(&, a, U a, u FU 8") stp(&, a, u 8, u F U 8"). We use the choice of and of n, respectively. We can conclude that tp(&, M*) = tp(&,M*) and also it is clear that !=cp[b,,b~,~,~,,a,]. So the hypothesis of (c) holds, hence its conclusion, i.e. tp(afl.l,M* U &) forks over M*.

But we have assumed (that for n the conclusion of (d) fails), i.e. tp(En,,, a, U a, U F U U 9") does not fork over a", hence (again by the choice of & first demand) tp(afl, ,, a. U a, U F U K U a") does not fork over en. As tp(e,,,"b,"&,M*) does not fork over aoUal U F U P , clearly tp(c,,,,M* U 6;) does not fork over a, U 8, U F U 9" U K ; so together with the previous sentence tp(cn,,,M* u K ) does not fork over a. U a, U FU a", but a, U a, U FU 8" C M*, so this contradicts the

Note that tp(g", ,

stp(&, 5, u a, u F U a") - -


previous paragraph; thus we have proved (d). Now working in CeQ :

E acl(a, U al u ,?), such that

(e) There are formulas 7 1 , ~ ~ and C ~ , C C

(el) 71[60, 6,,!o,a1, El], (e2) if I= ~,[6,, bi , a,, 4, El] then for every n large enough

!= 7, [a .1 , a,, a1, q, 6;1, PI, (e3) for any ah, a;, ~; , -6 ;1 and E’, such that tp(a’, q5) = tp(8, q5) =

tp (P , q5), ( ~ ~ ( 3 , ah, a;, F;, b;l, 8’)) U {7(z, a’) : I= 7 [ ~ ~ , ~ , 8’7) fork over 8’. This is easy by 1.3, 111, 2.5 and compactness (relying on (d), of

course), and w.1.o.g. c1 = r a n d we return to a. ( f ) W.1.o.g. Go = a, b, = 6, P E M*, tp,(M*,N) does not fork over

P and tp(&,,P) does not fork over M.

-

As {en, cn : n < w } is an indiscernible set based on a, A 6. By 111, 2.5 T~ is a first-order formula which is almost over aOn6,, hence 73 = T ~ ( z ~ , Z, g,@) for some 6, c 6; c N, and choose at, Go C a? E Nf, t p ( @ , P ) does not fork over at, and w.1.o.g. at C 6:. Clearly by (el) and (e2) above I= 7Ja1, C, a,, 63 and (by (e2)); for any 6, if I= 7,[6,, 5, a,, al, c], then I= ~ ~ [ a ~ , C, 6;, 6*]. Let

so

6, =

74(z1, 5, 6t, G ) = (vg2) [71(603 g2, 51, @ +73(z1,5, g2,6:)]*

(e4) I= 74[a1, C, 6:, GI. Let FI(YO> z>zi* a01 = ( ~ Y Z ) [P(goo, Y2, 5, z1,ao) A 71(goo, g2, Z1723I

Clearly tp(6:,M*) does not fork over a:, hence there is a formula q2(z, z1) with parameters from P such that :

(*) for every c‘, a; EM* kq,[~’ ,a; ] iff kq1,[6,,~’,n;,a,] A 7,[a;,af,6t,$1.

So clearly !=q2[C, all [by (e4) I= ~ ~ ( a ~ , C, 6:, a:) and using &’for g,, by (a) and (el) I = q l [ b o , ~ , ~ l , ~ o ] , use (*) above] and M c , M U al [as M G ~ P , ~ E P , tp(a,M) = tp(al,M)] and tp(a1 ,P) does not fork over M [see ( f ) ] , so we can apply 3.6(2) and get @ ; E M such that

(i) (3qqz(5,a3 (remember vz is over P), (ii) R[$(g,a;) ,L, 001 < a (we repeat the argument in 3.10A’s

proof).


By (i), for some F’EW, (iii) C ~ , [ F ’ , a;].

(iv) l = p J 6 0 , t f , ~ ~ , ~ o ] and t = ~ ~ [ ~ 1 ; , ~ ’ , b ; l l , a 3 .

(v) t ~ [ 6 ~ , &, F‘, a;, aO] and l= ~,[6,, &, ao, a;, a’].

(vi) I= T ~ [ & ; , F’, b;, @I.

(vii) (3%) 72[Cn,n,l, ao, a;, F’, &, a”].

(viii) (3”n) s.t. tp(Cn,,,M* U 6) forks over M*. But this means tp(Cn, ,,&I*), tp(&,M*) are not orthogonal, but

the latter is parallel to tp(&,W) and the former is orthogonal to any regular type orthogonal to M. We conclude that

(ix) tp(&,W) is not orthogonal to M, hence b;$W. But by (v) and the choice of p (see (a)), I= $[&,a;], so by (ii),

Hence,

By pl’s definition for some & E N ,

By (iv) and 74’s definition, and (v),

By T ~ ’ S definition,

BY (e3),

R[tp(&,M),L, 001 G W@,a’ , ) ,L , 001 < a, and by (ix) &$M and of course & E N . So & contradicts the choice of a. So we prove a = B (i.e. Fact 3.11A), hence Ax(C2).

We can continue:

FACT 3.1 1B : tp(6,M) is regular.

This is just like 3.10B [i.e. in the end, we assume F = 6, get F ; E N , F;$M, such that R[tp(F;,M),L, 001 < a, tp(6,W U F ; ) fork over M. So tp(F’,,W) is not orthogonal to tp(6,W), hence to tp(b,M), hence is not orthogonal to M. So F; contradicts the definition of a].

CLAIM 3.12: Suppose T is superstable and D(T) is scattered. Then (Tho, she) satisjies

(1) Ax(C2+), (C3), (C3+). (2) Ax(D2).

Proof. (1) Like that of 3.10, 3.11; as the changes are similar, we concentrate on Ax(C3+). For very finiteB and m, Sm(B) is a scattered topological space, hence the Cantor-Bendixon rank CB is defined. Among all pairs

{ (6, d) : 6~ N , 6$ A , d EM)


we choose one with minimal pair

(a,, a,) = (R[tP6, w, 001, CB(fp(6, J ) )

8(z, d) = tp(6, d), R[tP(6,J),L, a1 = R [ W , W , 001,

(by the lexicographic order). Then choose 8 ( ~ , 2) which witness it, SO

r # tp(6,J),8(z,d)ErESm(d)*CB(r) < CB(tp(6,J)).

tp(6,M U a) not almost orthogonal to M ) one with minimal We similarly choose among { (6, a) : EN, 64 A , @E A , I= 8[6, d] 2 E a,

(/f,,/fl) = (R[tp(6, a),L, a], CB(tp(6, a)) and let @@,a) witness it.

FACT 3.12A: a, =Po.

This is proved like 3.10A, the only difference is the way we replace the use of (b) and (b)’ in showing that @(Z, P), 6* contradicts the choice of (ao,al). So in choosing P we replace (b) by

(b)’ R [ W , @(%V), No1 s R[@(% a), @@, g ) , &I. This is easy to do. W.1.o.g. (*) R(@@ a), @, No) -c R(tp(6, d), @, No)

(by 111, 4.1, 11, 2.1). (If n = R[@(Z,a), @(Z,g), KO], then by Exercise 11, 3.7 (p. 55) for some k < w , n = R[@(Z,a), @(z,Y), k]. By 11, 2.9(2) for some formula a,, for every 8, n = R[@(z, E) , @(z, q), k] iff I= al[~]. So for (b)’ to hold C a1[6*] is enough, as always R[@(z, a*), @@, g ) , k] 2 R[@(z , a*), @(Z, g ) , No]. We choose, as there, 6*, clearly 6* E N , 6*CA; now by the choice of (ao,al), O(z,d), necessarily tp(5*,J) = tp(b, d), tp(b*,M), tp(6,M) does not fork over M, hence R(tp(6*,M), @,KO) = R(tp(6,M), @, No) (see 111, 4.1), hence R(tp(6,M), @, No), WP(~*,*,), @, KO) < R(@(z, 6*), @, KO) < R(@(E, 6*), @, No), contradicting (*).)

FACT 3.12B: tp(6,M) is regular.

A similar change made the proof of 3.10B work.

Proof of 3.12(2). Combine the proof of 3.14 below and the above.

Remark. Really we use:

CH. XI, $31 SPECIFYINQ THE AXIOMATIC TREATMENT 585

FACT 3.12C: Let T be superstable, D(T) scattered.

Suppose 8 is an ideal of m-formulas over A . Then for every m- formula p ( ~ , a) # 8 over A there is an m-formula +(z, 6) over A such that :

(a) +(% 5) I- v@, 61, ( c ) m, 5) # 8, (b) ~ E A ,

(d) +(~,6) has no extension in Sm(A) disjoint to 8 which forks

(e) +(Z,6) has no two contradictory extensions over 6 which are over 6,

not in 8.

Proof. Clear

CLAIM 3.13 : Suppose T is T:, K 2 K(T) , or Ti, T totally transcendental, then Ax(D1) holds. If in addition (T, C *) satisBes Ax(B3-), (B5), then Ax(D2), (D2-), (D2--) hold.

Proof. Immediate (see IV, 2.15-2.18 for those and other such theorems). As for Ax(D2), see 2.7( lo), and as for Ax(D2-), Ax(D2--), see 2.7(5).

CLAIM 3.14 : suppose (T, C *) is ( A 2 K,(T), K

E ;) A 2 K, (T) or (Ti.+,), c ;, K ) .

K,(T) or (Tho, Ee), K(T) = 8,. Then Ax(D2-) holds.

Proof. Let B = Domp and (as T = T:r(T)) w.1.o.g. p has no extension over A which forks over B. Let FEN realize p; so assume tp(r,A) is not almost orthogonal to M , and we shall eventually get a contradiction, thus finishing.

Let 1 < n be minimal such that tp(c,A) is not almost orthogonal to 4.

We can conclude thatlll, E *A U C. [If 2 = 0, by a hypothesis4 E * N , hence by Ax(A2) 4 G *A U c. If 1 = rn + 1, then tp(r,A) is almost orthogonal toM,,, by [choice of 11, and tp,(A,q) is almost orthogonal to M,,, [by an assumption in Ax(D2-)], so together tp,(A U ~ , q ) is almost orthogonal to K , By a hypothesis [of Ax(D2-)] the last phrase implies Ml c *A U C.]

As we can replace p by any p ' , p E p' E tp(c,A) of power <K,(T) , we can assume that tp(r,q U B) is not almost orthogonal to B nM,,


and tp,(B U F,&) does not fork over D z * B n&. By 2.7(12) there are F’,B,,i?, such t h a t 4 U El realizes stp,(B U F,D), tp,(B,,D UB U C) does not fork over D , c‘ realizes p but tp(F’,D U B U B, U c1) forks over B; and let p ( ~ , a) E tp(F’,D U B U B, U F,) fork over B. [Strictly speaking, in 2.7(12) B should be finite; this is anyhow the most important case, and we can imitate the proof generally.]

Case I. (T, E*) is (Tir(T), rlf,J.

Let 6’ be a reduct of 6 to the predicates and function symbols appearing in p, or p(2,d). This reduces Case I to Case 11.

Case 11. (T, E*) is (Tir(T), clf).

AsM, = * A U F,B U C E A U C, and I.R U < A, PI < < h (and the definition of ~ l f ) , clearly stp(B U c,D) is realized in &. As tp(B,&) does not fork over D, w.1.o.g. B, U t ~ , E &. So p U { p ( ~ , a)} is realized by F’, has power <lpl+ 1 < K,(T) and its parameters are from A E N ; hence it is-realized in N (N is T-saturated by the hypothesis of Ax(D2-)). So w.1.o.g. E’EN, F’ realizes p and tp(F’,A) forks over B, a contradiction.

Case 111. (T, E*) is (Tko, E , ) , K ( T ) = No.

So w.1.o.g. B = b,B, = bl ,D = d. As tp(c‘,Z U b U b, U F,) forks over 6, d E b clearly tp(c‘- 6, d U 6, U i9) forks over d. So for some p I= p[F’, 6, Fl, a,, d], p(Z,g, I?,, b,, d) forks over d. We can find d* E& and + such that for every a;,& from &,t(3~)p(z ,6 ,~’ , ,b; ,d) iff + +[I?;, b;, d*]. W.1.o.g. p ( ~ , 6 , ~ , Z, d) t- p ( ~ ) , so as tp(bln F,,& u 6) does not fork over&, clearly I= +[q, 61, d*]. Now& c e& u 6l U I?, (as 61ncl realizes tp(b^F,M), and 4 c e A U F , ~ E A), hence by the definition of te, there are ~:,c?E& such that I=+[i?:,6:,d*], and p(Z,g,~:,6:,d) forks over 8. By the former and the choice of +, k(32)p(z,b,~:,b:,d), hence for some F ” E N ~ = ( F ” , ~ , , ? , ~ : , ~ ) . This, together with “p(Z, 9, F;, 6;, d) forks over d” , implies tp(.YA 6, F: U 6: U d) E tp(F”^ 6 ,M, ) fork over d. But tp(6,q) does not fork over d, hence (by 111, 0.1) tp(c”,MU6) forks over d’u 6, hence tp(F”,A) forks over b. But by the choice ofp, F’‘ realizes p, so we get a contradiction to “ p has no extensions over A forking over B”.

CH. XI, $31 SPECIFYINU THE AXIOMATIC TREATMENT 587

CLAIM 3.15: Suppose T is TLo, T superstable with the (< CQ, 2)- existence property (see Def. XII, 4.2), or Ti (cf A 2 K,(T)) , or Ti, T totally transcendental. Then Ax( E 1 ), (E 1 +) hold.

Proof. T:,Ti follows by 2.7(6), 3.13, 3.8(6). Then case TL, is by Definition XII, 4.2.

CLAIM 3.16: (1) A(T:,., Cr,,) < x hold if: IAl < x implies IT:,.(A)I + lT;,.(A)l < x or T stable in x.

(2) h(TLo, cef) < ]TI if T is superstable; A(TLa, G e f ) < ITJ+; A(Tko, cf) < PI+, and Wk,, ze) < PI.

(3) The same results hold for A,.

Remark. (1 ) We can easily apply ( l ) , and also generalize sef to =if, etc.

(2) In part (1) of 3.16 we can use Ti,,,, T:,,*.

Proof. Now part (2) (and the corresponding half of part (3)) is trivial. Also, part (1) follows from (3). So let us prove (1) for A8. Let A,(i < x') be increasing continuous, lArl < x. Let S be the set of 6 < X+ such that :

(i) if T is stable in x , i < 6,peS"(A,) , p realized in U,A,, then p is realized in A,,

(ii) if i < 6, ITi,,(A,)I < x, then every peTT,,(A,) realized in U,A, is realized in A,,

(iii) similarly for T,",,,Tka (note that necessarily in all cases

(iv) if IT]'" < x, L, E L(T) , ILfI < K , 6 rLf is stable in x, then every ~ E S ~ , ( A , ) (i < 6) realized in U,A, is realized in A,.

Clearly S is a closed unbounded set. We have still two points to clarify why S works, i.e. why for every ~ E S of cofinality cf x (the problem for T:,. are parallel) :

IT1 < XI,

(A) If cf x < A there may be ~ET?,,(A,) not in ui<AT:,K(At). However, if cf x 2 K ( T ) we can repeat the argument in 111, 3.11.

If cf x < K ( T ) and cf x < K or even x'" > x, we can contradict IAl < x 3 IC,.(A)( < x] as in the proof of 111,3.6. So K < cf x < K(T) , 2'" = x and by the latter it is enough to deal with each 6 rL,, L, c L(T) , lLll < K , and if cf f 2 K(Th(6 rL1)) we can again finish and otherwise contradict [IAI < x 3 ITi,.(A)I < x].

(B) If x = a, for B c A,, stp(6,B) is not necessarily over A,.


However, for 6 ~ 8 , A, 5 tu,<x+ A, (by (iii)), hence for 6~ u, A,, EEFE"(A,) there is 6'~A,,bE6' [otherwise choose by induction on n, 6,, €A, , At<,, ~6~ E6,,], so for the types we are interested in this does not occur.

CONCLUSION 3.17: (1) If T is superstable, then (Tko, =go) and (Tho, ~ f , ~ , ~ , ) satisfy Ax(Al)-(A6), (Bl)-(B7), (B4+), (Cl+), (C2), (C3+), (C4), (D2-1, (E2), and A(Tk0, =go) < A(T),h(TL0, ~ f , ~ , ~ ~ ) < 1Tl+2xo.If we assume the ( < co, 2)-existence property, then Ax(El+) holds too.

(2) If T is superstable, D(T) scattered, then (Tf, , =Lo) satisfies Ax(Al)-(AG), (Bl)-(B7), (B4+) (Cl+), (C2), (C3+), (641, (D2), (E2) and h(Tk0, =i0) < ITl.

If we assume the ( < a, 2)-existence property, then Ax(El+) holds too and if T is totally transcendental, Ax(El+) and also Ax(D1) holds.

(3) If T is superstable, then ( T k , E,) satisfies Ax(Al)-(AG), (Bl)-(B7), (B4+) (C1-3, (C2), (C3), (C3+), ((341, (D2-), (W, and A(T:o, =,I < PI. If the hypothesis of 2.7(7) holds, the Ax(Cl-), (El) hold. (4) If T is superstable, then (Tio, zgo) satisfies Ax(A1)-(AG),

(B1)-(B7), (B4+), (C1+), (CZ), (C3+), P I ) , (D2), (El+), (E2+), and h(Tgo, rgo) = h(T) .

Proof. For Ax(Al), (A2), (A3) see 3.8(1), for Ax(A4), (A5) see 3.8(2), and for Ax(A6) see 3.8(3).

For Ax(Bl), (B2) see 3.8(4), and for Ax(B3), (B4), (B5) see 3.8(5), (6), (7), respectively.

For Ax(CI+) see 3.9(1) for (l), (2) and (4). For Ax(Cl--) see 3.9(2) for (3). For Ax(C2) see 3.11 for (1), (3), (4) and 3.12(2) for (2). For Ax(C3+) see 3.10 for (l), (3), (4) and 3.12(1) for (2). For Ax(C4) note that it follows from Ax(C3*) (by 2.7(4)). For Ax(D1) see IV, 2 for (4). For Ax(D2) see 3.12(2) for (2), and 2.7(6) for (4). For Ax(D2-) see 3.14 for (1) and (3), and see 2.7(5) for (2), (4). For Ax(El), (El+) see 3.15. For Ax(E2) see 2.7(9). For A(T, c*) see 3.16.

EXERCISE 3.1 .- Let T be countable No-stable. (A) If T has the dop or is deep, then I (h ,T) = 2A for h > No.


(B) If T is shallow of infinite depth without the dop, then for a > 0,

(C) If T is shallow without the dop and of finite depth, then either for every a 2 w , I ( N a , T) = 3Dp(T)(Ia(), or for every a 2 w , I ( N a , T ) =

aDp(T)(14n0) and for a < w , I(No, T) is < a D p ( T ) ( N O ) , or < a D p ( T ) V o ) , respectively.

[Hint for (C) : Work as in Chapter X using (Tio, E ,)-decomposition, 3.17(2) and the decomposition theorems of Section 2 ; there is no new point.]

EXERCISE 3.2: Prove a parallel theorem for T totally transcendental not necessarily countable; in (B), (C): for a large enough in (A) for A 2 lq + N,.

EXERCISE 3.3: (A) If T is unsuperstable or with dop or deep, Iio(A, T) =

(B) If T is shallow of infinite depth without dop, then for a large for A 2 N, + ID(T)I.

enough l i 0 (Na , T) = a D p ( ! r ) ( t a l ) (really L p ( T ) ( I a l ) < I io (Na, T) G aDp(,)

((la1 +2)2IT1). (C) If T is superstable of finite depth without dop, then for some

K G (2ITI)+, and n, k, n + k = Dp(T) for every a large enough

Iio(Ha, T) = a c U a I ~ ) *

( p < x 1 (D) Determine the “large enough’’ in (C), (B). [Hint: For (C) see XIII, Section 4.1

EXERCISE 3.4 [T superstable without dop, axioms as in 2.81: (1) If ( N , , , i ~ ~ : q ~ l ) is a (T, c*, A)-decomposition of M , J E I is

non-empty closed under initial segments, N G M* is T-prime, T- atomic and T-minimal over u,,,,N,,, then M* is T-prime, T-atomic and T-minimal over u,,N,, U N.

< a, N E *N, E *MO E *M’ for i < a, {Ni :i < a} independent over N, for i < /? Ni is an (N, 6)-component in MO and in M,, 6 EM“, N’ is an (N, 6)-component in Mo, tp(6, u,, i,,y) does not fork over N , then N’ is an (N, 6)-component in M.

(2) If

CHAPTER XI1

THE MAIN GAP FOR COUNTABLE THEORIES

XII.0. Introduction

We shall concentrate on superstable T. The chapter ends with the proof of the main gap theorem for countable theories. The new missing part is, by Chapter XI, the existence of a (P%, c:, ,~,) decomposition, and for it the missing part is the (A , 2)-existence property : if M < M , , 4 , tp,(M,,q) does not fork over M , then over MI U 4 there is a primary model. For this end we prove: if T does not have the (Ko,2)-existence property, then it has the otop (=omitting type order property). As the otop implies T has many non-isomorphic models, we can concentrate on T with the (No,2)- existence property. In Section 5 we prove that such T has the (A, 2)- existence property. Though we are interested in diagrams consisting of three models, we need to consider for each n < w a “stable system ” (4: 8 ~ 9 ” - ( n ) ) , 112yII = A, and want to prove by induction the existence of primary models over u 8 M 8 (in fact there are more complications). For more on such diagrams see [Sh 83al.

In Sections 1-3 we do some preparatory work. In Section 1 we deal with two new F’s: Fio,FAo, both strengthening FL,, and are useful for superstable countable T. Note that if (A, (ai : i < u)) is an FL,-construction, A E,B, then (B, (ai: i < a)) is an Ffro-construction. In Section 2 we prove all we need on stable systems. In Section 3 we deal with good sets: A is good if for every przS”(A) there is B G A, PI < K,(T) such that p does not fork over B and p is the unique extension of p r B in &“‘(A) not forking over B. So a Fzo,-saturated model is a good set. We prove that for any system ( l Q , : s ~ B - ( n ) ) of FZr,.,-saturated models u8g is a good set. This helps in Section 5 to show how to go from the “non-(KO, n)-existence property ” to the “non-(KO, 2)-existence property ”.

590

Notation. In notions like Fko-isolated, Fko-primary, Fko-prime, we omit the “Fk,”.

XII.l. On F,k and F{

(Hyp) T stable.

DEFINITION 1 . 1 : Let (see IV$1) Fi = ( ( p , B ) : p ~ s ~ ( A ) , B is a subset of A of power smaller than A, and for each @ = @(z,Y), for some formula Q,+(z, a+) which is in p and is almost over B, Q,+(z, a+) I- p f @}.

LEMMA 1.2: (1) ( p , B ) ~ F ; i f l ( p , B ) ~ F f and (p ,Domp)~FL~. (2) Suppose T is countable and superstable, then for every A , ~ E A

and Q, such that != (3zb(z, a) there is p €Srn(A) which includes Q,(z, 8) and is F;,-isolated (and also F&-isolated).

(3) If (A, C ) satisJies the Tarski-Vaught condition, tp(a,A) is Fie- isolated, then so is tp(& C). I n fact, the same formulas witness it and tp(a, A ) I- tp(a, C). The same holds for FLo, Fko. (4) In (3) i f tp(C,A) is not isolated, then tp(a,C) too is not

isolated. (5 ) A(F; ) = Ho and F:, satis@ Ax (I), (11, 1 , 2, 3, a), (111, 1, 2),

(IV), (V, 11 21, (VI), (VII), (IX).

ProoJ (1 ) The implication * is obvious: Let ( ~ , B ) E F ~ , be exemplified by Q,,(Z, a+) ($ = @(z, ~ E L ( T ) ) , F realizes p , then

StP(GB) I- {Q,*(z, a+) : @EL(T))

P = u P r @ ,

(see end of remark to Def. 111,2.1) and clearly {p@, a*}: @EL(T)} I- U,P r @ but

* hence stp(C,B)I-p, so by IV, 2.1 ( p , B ) € F f , Now the rpJz,@*)’s exemplify (p, Dom p ) E FL,.

Also, the other direction is easy. Let ( p , B ) E F;, ( p , Domp) E FLo, and Frealizes p . We know by IV, 2.1 that stp(F,B) I-p, and there are Q,+(Z, a+) E P ( @ E L ) such that ~,+(z, a+) I- Q,& a*), Q,,(z, a*) I- p r @. As stp(F,BB) t -p ,~,+(z , a+) ~p there is EEFE(B) , xE~I-rp+(z, a*), and let {dl , . . . , &} be representatives of the E-equivalence classes and

Q,*(x, * - a*) -* = v {zEd, : xEJl I- rp&z, a*)}.

592 THE MAIN QAP FOR COUNTABLE THEORIES [CH. XII, $ 1

Clearly, p,$(Z, a$) is consistent with p (a formula equivalent to ZEF appears in the disjunction), is almost over B (as a disjunction of such formulas) and is equivalent to a formula p;(~, a;) over A (as every automorphism of P over A maps it into an equivalent formula). As

definitions), clearly p&~, $) I- p r @. As p$(z, $) is consistent with p , also p;(Z, a;) is consistent with p , but as a; E A , p is a complete type over A , clearly q$(z, a;) ~ p .

(2) Choose +(z, 6) such that 6 c A, I= (~z)+(z, 6), +(x, n) l-p(z, a ) and (under those restrictions) Rm[+(z, b),L, co] is minimal. We know that every p , +(z, 6 ) e p e S m ( A ) , is Fio-isolated and (see IV, 2.18(4)) there is such p which is Fko-isolated. By part (1) we finish.

(3), (4) Also this part is easy. ( 5 ) See IV, $2 (on Fi0 and Ff, ), straightforward by (1).

p;(z, a;) b$W, q), v$@, I-pJz, a+), p+(z, a+) I- p r + (by their

CLAIM 1.3: (1) If A E B, p € S " ( B ) , p is Ff-isolated and does not fork over A , then p r A is Ff-isolated.

(2) I n fact, if C E A c Dom p , (p, C) E Fg0, then (p r A , C ) I= F:,.

Proof. (1) By Theorem IV, 4.3(2) p r A is Ff-isolated, and by IV, 4.3(5) p r A is Fko-isolated. Now use 1.2(1).

(2) Same proof.

Remark. The parallel of 1.3 holds for F:, Ff(A 2 K,(T)), F;(A 2 K,(T)), and F: (by IV, 4.3).

CLAIM 1.4 : Suppose <q 7 E I ) is a non-forking tree of models, and A is FL0-constructible, OT just Fko-atmic over u,,e,N,.

(1) If T does not have the dop, then every type not orthogonal to A is not orthogonal to some 4.

(2) For every ~ E A , n#u,,,N,, and V E I , tp(n,U,,,N,) forks over N,,

Proof. Like the proof of XI, 2.12, 13 and X, 2.2. Let A > 1-41 +&, llyll be regular.

Choose {M, : 7 EI) a non-forking tree of Ff-saturated models, N, c M,, tp*(M,,,N, U u {M, : not 7 4 v}) does not fork overM,- U &, (for 7 of length zero omit M,-). By 2.10, 2.3 such M , ' s exist and (u,,,N,, U,,,M,) satisfies the Tarski-Vaught condition, hence (see 1.2(3)) A is Fko-atomic over u,,,q. So A is Ff-constructible over u,,,M,. Now there is an Fin-primary M over u,,elM,, A c M . By X, 3.3 we

know that every type not orthogonal to M is not orthogonal to some 3. So if r is not orthogonal to A it is not orthogonal to some tp(c,%). As tp*(%, A) does not fork over 3, necessarily tp*(c,%) is not orthogonal to N, (by X, 1.1). ( w log tp(Dom r , M) dnf over A.)

This completes the proof of (1). As for (2), a failure of it implies there is aEM,a#U,,,q, tp(a,U,,rM,) does not fork over A!,,. But tp(a, Uv,r%) is F:-isolated, hence tp(&,M,,) is F,"-isolated (by IV, 4.3(2)). But a#M,,,a is Fio-saturated, a contradiction.

FACT 1.5 : Suppose V(Z, a) I- tp(c, a), ii E A E B, (A, B ) satisjies the Tarski-Vaught condition. If {~I(z, a)} has no extension over A which forks over a, then {~I(z, a)} has no extension over B which forks over a.

Proof. Suppose ~ E B , ( ~ ( 2 , a), 8(z, d ) } is consistent and forks over a. We can now define by induction on n, a distinct p n ES;JZ(A) such that { ~ ( z , a)} U p , is consistent. Hence for some n :

Rrn[tp(E, q, 6, KO1 = R"[{V(% a)), 8, KO1

> R"[{v(z, a)) U pn, 6, No] = Rm[tp(c, a) U pn, 8, KO].

By 111, 4.1, this implies that tp(c,&) U p , forks over ii, a contradiction.

DEFINITION 1.6 : Fi0 = {(p, B) : IBI < KO and letting A = Domp, p € S " ( A ) , B G A and the following holds: p r B has no extension over A which forks over B, and for every 6 for some e8(z, 6*) ~p rB, there are only finitely many rES;JZ(A) which are consistent with {@& k9N.

FACT 1.7 : Suppose B c A , p has no extension in Sm(A) forking over B, and p € S " ( B ) . Then for every 6 = 6(z,g) the following conditions are equivalent :

(a) for some @ ( z , b ) ~ p , there are only Jinitely many rES;IL(A) consistent with {@(z, 6)},

(p) for some @ ( z , 6 ) ) ~ p , for every rE{rES;JZ(A): r consistent with @(z,6)} the type p U r does not fork over B (equivalently, p U r is consistent).

Remark. We shall use this fact freely when dealing with Definition 1.6.

594 THE MAIN GAP FOR COUNTABLE THEORIES [CH. XII, $ 1

Proof. (a) * (p) . Let $ o ( ~ , 6 0 ) ~ p exemplify (a), and & = {rrzSF(A): r is consistent with $,(z, 6)}. For each YE& let p, be a finite subset of p such that if p U r is inconsistent, then p, U r is inconsistent. Let $(Z, 6) = $,(z, 6,,) A ArEro(Apr). As 6, E B, each p, is c p, hence is over B, clearly 6 E B . As & is finite as well as each p,, clearly @(%,6) is a first-order formula. Clearly, $(z, 6 ) ~ p , (p, E p, @,(Z,~~)EP, hence pl-$(Z,6), hence $ ( z , 6 ) ~ p ) , We now show that $(z,6) exemplifies (p). So let r~ r = {reSF(A) : r consistent with $(z, 6)}. Then p, U r is consistent (as $(z, 6) I- A p,) and r~ & [as $(z, 6) I- $,, (Z,6,)], hence p U r is consistent (by the choice of p,) , hence p u r does not fork over B [as p has no extension over A which forks over B, by the hypothesis on p and the extension property for non- forking].

(p) e- (cc). Let @(z, 6) exemplify (p) , and r = { ~ E S F ( A ) : r consistent with @(Z,b)}. We shall show that r is finite, thus finishing.

As @(z, 6) exemplifies (p) , r c 4 z ' { r ~ S r ( A ) : p U r does not fork over B}. By Theorem 111, 4.1 for every r E r (clearly T E 4 hence) :

R"k , 8, No] = T"b U r , 8, No].

Hence 14 < Mltk,8] < w . So we finish.

CLAIM 1.8: F&, E F;,.

Proof. Suppose ( p , B ) EF&,. As p rB has no extension in Sm(A) which forks over B, clearly (p, B) E F:,, so by 1.2( 1) it suffices to prove that p is FL,-isolated. So let 8 = 8(z, q) and we have to find @*(z, 6*) ~ p , $*(it?, 6*) I- ( p r 8). By Definition 1.6 there is $(z, 6) ~p for which r = { ~ E S ~ ( A ) : r consistent with $(z, 6)) is finite. For each T E r - { p r 8) choose a formula rp,(z, 6,) E r,tp,(z, 6,) $p (hence lrp,(z, F r ) € p , and rp, is 8 or 78). Let $*(z,6*) = $(z,6) A A{rp , (~ ,&, . ) : r~ r- { p r 8)). Clearly, $( z, 6) E p , $( z,6) I- (p r 8). CLAIM 1.9 : If (p, B) E F& , p E &"(A), (A, C) satisfies the Tarski-Vaught condition, then :

(a) p has a unique extension q E Sm( C), (b) (q ,B) EF&, (the Same witnesses @Jz, 6*) works), (c) if p is not isolated, then q is not isolated.

Proof. (a) By 1.8 (~,B)EF;,, so apply 1.2(3). (b) Trivial, as if there are exactly k types ~ E S ~ ( A ) consistent with

$(z, b), ( ~ E B ) , then there are exactly k types rESp(C) consistent with $(~,6).

(c) Trivial.

FACT 1 . 1 0 : If (p,B)EFio,prB E qES"(Domp), then Domq = Domp and (q,B)EFio.

Proof. Trivial (check the definition).

LEMMA 1.11 : FA, satisfies the following axioms: h(F) = No, Ax (I), (11, 1, 2, 3, 41, (111, 1 , 2), (IV), (V, 1, 2).

Proof. Let F = Fio,h(F) = No. Trivial. Ax (I), (11, 1, 2, 3, 4), (111, 1, 2), (IV). Trivially. Ax(V, l ) . I fq= t p ( a n 6 , A ) , p = tp(a,A U b),BGAand(q,B)EF,

then (p,B U 6) E F. First we have to show that tp(a, B U 6) has no extension over A U b

which forks over23 U 6. Suppose tp(a', A U 6) extends tp(a, B U 6) and forks over B U 6. As tp(6,A) does not fork over B, by 111, 0 .1 (3 ) , tp(a' b,B), a contradiction.

Now let 9 = 8(z;g) and we have to find a suitable $(z,.T)E tp(a,B U 6) (which is consistent with just finitely many re S r ( A U 6)). Now we can easily find E < w and 8,(z; g ; 23 (for 1 < k) such that l (g) = Z(6) and for every CE(A U 6) for some F'EA, and 1 < k 8(Z, 6,B) such that 4 is finite where 4 = { r : T E S ~ T . $ ) ; ~ ( A ) is consistent with

Let $(z,c) = A , < b $ , ( ~ , 6 , J , ) and r= {rES&;B)(A u 6): r is consistent with $(Z, .T)}. Clearly, Ill < n,<b 4; by the choice of the 8,'s, E is finite, and each 4 is finite (by the choice of $, (z ,g ,d , ) ) . So we finish.

6 , A ) forks over B, and clearly it extends tp(a

= 9,(z, 6, c'). For each 8, there is $,(z, g, d,) E tp(a

$& 8,dl)).

Ax (V, 2). As h(F) = No it follows from (V, 1 ) .

LEMMA 1.12: (1) If T is countable and superstable, then Ax(X, 1 ) holds for Fh,.

(2) If ( A , B ) satisfies the Tarski-Vaught condition, C is Fio-atomic over A, then tp*(C,A)Ftp,(C,B) and (A U C,B U C) satisfies the Tarski-Vaught condition.

(3) Suppose ~ E S " ( A ) , (p, B)€Fio ,p rB has a unique exteneion in

596 THE MAIN QAP FOR COUNTABLE THEORIES [CH. XII, $1

Sm(A) which does not fork over B. Then for every I9 = 6(z, y) for some

(4) If B E A,IBI < Ko,pESm(A),prBI-p and p is Fko-isolated, then (p, B ) E FA,.

( 5 ) Suppose p E Sm(A) is realized in some set FAo-constructible over A. If for some Jinite C c A, p is the unique extension in Sm(A) of p r C which does not fork over C, then for someJinite B E A , (p,B)€FAo and

(6) If ( A , ( a t : i < u),(&:i < u)) is an Fko-construction, A GtAl , then (Al,(a, : i < a), (4 : i < u)) is an Fko-contradiction too. Similarly

+=p rB, @ I- (P r 191, and w) E F A ~ .

PrBI-p.

for Fi, G), (FAo, G J F k 0 , = t ) .

Proof. (1) Let A, I= (3Z)cp(z, a), and we have to find p ES"(A) such that rp@, a) ~ p , p is Fio-isolated. W.1.o.g. {rp(Z, a)} has no extension over A which forks over a. Let {8,(z, yl) : I < w } list the formulas of L. We define by induction on n < w,rp,(z, a) such that po(z, a) = rp@, a), Q),+~(Z, a) t-rpn(z, a), I= (32)~,(2, a), and for each n there are only finitely many r6Srn(A) consistent with Q),+~(Z, a),

So rpo(z,a) is defined. Suppose rp,(z,a) is defined and we shall choose v,+~. Among @ = {@(z, a) : I= (~z)@(z, a), @(z, a) t-p,(~, a)}, choose one, g~,+~(z, a) with minimal Rm[$(Z, a), a,, KO].

Let r = {rES" (A) : r be consistent with P)~+~(Z, a)}. The only remaining point is " r is finite ". We shall prove that for

every r E r, R'"[{~n+l(z, a)} U r, en, 8 0 1 = R"[Q),+~(z, a), @,, KO],

and this will show that Irl 6 Mlt[rpn+l(z, a), S,], but the latter is finite so we shall finish.

For each r E r , choose qESm(A),{rpn+l(z,a)} U r E q. As r p ( z , & ) } ~ q [because Q),+~(Z, a) I- p0(z, a), po = 931, clearly q does not fork over [as rp(Z,@) has no extension over A which forks over A]. Hence,

R"[(rp,+,(z, a)} U T , a,, No] < by monotonicity,

R"[q, a,,, 8 0 1 = by 111, 4.1 as q does not fork over ii,

R"[q r a, a,, N,I = by the choice of cpn+l(~,a),

Rnr[rpn+1(z, a), a,, No1 < by monotonicity,

R"[{F,+,(z, a u r , a,, &I.

So the desired equality follows. (2) Easy. (3) Let p 1 = { p ~ p : g , is almost over B } ; as ( p , B ) ~ F i ~ , p ~ I - p . If

p r B $ p l , then for some cpEp',prB#v, hence by 111, 2.6(1) ( p r B ) U {lv} does not fokk over B , hence there is a qESm(A) extending ( p r B ) U { lv} which does not fork over B . Now p,q contradict by a hypothesis of Fact 1.12(3), hence prBI-p1 . As ( p , B ) eFi0, p is Fio-isolated, hence for every 9 = 9(z,g) for some $ E p , $ I - p r Q , but a s p r B t - p l I - p we can choose + E p r B .

(4) For every 9 for some + ~ p , + I- p r 9, but p r B I- +, hence some $ ' ~ p rB, +' I- +, hence I- p 19.

(5) By IV, 1.4(2), as h(FAo) = No, there is an Fio-construction (A,(c,: i < n ) , (4:i < n ) ) such that p is realized by some F S {ct: i < n}, and tp(c,,A U {cl: I < k } ) ~ F i ~ ( & ) (so & is finite). By IV, 3.2 {ct:i < n} is Fio-atomic over A, hence p is Fio-isolated. Let B =

C u (U,,,B, n A), clearly it is a finite subset of A. We shall prove p r B I- p ; a s p is Fi-isolated by 1.12(4) ( p , B) E FCo. As FC, S Fi0 S Fi0 (see 1.8, 1.2(2)), (A,(ct:i < n ) , (8: i < n ) ) is an Fio-construction over A, hence {ct : 1 < n} is Fio-atomic over A, hence stp(F, B ) I- p (remember p = tp(c,A),FE{c,:Z < n}) . So it suffices to prove (as C G B ) :

FACT 1.13 :If B G A , p = tp(q A ) E Fi (B) , p r B has a unique extension in Sm(A) which does not fork over B , then p r B I- p .

Proof. We know that stp(c,B) I - p . Suppose 9(z ,d ) ~ p , and we shall prove that p r B I - @ ( z , d ) . As s tp(c ,B)I-p for some EEFE(A), xEd- 6(2, d). Clearly, xEF is almost over A, hence by IV, 4.1 there is a formula p(z, 6 ) , 6 ~ A , which is almost over B , and (*I for every formula $ ( z , a ) , a ~ A , I-zEF-+ $(z, a) iff t-v(z,

As xEFis consistent with p , necessarily v(z, 6) is consistent with p , hence v(z, 6) ~ p . Also p f B I-v(z, 6) as otherwise ( p rB) u {lcp(z, 6)} being consistent does not fork over B (as v(z, 6) is almost over B , by 111, 2.6(1)), hence there is q€Sm(A) extending (PrB) u {-tp(z,6)} which does not fork over B , and clearly q # p . This contradicts the hypo thesis.

S o p r B F q ( z , 6 ) , andrp(z,6)I-9(z,d) (by (*) andasxEiA-9(z,d)). So p r B I - @ ( z , d ) , and as 8(z,d) was any formula in p , we have proved p r B I- p , thus finishing.

6) -+ @(Z, a).

598 THE MAIN GAP FOR COUNTABLE THEORIES [CH. XII, $2

(6) Left to the reader.

Remark. Note we have observed:

FACT 1.14 : (1) If p E Sm(A) is Fip-isolated over a set B s A , then there is q E p , q = p , which is almost over B.

A , {q(Z, a)} is consistent but has no extension over A which forks over a, ~ E A , { ~ ( x , a), $(z, 6)} is consistent. Then V(Z, a) A $,(Z,bl) is almost over a for some & E A , $l(2?,bl) E $(Z,6) and

( 2 ) Suppose

b(3Z:) [V(z:, a) A 1G.l(Z, it1)I-

XII.2. Stable systems

(Hyp) T is stable.

DEFINITION 2.1 : We call S = (M, : s E I) a stable system if ME E 6, I is a family of finite subsets of UI closed under subsets, s c t *Ma E 4, and for every ~ € 1 , tp,(M,,U,&&) does not fork over A: (see 2.2( 3) below).

Notation 2.2. ( 1 ) P ( s ) = {t : t E s, t # s}. (2) For a stable system S, let S = (w : S E F ) . (3) If s is a finite subset of U I , P ( s ) c I let A: = U s c t v .

s E t *A: E A:. Also, for J E I let AS, = u t e J e . (4) We omit the superscript S when its identity is clear.

LEMMA 2.3: (1 ) If I = {sa: CI. < q,}, s, E sa*a < p; M, < 6 and tp,(M,=, u,,,M,) does not fork over A, , then (M,:seI ) is a stable system of modes.

( 2 ) If (M,:s~I)isastablesystem, J E I , a n d s d A s E U J * s E J , then u,,,M, C t u,,,M,.

(3) If S = (M, : s E I } is a stable system, N, < 4, tp(N,, A:) does not fork over ut,,&, and s E t *N, E &, then ( N , : ~ E I } i s a stable system and U E , , N , SUS.IM,.

Proof. Essentially like [Sh 83a, 3.51, but we shall prove it. We first prove some facts.

FACT 2.4: If S = (M,: S E I ) is a stable system, and for 1 = 0 , l J I c I ,

CH. XII, $21 STABLE SYSTEMS 599

4 is the closure under subsets of 4, then tp,(U8,J,M,, u8,Jl&) does not fork over U {{M, : s €4 n I,}.

Proof. W.1.o.g. J1 are closed under subsets, and let J = J1 n J,. We can find a list (s,: a < a*} of I such that s, E sa a < /3, J = {s,: a < a,}, J, = {s,: a < a,}, J1 = {s,: a < a, or a, < a < as}. Clearly, for a < as, a 2 a,, tp,(M,., U,<,&p) is included in tp,(&., u {I@ : s, $t= t € I ) ) , and hence does not fork over A, c u{&& :/3 < a, or a, < /3 < a}. So tp,(M,., UB<”&. does not forg over d{&,:/3 < a,, or a, < ,8 < a}. By IV, 3.2(1) (applied to Ff) we can conclude that tp,(U{& :a2 < a < a, or a < a,}, uB<,sM8J does not fork over ua<,lM6,. But’this is as required.

FACT 2.5 : If S = (4 : S E I ) is a stable system, al E&(, ) (I < n) , t C UI, and Cq[a0, . . . , a,,-,], then we can Jind E&(,) ,, such that I=Q,[~;, . . . , ah-,] and s(l) E t - a; = al.

Proof. W.1.o.g. s c s(l) s ~ { s ( m ) : m < I } , the s(I) (I < n) are distinct, { I : s(Z) 5 t } is an initial segment of {I: 1 < n} and s(n) $ s(l) for 1 < n. We prove the assertion by induction on n. For n = 0 there is nothing to prove, and for n = 1 note &,,, ,, is an elementary submodel M,,,,. So suppose we have proved for n and we shall prove for n+ 1, i.e. for given a,~&(,) (I < n+ l) , t E UI and Q, satisfying the “w.1.o.g.” above. If s(l) t for every I , let a;= a,; so we assume (by the (6 w.1.o.g.” above) that s(n) $t= t . As tp(a8(,), U,<,&(,)) does not fork over A:(,,, clearly rp(ao, . . . , a,,-,, Z) does not fork over A:(,,, hence it is realized in every model which includes A:(,). So there is type p = p ( ~ ( ) ~ < , over A:(,, (in infinitely many variables) such that p ( z o , . , . , zi, . . .) I- Vi<”tp(a0,. . . , a,-,, zit). So for some 6 c A:(,,, and $ = $(zo, zl, . . . , zk, 6), and k < w

(i) I = ( ~ E ~ , Z , , . . . ,zk)$(z0,. . . , zk, 6), (ii) $(z,,. . . ,zk,6) t- VtGkq(a0,. . . , a,,-,,~‘). As br A:(,,, and (Vs c s(n)) [ s ~ { s ( l ) : 1 < n}] , w.1.o.g. 6 =

iion 6, h . . . 6, empty]. Now apply the induction hypothesis to a, A 6, EM,(,) (for l < n) and the formula :

6,-,, 6, c_ A&), and [s(Z) $ s(n)

- @To,. . . , E k ) $ ( Z 0 , . . . , Z k J 0 , . . . ,b , - , )

r 1

A (VZ,, . . . , Zk) L$(Zo,. . . , Ek, 6,, . . . , 6,-,) + v (p(Go, . . . , a,-,, zi) .

,, t ( l < n) satisfying the above formula and t < k 1

So there are 6


as in 2.5's conclusion; remembering that s(1) $ s (n) 6, = ( ), clearly 4 - K h . . . such that t=$[c,, . . . , ck, 6, . . . , G-']. So for some i < k t= ~ [ a ; , . . . , ah-', ~ $ 3 . So a;, . . . , a;-2,

n -t

bn-' c w(n) Hence there are c,, . . . , ck EM,(,)

def a; = q are as required.

Proof of 2.3. (1) An exercise in non-forking. (2) Follows from Fact 2.5. (3) First we prove that S' = ( N , : s E I ) is a stable system. For

every s tp,(M,, u (4 : t E I , s $Z t}) does not fork over u {4 : t c s}, hence (as N, c M,) also tp,(N,, u {@ : t €1, s $Z t } ) does not fork over u (4 : t c s}. But tp,(N,, u (4 : t c s}) does not fork over u {&: t c s}, by the hypothesis. So by 111, 0.1(2), tp,(N,,U{q:tEI,s $ t}) does not fork over u {& : t c s}. As & c 4, by monotonicity of non- forking we get the stability of the system (s E t *N, E & was assumed, and we know 1 is as required).

Then A;' G t A ! follows by Fact 2.5 and the following fact. Let j # u I , J% U {s U { j} : s~ I ) and N,uc,)E*M,,

FACT 2.6: (8: SE J ) is a stable system (J ,& as above).

Proof. Let s,(a < a,) be as in 2.3(1) for S, and define t,(a < 201,) by: t,, = s,, t2,+' = s, U {j}. Clearly, J = {t, : a < 201,) and t, c tfl - a < /3. Now use 2.3(1). For a even (=2/3) remember we have proved tp,(N, , u {M, : s E I , s $ sP}) does not fork over u {N, : s c sfl, s # sg}) and tkis implies what we need. For a odd (=2/3+ 1) remember tp,(M,,, u,,<flM,7) does not fork over u {M, : s c sfl}). As N,, c by 111, 0.1(3) this gives tp,(&=, uycu& ) = tp,(M8/, U,,,M, UN, ) does not fork over u {M, : s c ss} UN,, = d{N, : s c t,}, and this' is wkat we need.

So we finish the proof of 2.3.

LEMMA 2.7 : Suppose (M, : s E I ) is a stable system IIl < K .

(1) If each M, is K-compact, p a type of cardinality < K in the variables xt ,Jt E I , i < it), and every Jinite subset of p is realized by an assignment sending each xt,( to a member of 4, then p is realized by an assignment sending each xt,( to a member of 4.

(2) If each% is Fi-saturated, K 2 K,(T), p a type which is almost over a set of power < K in the variables xt, ((t E I , i < it < K ) and every Jinite subset of p is realized by an assignment sending each xt,( to a member

CH. XII, $21 STABLE SYSTEMS 60 1

of 4, then p is realized by an assignment sending to a member of 4.

Proof. (1) Clearly, K = No is a trivial case ; so w.1.o.g. K > No. Now we can replace 6 (and all M,) by their reduct to a language containing only the predicates (and function symbols) which appear in p, without affecting the hypothesis or the conclusion. So w.1.o.g. IT1 < lpl+ No < K , hence K,(T) < K , but then our desired conclusion follows from (2).

(2) So let p be admost over A , 1.41 < K . As K,(T) < K there is B c UsEIM, , IBI < K,(T) + [A[+ < K such that tp,(A, u,,,&) does not fork over B. Clearly, by IV, 4.2, (*) for every ~ E A , and p = cp(Z, @) there are 6 , , ~ u , , ~ M , and +, = ll.,(z, 9) such that :

(i) $v is almost over B, (ii) for every F E U , M , , ~ ( C , = +,(c, 6,).

So w.1.o.g. p is over Use,&, and almost over B. Checking the demands on K ( K 2 IBI++K,(T)+III+) w.1.o.g. K is regular. Looking a t the demands on p it is clear that for every p, p U {p} or p U {-p} satisfies the demands. Also we can allow more variables from xt,t ( i < it, t E l ) , and take increasing union of length < K . So w.1.o.g.

p = stp,((a,,,:i < i t , t E l ) , B ) , and let B, = B n &&,$zp{a8,t:i < i t } .

Now it suffices to prove that w.1.o.g. (**) for each t ~ l , tp,(4 U 4,U{B, U $ : t $ S E I ) ) does not fork

over U{B, u 4 : s c t } . Because then we shall choose a list (s,:a < ao) as in 2.3(1), and

then define by induction on a, U ~ = , ~ E M , (i c i t ) such that stp,((a, ,1:

i c i ta ,P < a) ,B) = stp((aia,r:i < i ta ,p 2 a) ,B) using ‘‘&a is b:- saturated ”.

Why can we make this assumption? First, w.1.o.g. tp*(B,AJ does not fork over A, n B,tp,(B,MJ does not fork over 4 n B (just increase B < K times). Then we define by induction on a < ~ , p , = stp,((a;,,:i < i ; , te l ) ,B,) such that :

(a) p, is the p we have, pa is increasing and continuous. (b) Each finite subset of pa is realized by some (b;,i: i < i ; , t ~ I ) ,

(c) (d) tp,(B,,AJ does not fork over 4 fl B, tp*(B,,q) does not fork

where b;, €4. forks over B, and B, c u,,,Mt.

over M, n B,.


For a = 0, a limit, there are no problems. For a = p+ 1 , we assume that pa does not satisfy (**). Choose t = tp with minimal Itl, andfinitew,, E i(,and6,,~B, n M,,suchthat (lettingap = ( a ( : i ~ w J , B ~ = B , nw, ~ f = { a f , * : i < i f ) ) , tp(apc6,,U{Bt u ~ f : t $ s ~ I ) ) forks over u {Bt U A t : s c t}. Hence there are dl =-(dt5 :j < jp , 1 ) E

Bfl U Atl for 1 = l , n , t $ s,eI, and qJp(z,g,dl, ..., d,,), forks over u {Bt U A t : s t t} and is realized by apn EP.

-

Let t (p , 1) = sp, n lfl, choose j (p , 1,j) (j < jp, 1 ) so that i t , , 1 ) G j (p, 1, [t(P, 11) = 7(P, 4) A ( L j J z ( L j z ) =.j(P, L j l ) f j) < itcp.1, + w , and

j(P,lZ,.i2)1* Now by 2*3(2) P u {qJKxtp,(:iEwp)’ ~ p , ~ , ( , , , 1 , , 5 ( , , , 1 , . . . . ) } satisfies (b). Then we can complete it to a stp((af,,:tEI,i < as required (notice it = if when s $ tp, and it < it+@).

If K = KO, choose t with maximal It1 such that t = tp for unboundedly many P’s . So if(a < w ) is eventually constant and we get a contradiction to K,(T) G K . So let K > KO, and remember that K is regular. Note that for every pair ( t , w), t E I , w is a finite subset of uai:, &,, = {P < K : tp = t , wp = w} is a bounded subset of K , (by K,(T) < K ) . Hence by the well-known properties of the closed unbounded filter, for some limit S < K , (Vt E I ) (Vw c Uu<, if) [A‘$, , _C 81. So for this 8, t,, w,, are not defined, a contradiction.

LEMMA 2.8: I n 2.7 we can replace (in assumption and conclusion) “xt,* assign to a member o f q ” by “each variable is assigned to a member of u,,,&”, provided that I is Jinite.

Proof. By the compactness theorem we can assign a model to each variable retaining the hypothesis and then use 2.7.

CONCLUSION 2.9 : Let (M, : s E I ) be a stable system ;

J E I , (Vt E I ) [t C U J + t E J ] .

( 1 ) If each & is K-compact, then u,,,M, ~:u,,~&. (2 ) If each & is Ff-saturated, K 2 K,(T), then u,,,& ~f us,,&.

Proof. Immediate by 2.5 and 2.7(1) [for ( l ) ] and 2.7(2) [for (Z)].

FACT 2.10 : For the stable system (4 : t E I ) and K we can choose j 4 u I , let J = I U {t U { j } : t E I ) and define&(sEJ-I) such that (M8:scI) is a stable system and for SE J - I , M, is F:-saturated.

Proof. Easy by 2.3( 1) and the properties of forking.

CH. XII, $31 ON GOOD SETS 603

CONCLUSION 2.11 : If (M, : s E I) i s a stable system of Ff-saturated models, c f K 2 K,(T), IJinite, then for any Fund an F,"-primary model M over U,,,M, U C, tp*(MU,,,M, U C ) i s FLo-atomic.

Proof. Suppose p = tp(J, USE,& U C) is Ff-isolated, and we shall prove it is Fio-isolated. First observe that w.1.o.g. K is regular > ITI. [Otherwise l e tp = K + + I T I + . Choosej#UI, let J = I u {t u { j } : t ~ l > . By 2.10 we can find F,"-saturated M, for s E J - I such that (M, : s E J ) is a stable system. By 2.9(2) u,,,M, cFuSeJM,. W.1.o.g. tp(Cnd,

does not fork over ussrM,, hence us,,& U c ~,"u,,,& U C). So as tp(&U,,,M, U C) is F:-isolated, also tp(&U,,,M, U C ) is Ff- isolated, hence F,"-isolated. By IV, 4.3(5) it suffices to prove that tp(a,U,,,M, U C ) is Fio-isolated.]

Now let tp = tp(z,g), and suppose for no finite q c p,ql-p r+tp

(remember p = tp(& u,,,M, U C ) ) . Then w.1.o.g. for some tp(z, Z, C) and for no @(z, a) ~p for every 6~ u,,,M, : tp@, 6, C) ~p implies @(z, a) t -q (Z ,b ,C) . Then r={T(Vz)[@(z,a)-ttp(z,5,C)] A tp(d,z,C)]:@(z,a)~ p} is finitely satisfiable in us,& [if n < w , @@, az) ~p for I c n let @(z,E) = A,,, $,@, aZ), so $(z, a) ~p hence for some ~EU~.,M,, p(%, b, Q EP but @(z, a) tf- p ( ~ , 6 , C) ; now substituting 6 for t is as required]. Hence every subset of r of power < K is realized in u,,,M,. But as K is regular > ITI, for some complete p1 c p, lpl( < K , p, I- p, so {~ (vz) [@(% a) +tp(Z, 5, C)] A tp(& Z, C) : @(Z, a) ~ p , } is realized in UserM,, say by 5. But as pl is complete, this means that p, ttp(z, i ~ , C) and Cq[& iZ, C], hence tp(x, C, @ ~ p . This contradicts p, I- p.

det

def

CONCLUSION 2.12 : For a stable system (M, : s €1) and s E I, tp,(M,, utcsMJ i s stationary inside u {M,: s $ t, t €4.

Proof. By 2.3(2), Def. 2.1 and XI, 1.2.

X11.3. On good sets

LEMMA 3.1 : Let T be stable, K = K , ( T ) ,

If & (1 < 3) are FF-saturated, 4 < 4, {&,M,} independent over 4, then M, U 4 is a good set, where


DEFINITION 3.2 : A set A is called good if for every p €Srn(A) there is B G A , PI < K, (T), such that p rB is stationary inside B (see Def. XI, 1.1) .

We shall prove 3.1 later (in 3.5).

FACT 3.3 : (1 ) Every F:-saturated M ( K = K ~ ( T)) is a good set. (2) I f A EEC,K = K,(T), C i s good, then A i s a good set. (3) If A i s a good set in a, then A i s a good set in Ceq. (4) If A is good, IBI < K,(T), then A U B is good.

Proof. (1) If PES"'(M) let C M , ICI < K , be such that p does not fork over C, let 6 realize p and let 6' realize stp(6, C) and belong to M (exists by the definition of F:-saturated). Now B = C U 6' is as required in Def. 3.2.

(2) Let ~ E S " ( A ) , and we shall find a set B E A as required in Def. 3.2. Let qeS"(C) be an extension of p which does not fork over A . As C is a good set there is B, c C, lB1l < K , such that q does not fork over 4, and q rB, is stationary inside C. Choose 4 c A , 141 < K , such that tp,(B,,A) does not fork over 4, and w.1.o.g. B, n A E& and p (which is in Srn(A)) does not fork over 4. Let f be an elementary mapping from 4 U 4 into A , fr4 = the identity, stp,(f(&), B,) = stp &,I&). Let B be the range off, and we shall show that it is as required. Clearly, 1B1 < K . Also p does not fork over B (as 4 E B ) , and as 4 c A E C, q extends p and does not fork over A , clearly q does not fork over 4.

So suppose p ' ~ 8 " ( A ) , p ' # p , p' extends p rB and does not fork over B, and let q'ESm(A) extend p' and does not fork over B. Now as p # p', clearly q' # q, hence q'rB, # q rB, so for some 664, *(z, 6) E q, +(Z, 6) E q'. But 6, f(6) realizes the same strong type over &,q,q' does not fork over 4, hence (see 111, 4.8(1)) 8(z , f (6))~q, +(z, f@)) Eq'. But q, q' extend p rB,B 2 f(B,), a contradiction.

(3) Suppose p = tp(6,A,Ceq), w.1.o.g. 6= b = a /E , E an equivalence relation in a defined by a first-order formula without parameters; now use the goodness of A for tp(&A). (4) Suppose p = tp(C,A U B ) ; for some C E A , ICI < K , we have:

tp(6', C) is stationary inside A for every 6' E B u F (C exists since A is good). Suppose c' realizes tp@, C U B ) , and tp(r,A U B ) does not fork over C; then tp*(C U B , A ) does not fork over C, hence tp,(c' U B , A ) = tp(C U B , A ) , hence tp(c",A U B ) = tp(C,A U B ) , as required.


A more general lemma than 3.1 is

LEMMA 3.4: Let T be stable, K = K,(T).

Suppose Do G Q(Z = 1,2),D2 = mi, 4 is F:-saturated, each Q is good, {D,,D,} independent over Do and L& E:D, . Then D, U D, is good.

Proof. Let qeSrn(D, U XJ. Let Frealize q, q does not fork over C, C c D, u 4, llCll < K . Also we can define C, = Q n C , w.1.o.g. tp,(C,,Do) does not fork over C,.

We shall work in aeq (see 111, $6, and easily we can transform the hypothesis to aeq, and the conclusion back to a).

Stage A: W.1.o.g. for some regular h > (21T1)+, 4 is FT-saturated, Do s;D,. Straightforward ; choose elementary mappingf,(i < A ) such

does not fork over Do, and fo is the identity on Dl. Next choose M,*, an F!-saturated model, 4 U uo,,,,f,(Dl) S M Z , and tp,(M,*, u,,,f(D,)) does not fork over uo,,,,f3(D,). Now the triple Do* = b,,,,,,,f,(Dl),D: = D: U 4,D: = w,*l is as required and easily 4 U D2 c:D: U DZ. So by 3.3(2) it is enough to prove 0: U D: is a good set.

that ~ o m j , = wt rDo = the identity, t p , ~ ~ ) , u,,,~cD,) u q),

Stage B: q~ij'"(D, U 4) is the unique extension of t p ( ~ , 4 U 4) ( in Srn(Dl U 4)) which does not fork over C, where 4 = acl(Do U C,) (of course A, G4). Clearly, q does not fork over 4 U Dl (by monotonicity of forking). It is enough to prove that q is the unique extension of tp(E,4 U 4) in Srn(Dl U 4) which does not fork over 4 U D, (by 111, 0.1(2)). So, by symmetry of forking (see 111, 0.1(1)) it is enough to prove that t p , ( 4 , 4 U D, U @ is the unique extension (complete over 4 U Dl U @ of tp,(*,A2 UD,) which does not fork over 4 U 4. As tp,(G, 4 U D, U @ does not .fork over 4 U Dl (by symmetry of non-forking, see 111, 0.1(1)) it is enough to show that t p , ( 4 , 4 U Dl) is stationary. But as {Dl,G} is independent over Do,Do E 4 ~ 4 , clearly tp,(@,A, u 4) does not fork over 4 (see III,0.1(3)), hence it is enough to prove that t p , ( 4 , 4 ) is stationary. But we are working in aeq and 4 = acl(Do U C.), hence 4 = a c l 4 (by 111, 6.2(4)), hence every complete type over 4 is stationary (by 111, 6.9( 1)).


Stage C: Let q1 = q r (4 u 4). We can choose &, c A , U Dl, 141 < 12'1 such that for every finite set A of formulas, Rn(ql, A , KO) = R"(q, r4, A , No), and Mlt(q,A, No) = Mlt(q, rBo,A, No), and C ~ 4 .

Choose 4, such that 4 G 23, G Dl U 4,141 < ZITI < A and 4 u cis (the universe of) anLI,,+,ITI+-elementary submodel of 'ill = (Dl U 4 U F,

Do9Q,&c,... ,d ,..., Q ) ( - ) , , . . ) ~ ~ ~ , ~ ( - ) ~ ~ ( ~ ) . That is, 4,4 are monadic relations, the members of F and each deC an individual constant and each Q) = v(x0, . . . , x,+~) an n-place relation, i.e. Q, = {<ao, . . . , an-J : a, E I'W = IDll U 141 U F, Remember also that L,TI+,IT,+ is an infinitary logic. Clearly, tp,(B, n 4,4 n 0,) does not fork over 4 n Do.

Now choose an elementary mapping j , Domf=& u c,,fr (4 (I 4) = the identity,fmaps 4 n Dl intoD, and it maps cinto&, and

There is no problem in doing so: first define f r ( 4 n 4) by the hypothesis on Do,Dl from Stage A. As t p ( 4 n 4,4 n Q) does not fork over Do n 4, clearly id(,l A, ) u ( f r (4 n 4)) is an elementary mapping. As 4 is F;-saturated we can define f(q suitably.

and 6 +q[ao, . . . , a,-,]>.

st~,(B, u ~4 n 4) = stp,(f(B, u c3,4 n 4 ) .

Stage D: For every extension q'eSm(Q U 4) of q r ( Q U C, U f(q) which does not fork over C,q'rB, = q r 4 . So suppose Q ) ( Z , ~ , ~ ) E

q r B , = tp (c ,&) ,bc& n 4 , d t 4 (1 9. AS 6c&nA, and we have defined 4 = acl(Do U C,), clearly for some e*eC, and EED, and 8eL(T) , Ic < o, the following holds :

k 8[6,8*, E] A (3%J)8(8, F*, C).

As F*, b E 4 n 4, by q ' s choice there is gas required which belongs to Do n 4. It is also clear that 8(EeQ, F*, E) c 4 n 4 (as 4 = ad$). As f is the identity on 4 n 4, clearly

W 8 ) [ ~ ( K F*, e3 +Q)(c, 874 3 V(f(c3, 8, f(J))l

@(x) 22 (V8) [@(8, F", e3 +Q)(% 8, a = Q,(f(c3,8, f@)l,

(just try every ~ E B , n 4 as this is enough). Hence the formula,

belongs to q (as c satisfies it) ; and its set of parameters is

F* u B u d u f(q u f ( d ) E c, u (4 n 4) u (4 n 4) u fm u f(4 n DJ G u c, u

(remember that f maps 4 n Dl into Do).


Stage E : For some C: E 4, IC:l < K , C, E C: and q r (D, u C, U f(q) is the unique extension ofq r (C: U C, U f(q) inSm(Dl u C, u f(q) which does not fork over C, U C:. We choose C: c Dl, C, E C:, IC:l < K ,

such that tp*(C, Uf(C)^c,,4) does not fork over C:, and every extension of tp*(C, U f(@ C, C:) which does not fork over C: and is complete over Dl is equal to tp*(C, U f(q^C,Dl). This is possible as D, is good and K = K,(T) is regular.

Suppose C: is not as required, then some F* realizes some Q*E

Sm(Dl U C, U f(q), where q* extends q r (C: U C, u f(q), does not fork over C, U C:, but q # q*. Then by 111, 0.1(3) tp*(C, U f(qâ*,B1) does not fork over C:; i t extends tp*(C, Uf(C)"c,CCy), but is different from tp*(C, U f(q^ c,D1). This contradicts the choice of C:.

Stage F: Conclusion of the proof of Lemma 3.4. We shall show that C, U f(q U C: satisfies the requirements, i.e. q r (C, U f(q U C:) is stationary ins ide4 U 4. Clearly, the power of this set is < K (C, - see the beginning ; C: - see Stage E). So suppose q' E Sm(Dl U 4) extends q r (C, U f(q U C:), does not fork over C, u f(q u C:. Clearly, q' does not fork over C, hence over C, U C:. By Stage E, q' r (Dl U C, U f(q) =

q r ( Q U C, U f(q). By Stage D, q'rB, = qrl?,. But by the construction in Stage C, & E B,, and so q' r& = q r&, and remember q,q' does not fork over C, and C E &.

We want now to prove that q' r (4 u 0,) = q /' (4 u Q), if this fails, for some T,(~,E),T(z,E)')EQ~(A, u D 1 ) , - p ( z , q q ' r ( 4 u 4). By the choice of &,

Rm[qr(4 u ~ , L T , N , I = ~ m [ q r & , ~ , ~ , i ,

i w q r (4 u ~ , ) , T , N , I = ~ l t [ q r & , ~ , ~ , i .

m'r(4 u ~ l ) , ~ , x o ~ = w~&,T,K,I .

As q' does not fork over &,

Let 4 = { T ~ : Z < a} be the set of T E S ; ( ~ ~ ~ Q ~ ) such that (qr&) u T

does not fork over &. It is known that a = Mlt[q rf&,rp, No] < w. For at least one l = I , , (q' r (4 U 4)) U rz does not fork over &, so, -g)(z, q E r z 0 , hence rzo+&, where 4 = {rES;(laeql):qr(A, u 0,) u T does not fork over &}. So 4 is a proper subset of { r l : 1 < a}, hence


MW r (A, u ~ ~ ) , v ~ xO1= IGI < I ~ I = ~ l t [ q rq lv l x,1, Contradicting the choice of 4.

Stage B, q = q‘, so we finish the proof of 3.4. So we have to conclude that q r (A, U Dl) = q’r(A2 U Dl) so by

CONCLUSION 3.5: Suppose ( & : t ~ P ( n ) ) i s a stable system of F:-saturated models, where K = K,(T) ( P ( n ) is { t : t G n = {0 ,1 , . . . , n- l}, It1 < n}, and on a “stable system”, see Def. 2.1). Then A = u t N t i s a good set.

Proof. We prove this by induction on n. For n = O,A = #, so A is trivially good. For n = 1,A = N#, and this is Fact 3.3(1). For n = 2, use Lemma 3.4 with

Do = %,Dl = q o p 4 = ql). F o r n = k + l , l e t D o = ~ { & : t ~ ~ P - ( k ) } , ~ = N , = ~ o , l , ~ ~ ~ , , ~ l , , and

Dl = U { & : t E P ( n ) , t z k}. Now D, = is a good set by 3.3(1), DolDl are good sets by the

induction hypothesis (for Do - clear : for Dl - note Dl = u {& (,) : t E 9 - ( k ) } and (4 : t E P ( k ) ) is a stable system of F:-saturated models). The remaining hypothesis of 3.4 (on Do,Dl) follows by 2.9(2). The conclusion of 3.4 says that A = Dl U D, is a good set.

XII.4. The otop/existence dichotomy

(Hyp) T is stable.

DEFINITION 4.1. T has the omitting type order property (otop) if there is a type p ( ~ , 8, Z) such that for every A and a two-place relation R on A, there is a model M of T, and c i a ~ M for a < A such that:

for any a < p < A : CCRP iff the type p(aa, cipl Z) is realized in M .

Remark. We can think of various variants; for T countable superstable without the dop they will be equivalent. For T stable, for example, it is natural to make z,g into infinite sequences.

DEFINITION 4.2. The theory T has the (A12)-existence property if,

CH. XII, $41 THE OTOP/EXISTENCE DICHOTOMY 609

whenever4 <4(1= 1,2), pendent over 4, there is a primary model over Ml U 4.

11411 = h ( 1 < 3) and {M1,4} is inde-

The main theorem of this section is

THEOREM 4.3 : A countable superstable T has the otop $'(a) or at. least (b) below hold :

(a) T does not have the (&, 2)-existence property, (b) there are &-, < & ( l = 1,2), {Ml,i&} independent over &, and B,

Fio-constructible over Ml U 4 but not atomic over Ml U 4.

Proof of 4.3 is broken down into a series of facts.

FACT 4.4: Suppose condition 3.3(a) holds. Then 3.3(b) holds.

Proof. As the (No, 2)-existence property fails, there are countable 4(1< 3 ) , 4 <&(1= 1,2), {illl,&} independent over 4 such that there is no primary model over MI U 4. Hence there are ii G Ml U 4, and rp such that l=(32)rp(z, &) but {rp(z, a)} has no isolated extension in Sm(Ml U 4) (remember that T and the Mi's are countable). As Tis superstable there is&*EMl U A$ andrp*,v*(z,a*) I- p(z, a), l=(3~&*(2, a*), and {Y*(z, n*)} has no extension over Ml U 4 which forks over a*. By 1.12(1) there is an Fio-isolated ~ , E S " ( M , U A&), such that p * ( ~ , a * ) ~ p . So (b) of 4.3 holds.

DEFINITION 4.5. Let [ be the minimal ordinal such that there are 4) ( 1 < 3) ii*, F = ( c l : 1 < n ) , F* G F and + + ( y , 6) such that:

(1) 4 < J f l , 4 <A&,

(4) F* c @(%,b),

(6) s is R1C@+(2, w, 003,

(2) {Ml,4} is independent over 4, (3) tp(c,,Ml U A$ U { c l : l < k}) is in FLo(&* u {cl: ; < k } ) ,

( 5 ) tp(F*,Ml U 4) is not isolated,

(7) @*,6 are from Ml u 4.

I ACT 4.6: y is well dejhed (and <a), am. w.1.o.g. 6= &*.

Proof. Clearly, by 4.4, condition 4.3(b) holds, and the derivation of an example for 4.5 is easy by 1.11, IV, 1.3(1) and IV, 1.4(2). We can define a* by Ax(III.2) from IV, $1. W.1.o.g. b = &* as we can enlarge both.


FACT 4.7 : We can f ind 1c/+, &(l < 3 ) p , a*, C, C* as in 4.6 such that (a) ~ = C * , p r @ * l - p , ( p , a * ) ~ F i ~ , lettingp= tp(C,Ml UMJ, (b) each Mz is Kl-saturated. Moreover, we can have for some A = A’o,

il&, is a saturated of power A , Ml and 4 saturated of power A+.

Proof. Let M,(Z < 3)p,a*,C,C* be as in 4.5. Choose A = hXo 2 xzt3 1 1 & 1 1 . We can choose saturated M; ( 1 < 3) ,Mz < M,*M$ < M,*, {M,*,M,*} independent over M,*, IIM,*II = A , llM:II = llM,*ll = A+, and for Z = 1,2 tp(M,*,M, U MZ-J does not fork over 4 U M,* and tp(M,*,Ml U 4) does not fork over (choose M,*, then M,*, and lastlyM,* and use II1,O.l). By 2.3(3) (Ml U 4 , M T U M,*) satisfies the Tarski-Vaught condition.

Clearly, tp((c,:l< k),Ml U 4) is Fio-isolated (by IV, 3.2), hence by 1.12(2), for each k < n(Ml U 4 U { c z : l < k}, MY U M,* U {c , : 1 < Ic}) satisfies the Tarski-Vaught condition, hence by 1.9(b) tp(c,,M: U M,* U {c,; 1 < k}) is F&-isolated. So there is a1 E M: U M,* such that for every k, tp,(c,,M,* U M,* U {cz: 1 < k } ) ~ F & ~ ( a ’ U {cz: 1 < k}). By 3.1, w.1.o.g. tp(C*,M,* U M,*) is the unique extension of tp(C,al) which does not fork over &*. So by 1.12(5),

tp(c*,al) I- tp(a*,M? u M ; ) ,

(13, a*) E Fi,.

So we can replace a* by g1 and a* by C.

FACT 4.8: There are countable MF,p, a*, $+ as in Fact 4.7(a) such that M,* E ,M,*(l = 1,2) and M:, M,* are isomorphic over M,*. Moreover, for some countable &,M,* sa&(1 = 1,2) , and 4 realizes p and & is Fio-constructible over M: U M,*.

Proof. LetM,*, p , a* be as in 4.7. Let& be Fio-primary overM,* U M,*, by a construction starting with a sequence C realizing p (see IV, Def. 1.2). It exists by 1.12(1)) and IV, 3.1(5)). By the saturation of the M,*’s, M,* E,M,* S,MZ,M,* G,M,* EOM:, and M,*,M,* are isomorphic overM,* by the saturativity conditions in 4.7. Everything is as required except the countability, but by the Lowenheim- Skolem argument we can find M,*,p, a*, as required.

FACT 4.9: Let &(1 < 3 ) , p , a*, 1c/+ be as in 4.8 (renaming M,* as &). There is MZ such that M: is countable, Ml U 4 G M:,M: is Fko- constructible over Ml U 4 and M: omits p .

CH. XII, $41 THE OTOP/EXISTENCE DICHOTOMY 61 1

Remark. Really for our purpose, “4Fio-atomic over Ml U 4” also suffices, but it is aesthetically nicer to have the Fio-constructible.

Proof. First note that there is no consistent p(~ ,6 ) , 6€M1 U 4 p ( ~ , 6 ) t- p la*. So by the omitting type theorem there is M: omitting p,Ml U 4 E M:. However, we need a special kind of M: so we have to be a little more careful. Clearly, it suffices to prove

SUBFACT 4.10 : Suppose A is countable, p an m-type over some ( jn i te ) F E A with no support over A [i.e. there is no consistent $(z, a, @, ~ E A , $ ( Z , & @ t - p ] and +(3y)p(y ,a) ,a~A. Then some b satisjes p ( y , ~ ) , tp(b,A) is Fio-isolated and p has no support over A U { b } .

Proof As T is superstable, w.1.o.g. {p(y, a)} has no extension over A which forks over a and c E a. We shall define a type q(y) over a, s.t.

(a) for every 6(y, @ for some $(y) ~ q ( y ) , {rESi(A) : r consistent with $(y)} is finite,

(b) there is b realizing q s.t. over A U {b} there is no support for p . (c) p ( y , a ) ~ q ( y ) and q is over a. This suffices. A sufficient (and necessary) condition for (b) is: (b)’ there is a model M,A c M, M realizes q but omits p ;

and a sufficient condition for (b)’ is: (b)” for every ~ E A and formula #(Z,y,d), if q(y) U { 9 ( ~ , y , d ) } is

consistent, then so is q(y) U {$(x, y, d), +(z)} for some $(z) ~ p . Let {9:(~, y, d,,) : n < w } list all formulas with 6 ( ~ , y, d), d~ A,

and {8i(y, Zn): n < w } list all formulas 8(y ,~) . We defined by induction on n < o a formula p,(y, a) s.t.

(‘1 + ( ~ Y ) v ~ ( Y , a), (ii) pJy, 8) = p(y, a), (iii) t - ( v ~ ) [ ~ ) n + l ( ~ , a) +Vn(Y, a119 (iv) g ~ ~ ~ + ~ ( y , a) is consistent with only finitely many rESi ; (A) , (v) either {Q)~,,+~(Y, a),

$(z, @ ~ p , for every p’(y, a) y, d,,)} is inconsistent or for some

+ ( 3 ~ ) [pZ,+l(~, a) A V’(Y, a)] + (39, z)

[ P ) Z ~ + ~ ( Y , a) A v’(Y, a) A ~ O , ( X , Y, dn) A +(z, @I. Clearly, if we succeed, then q(y) = {p,,(y, a) : n < o} is as required :

(iv) takes care of (a), and (v) takes care of (b)”. For n = 0 there is no problem. For n = 2m+ 1 act as in the proof of 1.12( 1 ) . So we are left with n = 2m + 2 . Let 9 = 9;.


If there is q’(y,a) s.t. {q ’ ( y ,8 ) ,qn- l ( y ,a ) } is consistent but {q’(y, a) ,~ , -~(y , a), 6(z,y, Zm)} is inconsistent, then choose q,(y, a) = q’(y,a) A q,-l(y,8) (easily (v) holds, as well as (i) and (iii)). So we assume there is no such 9’.

So, in particular {qnw1(y,a) A i3(z,y,Zn)} is consistent. As p ( z ) has no support over A and a, Z, E A, clearly for some $(z, C) ~ p ,

8) A 8(Z, y, d,) A i$@, C)} is consistent and it implies ~ , - ~ ( y , a), and its parameters are from A. By the choice of q and by (iii), clearly {p*(y)} does not fork over a. Now apply IV, 4.2 with q*(y),a, here standing for P(Z, a), C, A there and get q+(y, a) (standing for $(z, 6)) s.t.

(A) {p*(y), lq+(y,a)} forks over 8, hence as q* (y ) t -q(y, a) and q ( y , a)’s choice, {q*(y) , l q + ( y , a)} is inconsistent so q * ( y ) l-p+(y, a),

(B) for every W(y, a’) (a’ E a), q * ( y ) I- W(y, a’) implies q+(y, 8) I-

By (A) we can replace “implies” by “iff”. We let q,(y,a) = p+(y, a), and as (iii) holds by (B) (as q* (y ) l-r~~,-~(y) by its definition) and (i) is immediate by (A) (as q* (y ) does not fork over A) we have just to prove that the second possibility in (v) holds. Suppose {q,(y, @),q’(y, 8)} is consistent, and it is enough to prove {qn(y, a), ~ ’ ( y , a), S(Z, y, J,), +(z, C)} is consistent. If not, then

def {q , - l (y , a), e(z, y, ?t,), l$(z, C)} is consistent. SO q*(y) = (32) [qn-l(y,

W(y, a‘).

qn(y, a) A v’(Y, a) I-- ( 3 ~ ) Y, dn) A -$@, GI, hence

hence

hence (by (B))

i.e.

Pnb, 4 A v’(Y, 8) ~-T*(Y) ,

P*(Y) I - l ( V n ( Y , 8) A ~ ’b , a)),

P+(Y, a) k l ( q n ( Y , 4 A v’(Y, a)),

q+(y, a) E-.(q’(y, a) A q’(y, a)), hence

{q+(y, a) A q’(y, a)} is consistent, contradicting the choice of q’(y, a).

THE CONSTRUCTION 4.11: Leti&(Z < 3)M$,@+(z,a*),p~S~(M, U 4) be as in 4.9, and let f be an isomorphism from Ml o n t o 4 over 4. Let 8: = a* f~ i& and w.l.o.g.f($) = @, t p (q ,&) does not fork over at.


Now let A and a two-place relation R on A be given. We shall define

For n = 0 we let N+, =A$,. For n = 1 let us choose for each i < h a model

by induction on n, for every t c A, (ti = n, a countable model 4.

isomorphic to Ml over&, ft an isomorphism from Ml onto = the identity, and {qf): i < A} is independent over 4. We let at = fi(a:),

such that f f

f; = f f of -l:4 +-4*), P,J = (f, u f2 (PI for a < 8. For n = 2 we have to define qu,s) for a < 8. We separate two cases: Case a: aR8. We can extend (f, U f;) to an elementary mapping

Case b: Not &/I. We can extend (f, U fj) to an elementary

For n > 2 : We let 4 be F1,o-constructible over u {N, : s E t , s # t } .

f , , s from 4, and we denote its range by qu,s).

mapping f',@ from M: and we denote its range by qu,s).

We call u {4 : t E h finite} by MR.

FACT 4.12 : For every finite t G A, It1 > 1 , and F E ~ , tp@, u {& : s c t , s # t } ) is Fko-isolated.

Proof. Quite clear.

FACT 4.13: (4:t c A , It1 < KO) is a stable system (see Def. 2.1).

Proof. Let { t l : i < a} be an enumeration of {t c h : t finite}, such that ti c t j implies i < j and for i < A , t l+, = {i}, and to = 8. We prove by induction on

=

y+ 1,y 2 A ; by the induction hypothesis and 2.3(2) (u{N,: s G t,, s # t ,}, u {qt : i < y } ) satisfies the Tarski-Vaught condition. Also by 4.12 for every F E ~ , tp@, u{N, : s E ty , s # t,}) is Fko-isolated. So by 1.2(3) this type has a unique extension over uf<,,4,, hence this extension necessarily does not fork over u {N, : s E t,, s # t ,}. So we finish 4.13 by 2.3(1).

< a that (4: t € ( t f : i < 8)) is a stable system. There are no problems for /I = 0 or limit or /I < A. So let

Remark. Observe we really have proved:

CLAIM 4.14 : If I .is a family of finite subsets of A, s E t E I - SE I , N,(sE I ) models, {q,,: {a} E I ) independent over N,, s E t *N, E 4, and t ~ l A It1 > 1 * tp,(q, u {N,: s E t , s # t}) is FLo-atomic, then {N,: S E ~ )

is a stable system and tp,(&, u {N,: s E t , s # t } ) I- tp,(&, u {&: t $ s a ) .


Obvious, as f a U f j is an elementary mapping, and 4.7(a) (which is inherited by 4.8, 4.9).

THE MAIN FACT 4.16: MR realizes pa,! r (Q U a;) if aR@.

Proof. If dip, then pa,! r (a: U a;) is realized by fa,&) for any C E ~

realizing p (C exists by the choice of k$). We now deal with the difficult half. Suppose "not dip". Then

qa,!} omits pa,!r (Q u a;) as M,* omits p ra*. We now prove by induction on n, 2 < n < w , that if {a, p} c t E A ,

It1 = n, then I$ omits pa,! (this suffices by 4.15). For n = 2 we know it. Suppose we have proved for n and we shall prove for n+ 1. Let {a,p} E t E A , It1 = n+ 1. Choose yEt-{a,p} and let s = t-(7). So pa,! is over N , and N , omits it. We also can easily show that & is Fie- constructible ove r4 U qy) (by 1.9(b), 1.12(2) and as any F&-isolated type is Fko-isolated). We also know that N, < N,,qyl and {&,qy)} is independent over NB. Let A = {dEI$:R[tp(d,N, U q y ) ) , L , 001 -= 0. However, if J E A , tp(d,N, U qy)) is isolated (for otherwise we get a counterexample to the choice of 6 in Def. 4.5). So A is atomic over N , U qyl. As 4.8 says 4 Gayy) by XI, 3.9(1) N, &,A.

Also, if a€&-&, l==llr+(d,a: "a;) ($+ - from 4.7 (and 4.5)), then r = tp(d,N, U qyJ, being realized in an Fko-constructible set over N , U qy), is also Fko-isolated ; hence for some formula 8* in it, 8*(x) I- tp,(d,N, U qy)) for 8(x , y) = [z = y]. Now r forks over& [otherwise it is finitely satisfiable in N,, hence 8* is satisfiable in N,, say by d' E N , , hence 8*(x) If x # d', d' EN, contradiction] ; hence,

R"tp(d,N, u qy}) ,L, 001 < Rl[tp(d,N,),L, a1 < Rl[=llr+(x,aZ "a;),L, co] = c.

We can conclude that (as N, E A )

$+(&,a: " a;) E A .

So if pa,! is realized in &, any sequence realizing it is included in A. But by XI, 3.9(1) 4 G a A , so if in A there is a sequence realizing pa,! r (a, U a!), then (remembering am U ap E 4,) U qfl & N , ) there is such a sequence in N , ; contradicting the induction hypothesis. So we have proved 4.16 and hence Theorem 4.3.


LEMMA 4.17: Suppose that (b) of 4.3 fails, T is a countable superstable without the dop (for (3) superstable without dop sufices).

(1) If A!, G M1,4, { M l , 4 } is independent over 4, and A is Fko- constructible over Ml U 4, then M is atomic over Ml U 4.

(2) If 4 are as above, M is Fko-constructible over Ml U 4 (and Ml U i&. EM), then M is minimal over Ml U 4 (i.e. there is no M*,

(3) If 4 are as above and are F;-saturated and M is Fio-constructible Ml u 4 r M * cM).

over Ml U 4 ( M l U 4 c M), then M is F;-saturated.

Proof. (1) We can find M Fi-constructible over Ml U 4, A E M; so w.1.o.g. A = w. Also if there is a counterexample, there is a countable one (by the Lowenheim-Skolem argument). By 4.3(b) (and the countability) there is M* c M primary over Ml U 4, hence atomic over Ml U 4. So if M is not atomic over Ml U 4, necessarily M # M*. So it suffices to prove (2).

However we first prove (3). (3) We can find an F:-saturated model M+ F;-constructible over

Ml U 4 , M C M+. If M = M+, we can finish, so assume M # M+ and choose CEM+-M with R[tp(c,M),L, co] minimal. By X, 3.3 tp(c,M) is not orthogonal to some M,, but 4 E .M+ trivially, so by XI, 3.11 there is CEM+-M such that tp(c,M) does not fork over 4, a contradiction (say to V, 3.2).

(2) SupposeMJ < 3)Mare a counterexample, Ml U 4 E M* c M , and let A > llMll+ (TI be a regular cardinal. We can find F;-saturated (= A-saturated) &(l < 3) such that 4 E 4, tp,(N,,M) does not fork over A!,, tp,(q,M U &) does not fork over&, U &, for 1 = 1,2. Now by 2.3 (Ml U 4 , N l U N,) satisfies the Tarski-Vaught condition, hence by 1.2(3) M is Fko-constructible over y U N,, hence there is N , Fko- constructible overN, U N,,M U Nl U N, G N. By (3)Nis F;-saturated. Now let ( c i : i < a) be a maximal sequence such that cf EN, ci # {c,: j < i}, and tp(c,,M* U & U N, U {c,:j < i}) is Fko-isolated. Clearly, a < IINII+. Also B = M* U Nl U N, U {cf : i < a} is the universe of some N* < 6, because every {cp(z, E)} (6 E B) can be extended to an Fko- isolated complete type over B, and this type is realized in N (as N is F;-saturated). Now for every ~ E B , tp(6,M* U U N,) is Fko-isolated, whereas (Nl .U N, U M*,Nl U N, U M) satisfies the Tarski-Vaught condition [because tp,(Nl U N,,M) does not fork over M*] . So N* n M = M* fl M = M*, hence N* #Nand by (3) N* is F;-saturated. So N,,N,,N,,N,N* contradict ‘‘Tdoes not have the dop” (see Def. X, 2.1).


Remark. Why have we not assumed in 4.3 “T does not have the dop”, thereby simplifying the proof? As the present 4.3 says more, e.g. it may help to prove “if T is superstable with the otop, then I ( K o , T) = 2’0”.

XII.5. From the (KO, 2)-existence property to the ( A , 2)- existence property

(Hyp) T countable without 4.3(b), T stable.

DEFINITION 5.1 : We call Y = (M, : s €1) ( I E 9 ( n ) is closed under subsets, n 2 2 ) a sp. stable system if

(a) ( M , : s e I ) is a stable system (see Def. 2.1),

(c) u,,,M, is F&-atomic over Jf,I-~oj U & I - { l f ,

(d) if SEI, (0, i} E s, then M, is Fio-constructible over A:,

(b) U I - {0} €1, U I - { 1) €1,

where we let A: = u{&:t~P(a)} .

Remark. Note that for I = 8-(2) , a sp. stable system is exactly a stable system.

DEFINITION 5.2: T has the (h,n)-existence property (for sp. stable systems) if for every sp. stable system Y = (M,: s € P ( n ) ) , A = CeeI llM,Il, there is over A: an Fao-primary model.

DEFINITION 5.3: A true sp. stable system Y = ( 4 : s ~ I ) is a sp. stable system which satisfies

(e) if seI ,{O, l} E s, then 4 is Fao-constructible over u,,,& (hence M, is prime over A:).

DEFINITION 5.4: (1) T has the true ( A , n)-existence property if for every true sp. stable system Y = ( & : s € P ( n ) ) , A = CseI llM,Il, there is over u {& : t E P ( n ) } an F&-primary model.

(2) T has the strong (Ko,n)-existence property, if for every true sp. stable system Y = (1M, : s~B- (n) ) , in which each M, is countable every Fio-constructible set over A: is atomic over A:.

FACT 5.5 : (1) Every true sp. stable system is a sp. stable system.

property. ( 2 ) The ( A , n)-existence property implies the true ( A , n)-existence

CH. XII, $51 THE (No, 2)-EXISTENCE PROPERTY 617

(3) I n Def. 5.1, 5.3(e) implies (d) and (d) implies (c). (4) If Y = ( M 8 : s € I ) is a sp. stable system, (0, l} E 8 6 1 , J c I i s

downward closed, s - {0} , s - { 1) E J, then Ma i s Fao-constructible over UtEJ&. If Y i s a true sp. stable system Ma i s F$o-constructible over U t E J q . Also, UaelM8is Fio-constructible over MUI-io, U MU I-il, .

(5) The strong (No, n)-existence property implies the true (Ho, n)- existence property.

Proof. Easy (in (3), (d) implies (c) as 4.3(b) fails and use (4)).

CLAIM 5.6: (1) T has the (N,,n)-existence property iff for every sp. countable stable system Y = (M, : s E P ( n ) ) , every type {cp(z, a)} over A: can be completed to a n isolated complete type.

(2) Similarly for the true (No, n)-existence property.

Proof. Immediate.

CLAIM 5.7: Suppose Y = ( M , : s ~ q - ( n ) ) i s a true sp. stable system h = EllM,II > No, Then we can define M;,a < A , such that

(a) I[M;II < KO + lal,M; i s increasing continuous with a , (b) M; n & = M t n t ,

(c) tp,(M;, u,, J&) does not fork over Ute JM; for every J E S-(n), (d) Y” = (M;: s E q - ( n ) ) i s a true sp. stable system, (e) Y a w p = ( M ; , f l : s ~ P ’ - ( n + 1)-{n}) i s a true sp. stable system,

where M;*p i s M; i f n $ s and M!-{,,) i f n E s , if /3 = h let M{+, = M8-(nl.

Proof. Easy: for (d) check the definition of a stable system and for (e) use 2.3(1) for an enumeration ( s (1 ) : l < 2,+l-2) such that s(21) E 9 - ( n ) , s(21+ 1) = 421) U {n}. For checking Def. 5.3(e) (hence Def. 5.l(d)) use Theorem IV, 3.3. For 5.l(c) use IV, 4.3.

LEMMA 5.8: Suppose n < o ,h > No and T has the true (No,n)- existence and the true (p, n+ 1)-existence property for every p < A , then T has the true ( A , n)-existence property.

Remark 5.8A. Lemmas 5.7 and 5.8 remain true if we omit the “true ”.

Proof. Let ( M , : s e P ( n ) ) be a true sp. stable system, llMaII E A. We should find a primary model over UM,. We use 5.7 and its notation.

618 THE MAIN GAP FOR COUNTABLE THEORIES [CH. $5

We define by induction on 01 < h an ordinal 5, and at(UB<a5B <

(1) for i < Q, tp(a,, A: U {a , : j < i } ) is isolated; moreover, some

(2) A$ U {aj : j < Q} is the universe of a model Mu. For u = 0, we know that Y o is a true sp. stable system. As 11x11 = No there is a primary model over A z ' , G and let =

A:' U {at: i < c0} be such that tp(a,,A:" U {a,: j < i } ) is isolated (e.g. 5, = w is necessarily alright). This gives (2) and somewhat less than ( i ) , but by 5.7(e), and 2.3(2) (A:',,:) satisfies the Tarski-Vaught condition, so (1) holds. For 01 limit let 6, = uB<a[B and there is nothing to do.

So let 01 = /3+ 1, it is easy to check that we can extend the system Y " - S by M, =Ma for s = (0,. . . , n- l}. We get a stable system by 2.3( 1) [let (0, . . . , n- l} be the last s] and it is a true sp. stable system - use 5.5(4).

Apply the true (la1 +No, n+ 1)-existence property (as above (1) holds by 5.7(e), 2.3(2)).

Finally, UachNa is a primary model over A:.

i < Q) such that

formula over A: U {a,:j < i } isolates it,

LEMMA 5.9 : Suppose T has the strong (No, m)-existence property for m 6 n. I f 9 = ( 4 : t ~ I ) is a sp . stable system, S E I , { O , 1) c s,lsl 6 n, then M, is atomic over A:.

Proof. If there is a counterexample, w.1.o.g. for some m 6 n, I = 9(m) , s = m, m is minimal and IlxII = No (as in the proof of 5.7.) As m is minimal, for every t G s, t # s, (0, l} c t implies 4 is atomic over AT, hence by the countability N, is primary over A;. So (4 : t E

9 - ( n ) ) is a true stable system. As @ is Fio-primary over A:, and the strong (No, m)-existence property holds, clearly ik& is atomic over A:, a contradiction.

Notation 5.10: Let n* be the minimal n for which the strong (No, n)- existence property fails if there is such n and w otherwise.

FACT 5.1 1 : n* > 2.

By the hypothesis of the section "T countable stable and condition 4.3(b) fails".

CLAIM 5.12 : Suppose T is superstable without the dop. Then n* = w.

CH. XII, $51 THE (Ho, 2)-EXISTENCE PROPERTY 619

Remark. Really "without the dop" is not necessary, this will be shown in a subsequent paper.

Proof. Suppose n = n* < w , and let S = (M, : s E S - ( n ) ) be a counterexample, i.e. ( M , : s E T ( n ) ) a true sp. stable system, llM,II = N,,M, is Fko-constructible but not atomic over A;.

Let I = P ( n + 1)-{n}. Suppose h = ( P o ) + .

FACT 5.13 : We can dejine M,(s E I - P ( n ) ) such that (a) ( M , : s E I ) is a sp. stable system, (b) if s E I - T ( n ) , then M, is A-saturated (in fact F:-prime over

U { w : n E t c s} when s # {n} ) .

Proof. Let I = {s(Z): 1 < 2"+l-2} be such that s(l) c s(m) * I < m, and P ( n ) = {s(l) : I < 2" - l}. So M&(l < 2" - 1) are defined. We define the rest by induction on I , such that the relevant parts of (a) and (b) hold.

Suppose we have defined for every m < 1. If O $ s(l) or 1 $s( l ) there is no problem, so assume (0, l} c s(Z). As ( w : t f z F ( s ( Z ) ) ) is a sp. stable system, Is(Z)( < n*, J&-(") is atomic over Utcs(l,-{n& (by 5.9) and (Utcs(z)-jn)&, Utcs(l)-ln& (")) satisfies the Tarski-Vaught condition. Hence by 1*2(3) tp*Ws(z)-{n), Utcscz,-{n)&) I- tP*(q(t)-(n), U t c s ( t ) - ( n l & u (n)) and SO (remembering II&(z)-(n)II = Ho)%(l)-(ni is Fko- constructible over Utcs(z)& and so we can find which is Fko- constructible over Ulce(z)&, hence over u(@: tcs(Z) , nEt}. By 4.17(3) it is A-saturated and by 5.9 atomic over u(4: t c s ( Z ) , n ~ t } . So the relevant part of (b) holds, and also (a) is easy, as w.1.o.g. tP,(M,(Z), um<14(m)) does not fork over Utc,(l)&.

So we have proved 5.13. Let us continue with 5.12. Let for sEB-(n),M,* = M,u(n), so (M,* : s E P ( n ) ) is a stable system,

each M,* is h-saturated, and by 2.3 (u,&&,u,M,*) satisfies the Tarski-Vaught condition. Hence by 1.9, 1.12(2) for some F E X , tp(c, u,M,*) is not isolated, tp(c,, u,M,* U {cl : Z < k } ) is Fko-isolated. By 3.5 tp(c, u,M,*) is the unique extension in Sm(UsM,*) of tp@, 6) which does not fork over b for some b C u,M,*. By 1.12(5) for some d~ u,M,* : tp@, d) I- tp(C, u,M,*), (tp(C, u,M,*), d) E Fie, hence tp(c, d) is not isolated, tp@, u,M,*) does not fork over d.

Let for s E P ( n - l),M,** = M* 8 u ("-1). Then ( M , * * : s E P ( n - l ) ) is a sp. stable system. Let t ( *) = n - 1 ( = (0, 1, . . . , n- a}). By 5.9, M$,


is atomic over u {M: : s E 9-(n - 1 )}, hence even u {M:, In-1) : s E

S-(n- l )} by 2.3, 1.2(3) hence M& is atomic over ~{M:*:sE T ( n - 1)). So, clearly u {M:* : s E S - ( n - 1 ) ) u d is Fio-constructible over u {M,** : s E S - ( n - l)}, and by the previous paragraph tp@, u {M:* : s E P ( n - 1 ) ) U d}) is Fi,-isolated. Together, u {ill,** : s E 9- (n-1)) U cnZ is Fin-constructible over U{M,**:sE9"-(n-l)}. By 5.9, tp(c"d,U{M,**:sEP(n-l))) is isolated; hence (byAx(V.l) for Fa,) t_p(c, u {M,** : s E F ( n - 1 ) ) U d) is isolated ; as it does not fork over d , also tp(c, d) is isolated, contradicting the previous paragraph.

CONCLUSION 5.14 : Suppose T is countable superstable and without the dop and the otop. Then T has the true sp. (A,n)-existence property for every A , n ; hence it has the ( A , 2)-existence property for every A.

X11.6. The book's main theorem

THE MAIN GAP THEOREM 6.1 : Let T be countable.

the otop, then for every uncountable A , I (A , T ) = 2A.

then for every a > OI(N,, T ) < 3,Jlal).

(1) If T is not superstable or ( is superstable) deep or with the dop or

(2) If T is shallow superstable without the dop and without the otop,

In fact, if Dp(T) 2 w I ( N , , T) = 3Dp(T)(lal), and i f Dp(T) is Jinite, then 3 D p ( T ) ( 1 4 ) T) G 3Dp(T)(lal(a19.

Remark 6.1A. In ( 2 ) also every model of T is prime and even primary over some non-forking tree of models ~ ( v E I ) , < 2Ko.

Proof. ( 1 ) If T is not superstable, this was proved in VIII, $2. If T is superstable but with the dop this (and more) was proved in X, 2.5, IX, 1.20, and if T is superstable with the otop the same proof works. If T is superstable and deep, more was proved in X, 5.1.

(2) By 4.3, condition 4.3(b) fails, so the hypothesis of Section 5 holds. By 5.14 the ( < co, 2)-existence property holds.

By XI, 2.4 for (Tko, E @) for every modelM there a non-forking tree { q : y ~ f ) of models, 11q11 < 2'0 so that M is prime over u,,,N,. So exactly as in X, 4.7 we get the upper bound (by X, 4.44) the depth is the same) and by X, 6.1, we get the lower bounds (for finite depth by the same proof, without marking levels).

CH. XII, $61 THE BOOK’S MAIN THEOREM 62 1

This way for Dp(T) finite we get only I(&, T) < &,p(T)(Icz(’~). Using also XI, 2.16, we can in (2) have (al’l instead of JaJ’r.

LEMMA 6.2 : If T is superstable without the dop and with the existence property, then the otop fails.

Proof. For such T we have a structure theory, contradicting the otop.

CHAPTER XI11

FOR THOMAS THE DOUBTER

XIII.0. Introduction

Let T be countable in the introduction. The aim of this chapter is to justify the assertion that XII, 6.1 is

“the book’s main theorem”. For this we deal with some related problems and show that we can solve them too.

In Section 1 we show that if T is superstable, shallow without dop and otop, then any model can be characterized by up isomorphism by generalized cardinal invariants of countable depth. If T is superstable deep without dop and otop, then we show a weaker result is the best possible. For other theories, strong negative results appear in [Sh 871.

In the second section we show that we can essentially compute IE(h, T) for countable T. The main difficulty is the case T superstable without the dop but with the otop. Why is there a difficulty which does not occur when T has the dop ? There the embedding preserves “dim ( I ,M) 2 A” (but not necessarily “dimI,M) 6 A ” ) ; however, we know quite accurately which indiscernible sets have large dimension (the and some types not orthogonal to 8) . Here realizing a type is preserved but not necessarily omitting a type ; and working just withEM(1, @), whereEM(1, @) I= [(3@ A p(Z, as, @t]if(8<t),

is not enoagh. So we repeat an analysis like the one in Section 4 of Chapter XII. So we have countable models q 5 ) ( i < A ) independent over A!,,, and we use an Fko-primary model M on their union. We look for i < j, only at types realized in {a: tp(c,qi) U q3)) has rank < t}, with 6 minimal so that continuum many types may occur. The result is that we have 2 N o possible types, but in M only countably many types are realized. So i t makes sense that if for each pair {i},{j} we make a random choice we shall get many pairwise non-elementarily embeddable models, and this we do (so the section has a combinatorial part).

622

CH. XIII, $13 MODELS CHARACTERIZED BY INVARIANTS 623

In Section 3 we prove the Morley conjecture : the function I (h , T) A 2 IT1 is non-decreasing in h except for T categorical in N, but not in No,A > p = No. Our strategy is that for many kinds of T we can compute ] ( A , T) for A > No and then see that it is non-decreasing (as 2’0 < I@,, T)). The remaining case is T superstable shallow of finite depth, without the dop and otop. Then in the main case we can decompose any model M of power p : i.e. it is prime over a non- forking tree, find a large splitting and then add more copies to get a larger tree so that the prime model over i t has power A. So we found a function from {M: IlMll = p} to { M : llMll = A}, which is near enough to being one-to-one, i.e. to preserving non-isomorphism to prove our conclusion.

If we cannot find a good enough splitting in the tree, we prove I(A, T) 2 21IMll for A 2 IlMll.

In Section 4 we compute I(&, T) for a large enough; this involves some finer analysis of the trees (mainly 4.4) and the result is 4.1 1. We finish in 4.15 proving that, e.g. I (Na , T) > 3, for some a implies I(&, T) 2 la + 1 I for every a.

XIII.l. Can the models be characterized by invariants?

In this section we try to show that in some sense a “set of invariants for models of T, which characterize each model up to isomorphism ’’ is possible for suitable countable T. The point is of course that for countable theories which are not “suitable” there are complementary results.

One such invariant is the L,,,-theory of a model of power A > 2’0 which is sufficient for T superstable without the dop and otop. For a model of power A, its L,,,-theory looks a very strong invariant, in particular the quantification may look too strong. So the negative complementary results (see [Sh 871) seem more convincing. How- ever, we then strengthen the positive results by replacing Lm,A by Lm, p(Qt+))p<A, where Q,” is a simple generalized quantifier, similar to a cardinality quantifier (saying a dimension is 2 p ) .

We may still complain on the use of sentences with arbitrary depth, but we can prove that this is necessary if T is deep. On the other hand, if T is shallow, then we can consider only sentences of small depth (a bound is determined by the depth of T).

624 FORTHOMAS THEDOUBTER [CH. XIII, $ 1

THEOREM 1.1 : Suppose h > IT1 + ZXo, and T is superstable without the dop, with the ( <~,2) -ex is tence property. Then any two Lm,A- equivalent models of T of power h are isomorphic.

This will follow from the stronger results but for this we have to define Q:.

DEFINITION 1.2 : Syntactically, Q; operates in the following way for a<o:(Q,BxO,xl ,..., x , . . . ) , ~ ~ (q,(xo ,..., x ,_ , ,g , ) :n<a) is a formula if each qn(x0 , . . . , x % - ~ , g,) is the quantifier bounded by the variables xo,xl, . . . , and

MI= ( Q f , xo, xl, . . . , x,, . . .),,, (qn(x0, . . . , xn+ c,) : n < a )

iff for every set A E M , IAJ p. In fact, this is the only way we use it, so we can demand in the definition that the q,’s define a dependence relation (as in V, 1.14). In this case if x > N,,L,,,(Qf) c d(LX,,(3”’x)) where 3 2 x “means” there are 2 p x ’ s such that . . . , and on the A-closure see [MSS 761.

CLAIM 1.3: Lm,K(Q,B+)PcA c Lm*A when K < A, i.e. every formula in the first logic is equivalent to one in the second.

Proof. Trivial.

THEOREM 1.4 : Suppose h > IT1 + 2’0 and T is superstable without the dop and with the (<~,2) -ex is tence property. Then any two Lm, ,:+ITI+(Q:+)p<h)-equivalent models of T of power h are isomorphic.

QUESTION: Can we demand only Lm, ,TI+(Q,B+),<A-equivalence ‘1

Proof. We shall use (Ti,, G go, xo)-decomposition inside the models. Let llMIII = 11411 = A,M,,& are Lm, ,:+ITI+(Q;lS+),,,-equivalent models. We shall define by induction on n a set 1, c A, models P?, ~ ( T , I E

I,-u, , ,Im,l = 1,2) and a function F, such that: (1) P9 C&. (2) (h$ $: T,I €4) is a (Ti,, c:,, ,o)-decomposition inside & (see

Def. XI, 2.4, 2.5).

CH. XIII, 4 13 MODELS CHARACTERIZED BY INVARIANTS 625

(3) (Ni , $ : 7 EI,) is maximal in the following restricted sense : there is no q ~ I , , l ( q ) < n and a, v = q - ( a ) # I , and@",@, such that (Nk, $: ~ € 4 U {v}) is a (Tio, decomposition inside Ml.

(4) F,, is an elementary mapping with domain ulernq, mapping 3 onto q.

def

(5 ) . . . 7 c, * * * , ) c s N : is L,,,:+ITI+(&r+)r<,-equivalent to (4, . * * 9

F n ( C ) , . ) c s N f .

( 6 ) F, E: F, for m < n. Clearly, if we succeed, then (@T:7~u,,<,IJ is a (Tio, ~ i ~ , ~ ~ ) -

decomposition o f q , so by XI, 2.8& is prime over u {@? : 7 E u, ,,I,}, and u,,,F, is an elementary mapping from u {q : q ~ u , I,} onto

So we can find an elementary embedding F of Ml into 4 extending u,,,F,. But X I , 2.8 also says that A!& is minimal over u {q: 7 E

u,,,z}, hence F is onto 4. So MI,&& are isomorphic (alternatively we use the uniqueness of prime models).

For n = 0 there is no problem. So suppose we have defined for n (and all m < n) and shall define for n+ 1. Clearly, it suffices to prove

U {q : 7 EU, 4).

FACT 1.4A: Suppose ~ € I , , l ( q ) = n , p € S " ( q ) is a regular type orthogonal to 3- ( i f it exists). Then we can jind a and W f l , a j , P f l , ~ j (/3 < a such that:

(A) tp(dfl,P1) is regular not orthogonal to p . (B) {$: /3 < a} C Ml is independent over (C) {$:/3 < a} is maximal (under conditions (A), (B)). (D) @q U $ G Pfl G M , tp*(@fl,Pv u $) is almost orthogonal to flq. (E) Pfl c A C M , tp,(A,N", U 5) almost orthogonal to Pq implies

N"a io, x,A *

(F) I IqI I < ITI+2No. (G) For each /3 < a, F, r q can be extended to an isomorphism Ffl

from A$ onto 9.

equivalent. (H) (Mi, . . . , C , . . - , ) , , N ; (M, ,...,F f l (c) ,..., )ct~;areL,, ,:+~Tl+(V,B+),<,-

Proof. Let 4 = define by induction on y < y ( * ) r: so that

type over in Ml.

tp(@,N1,) is regular not orthogonal to p} . We

is the set of & € I , realizing some complete L,, ,:+ITI+(Q~+)p<l- (i)

(ii) The dimension of r: over (uj<yq,q) is < A.

626 FOR THOMAS THE DOUBTER [CH. XIII, $ 1

There is no problem to do this and to define corresponding $. For each y we can find maximal subsets of q, independent over (u,,, q,flv), {q,a:/3 < ut}. We can assume = u; (all call i t ay) by (H) for n.

By the definition of the q ’ s , the dimension of 4 over (U,<,(*)G, @,) is A or zero. In the first case let {%: /3 < A} be maximal independent (over (uley(*) G,Nf,)) subset of 4.

We now define by induction on /3 < A $(l = 1,2) such that (i) ~ $ E J , $ does not depend on Uy<,(*)l)‘ U {$: 6 < /3} over N : .

(ii) (MI, . . . , C, . . . , $4,,,: and (4, . . . ,J’,(c), . . .

(iii) F depends on {at : [ < 2/9} U uy<y 4.

If we cannot continue, we contradict the maximality of y(*) . If the dimension of 4 over (u,,,,,,~,lVl,) is zero, we do not define any 4.

After doing it we have defined dp(/9 < a) as required (with a different indexing) and there is no problem defining Pq.

areLm,,:+l,l+ (Q,”.),< ,-equivalent.

(iv) bs 4 depends on {q: [ < 2/3+ l } U d;<,(*) 4.

THEOREM 1.5: If in 1.4 we assume in addition that T is shallow, y = 2Dp(T), then any two Lm, (~,)++ITI+,y(Q~)/l,y-equivalent models of T of power A are isomorphic, where:

DEFINITION 1.6: Lm,A,y(&f)per is the set of sentences ofL,,,(Qf),,, of quantifier depth < y .

Proof. The same as the proof of 1.4 but for each r] we demand only :

2Dp(i~~,i’(-) if l ( g ) > 0, yq = 2Dp(T) if 1(q) = 0. ( 4 7 C ) c e N o , ( 4 9 . f q ( ‘ ) ) c ~ N ; are Lm,3:+(TI+.y(q) where yq =

THEOREM 1.7 : In 1.4 if we restrict ourselves to Fi0-saturated models, then we can omit the requirement that “ the ( c co ,2)-existence property ” holds.

Proof. Just use X, 2.1 instead of XI, 2.8.

THEOREM 1.8 : Suppose T is deep and superstable, A > No. Then the L,,,-Scott height of models of T of power ,u 2 A + A ( T ) can be any ordinal <p+.

Remark. If ,u 2 A++A(T) of course there are L,,,-equivalent non-

CH. XIII, $21 ON HAVING MANY MODELS 627

isomorphic models of power p, in fact F:-saturated (except, of course, when T is unidimensional).

Proof. Let a < p+ be limit; it is known that there are 6 < 5 < p+, t > p, such that (6, <), (5, <) are L,,,,,-equivalent (but of course not La,,-equivalent) (by Kin0 [K 661).

Let I = { r ] : 7 a strictly decreasing sequence of ordinals and q ( O ) , if defined, is 6 or 0. So in ( I , d ) , ( 6 ) , ( c ) satisfies the same La,,,=- formulas.AsTisdeepthereare4, a,, s.t. 141 = No,& c a,EA+, t p (a , , 4 ) is orthogonal to stipulating, A-, = 9, and tp*(.4+1, 4 U {a,}) is almost orthogonal t o 4 for n > 0. We can find elementary mapping ~,(T,IEI), Dom f, = L&,,, fVrk E f,, such that letting A, = Rang(!,), (A,,: r] E I ) is a non-forking tree. Let M be Fgo-prime over u,,,%. By [Sh 871, letting A1 = {b , : n < w } , the sequences (jct>(b,) : n < w ) and ( ftt>(b,): n < w ) realize the same L,,,,,-type in M ; but as in X, 5.1's proof, not the same L,.,-type.

XIII.2. On having many models, no one elementarily embeddable into another

THEOREM 2.1 : Let T be countable. (1 ) If T is not superstable or superstable with the dop or otop, then for

every regular h 2 Zn, (as in 2.9) IE(h , T ) = 2,. (2) If T is superstable without the dop and otop, but is deep, then for

every A , h < K~ (=smaller than the first beautiful cardinal (see Silver [Si] ; the relevant facts are summed up in [Sh 82, Theorem 2.4, p . 1941) IE(h , T) = ZA, and for every h 2 K ~ I E ( A , T) = K;.

( 3 ) If T is superstable without the dop and otop, and is shallow, then for every h,IE(h, T ) < I,,(,, < awl. Proof. Most parts appear in Chapter X or have similar proofs. The only missing part is (1) when T is superstable without the dop but with the otop. As the dop fails by XII, 6.1 the (< 00,2)-existence property and even the (No, 2)-existence property fail.

Reflection will explain the difference with the non-dop: there we know exactly which types have uncountable dimension, whereas here we do not know when p(z,a',a' ') is realized. So we have to rework somewhat XII, $4.

We assume the following combinatorial principle (and prove the relevant cases later).

628 FOR THOMAS THE DOUBTER [CH. XIII, $2

DEFINITION^.^: (CP,,x,p,K,B ). There are ordered sets4 3 ( A , < ) (a < x) each of power A, and g, a two-place function from I , to 19, such that :

unembeddable into IF, where +(. . . (xi,yt), . . . ) l < B = “the ga(xt,yt) are distinct for i < 8’’. This means: let L:,K be a vocabulary {Fa, i : a < p, i < K } Fa,( is i-place M,,.(IB) is the free L:,.-algebra which {t : i E IB} generates, so we demand :

(*)

Suppose a # p < x (see [Sh 831), 1, is +((xo, yo), * * . , (x<, y*>, * * -)(<a-

iff:Ia+Mp,K, then we can find (a i ,a i : [ < 9.) in I, s.t.

(a) f (a t ) = ~ ‘ ( y ; ) , T~ an L;,.-term, and for [ < 5 < 0, $-y;,ylny; realize the same atomic type over {i : i < 8) in IB,

(b) (g,(ai,ai):[ < 8) is with no repetition.

LEMMA 2.3 : Suppose (CP,,,, ,,, T is countable superstable without the dop, but with the otop).

N,). Then IE(h, T ) 2 x (assuming

Remark. We can get the conclusion also for T with the independence property or with the dop (T stable).

The following is closely related to XII, 4.4.

DEFINITION 2.4: Let 5 be the minimal ordinal such that there are % ( 1 < 3)a*€M1 U K , p , , ( q ~ ” 2 ) and f(y,@*) such that

(1) 4 i % ( l = 1,2) and {Ml,&} is independent over 4. ( 2 ) P,ES”(M, U 4) are all Fio-isolated, not isolated and distinct. (3) If F realizes p,, then F E ++(a, a*).

(5) Each p , does not fork over a*. (4) 5 = R [ f ( y , @*I,& 001.

FACT 2.5: (1) is well defined (when T is countable, superstable without the dop but with the otop).

(2) In the definition, w.1.o.g. each 4 is countable and 4 G,#~ and Ml is isomorphic to 4 over 4. (Of course 6 is well deJined and < 00.)

Proof. (1) By XII, 6.2 the (N0,2)-existence property fails, so for some % ( 1 < 3) satisfying 2.4(2) and EM, U A& and Q, I= (~z)Q,(z, a), but q(Z, a) has no isolated extension in Sm(Nl U 4). Now we can easily find the pi's, using @+ = (y = y).

( 2 ) Just like XII, 4.7, 4.8.

CH. XIII, $21 ON HAVINQ MANY MODELS 629

FACT 2.6 : If 4 < qil(i < a} independent over M,, and M is FLo- constructible over u,, qrl and A = {bEM: R[tp(b,UM{,,,L, a303 < 5), then A is atomic over bic,M{fl.

We shall later prove it for a = 2.

Suppose not, then, as in the proof of XII, 4.7 w.1.o.g. each ib&, M,, are &-saturated. By XII, 4.17(3) M is K,-saturated too. So we can define by induction on /3 < a a model N,, such that No = q,N, = u,,& and Ny,, is Fie-prime over N y U qyIIN,+, -C M . As T does not have the dop, M = N,, and so there is a minimal /3 such that

is not atomic over Ui,sqtl. It is easy to see that for t9 limit A, = uycsA,,, and that for F E A, or

even FEN^, tp@, Uf,y$rJ I- tp ( i? ,u ics~t l ) . Hence our /3 is not limit, and trivially not zero, hence /3 = y + 1. Let FEA,, tp(c, u,,,qiI) not isolated. But (as we have assumed the fact for a = 2) tp(c,q,, U N,) is atomic, and Ay is atomic over ui<yJ!.(fl.

We now prove (*) tp*(Ny,A,, U qyl U does not fork over A,,.

This suffices, since by symmetry this proves tp(F,N, U 4) does not fork over A,, U hence by IV, 4.3. tp(F,A,, U ~ , + ) is atomic, and as 4 is atomic over ui<,ql1, hence over u , c , ~ , l , we shall finish by IV, 3.2(4).

Let us prove (*). So let EN,, choose 6 ~ 4 , C E ~ , tp(d,%) does not fork over 6,

t p ( 6 , q ) does not fork over a, a G 6. Let and we shall prove that tp(& 6 U F U q does not fork over

5. W.1.o.g. for some $t? E $(a, a,, c), where 6, E 6 n (u - ,-=Y A#&), R[@(z, 6,, 6 , L, a303 < 6. Suppose tp(& 6 U F U iT) forks over b, then for some q kT[J, E, c, 61, and for every E’, F’, tp(& 6) U q(g, c’, F’, 6) forks over 6 (see XI, 1.3). We can find E * E ~ , realizing stp(E,;) [because M,, is HI-saturated]. Now N,,J!.(,, are independent over M,,, hence Z-E-6, Jna*”6realizes the same type. So R[@(x, K1, a*), L , a33 < 6 , and there is F* EN^ such that A A, @[c:, 6,, e] t=q[d, c*, F*, 61. Clearly, F* c A,, and tp(d, 6 U F* U a*) forks over 6, hence tp(d,A,,) forks over 6, a contradiction. As we may increase 6, we proved (*).

We are left with the case a = 2, and w.1.o.g. the J& are countable,

630 F O R T H O M A S THE DOUBTER [CH. XIII, $2

and by the choice of E we can find B atomic over Ml U 4 (hence B # A ) , MI U 4 G B E A such tha tB ER~A. We then can find amodelN which is FiD-constructible over B, now A $2 N. But as A E,B, N is Fio-constructible over A too, hence we can find M Fio-constructible overN U A, hence over A , hence overM, U 4, and asN n A = B # A, N a proper submodel of M . This is a contradiction to "T does not have the dop" by XII, 2.8.

2.7. ProofofLemma2.3: Letik&p,(l < 3 , 7 ~ " 2 ) beasin2.4,2.5,soby 2.5 we can assume that there is an isomorphism f from MI onto & Over 4. Let Ia,ga(a < X) exemplify C P , , x , ~ I , x , x , and for a c x we shall define a model Ma. Let M i = 4, for ~ E I , f: an isomorphism from Ml onto M&, f ; r&, = the identity, {M&: t ela} independent over Mi. Let f:,j=f: U ( f ; o f - ' ) : M 1 U MZ-tM;) U Mrj), pf , j=f ; , j (p , ) SO

P v S m ( q i ) u 4,)). Let {qa: a < wl} be distinct members of "2. We define by induction

on n for every finite subsets t of 4 of power n a model MT. For n = 0 , l : we have defined. If t = {i,j} i < j, let M; be Fio-constructible over 2y U realizing

If n > 2,'M; is FXo-constructible over u {M, : s G t , s # t}. We make 4 depend on no more than is necessary, i.e. it depends on f r t , not on a.

Let Ma = utGIaM; . Clearly, (4: t E I , ) is a stable system, and

def p j y , where q(i , j) = q"(i,j) = Y g a ( i , j ) .

FACT 2.8: For every countable A E Ma, only countably many types p € S " ( A ) such that p I- A,,, $(xl, b), where EEA, R[$(y , E ) , L, m] < 6 , are realized in Ma [this is because for some countable t c A , (letting M; = u {Mi : s E t , s finite}) A E M; and for every such type realized by a~M,tp(c,M; U Uy,A,y,+riM/) is atomic (by Fact 2.6 as in the proof of XII, 4.16)].

So we get an easy contradiction to the combinatorial principle we have assumed.

LEMMA 2.9: Suppose A i s >y<",y+ 2 K , K < 9 < A , 9'" < A, 9, K

regular and h is not a strong limit of cojinality < K , and l (3u c A ) [r strong limit A cfu < 9 A r < A < 27. Then CPA,2~,p,K,B holds.

Remark. We can get the "full strong'' properties as in [Sh 83, $21.

CH. XIII, $21 ON HAVING MANY MODELS 63 1

Proof. Our proof is split into some cases. Case A. A is regular and (Va < K)(V~ < A)[Jil’”l < A ] . Let S, E (6 < A : cfS = 9} (i < A) be pairwise disjoint stationary

sets and for each SEU,,,S, let q8 be an increasing continuous sequence of ordinals < S, of length 9, each of cofinality <a,& = u,,,q8(a). For any A c A we define a two-place function g A : A+* as follows:

gA(S,j) is Min{a < 9:qa(a) 2 j},

g,(S,j) is 0, otherwise.

if SE U S,,j an ordinal <S, , € A

Now if 9 is a family of pairwise incomparable subsets of A, 191 = 2,, then {(A, g,) : A ~ 9 } exemplifying the desired conclusion.

For suppose A , B E A, A -B # $3, f a function from I, into M(I,) (IA = ( A , g A ) , I, = (A ,g , ) ; for the definition ofM(1,) see [Sh 831 1.1). Let i(*) E A -B. For each i < A letf(i) = 7*(at), f i t a sequence of length < K from I,, 7, a term in LXK.

We can find a stationary S c S,(,) such that (remember pCK < A) : (1) for every i ~ S , 7 , = 7*,fi, = (a,,,:a < a,), a, < K, and w.1.o.g.

Similarly, w.1.o.g. : (2) the truthvalueof

a,,, = 2 .

>, < at,y”, “ U ~ , ~ E U ~ ~ , ~ ’ ’ , “q,Jj) < a,,p ,

for b, y < a0, 5, j < 9,”1s the same for all ~ E S (as there ar~”<91uol < A possibilities).

6 6

Tat, (3.) = %J”’ “Val (3.) ’ a,$’’ and also c = Min{S: qa (5) 2

By Fodor’s lemma, as ( V i < A ) [l i l i”ol < A] w.1.o.g. : (3) for every ~ E S , /3 < a. if a,,B < i, then a,,p = u , ~ , ~ , where i, =

MinSandalsoq,(,l,(~,,B- l)isconstant, where&B = Min(6: q,Jc) 2 i} (if i t is well defined).

Also, w.1.o.g. : (4) if i < j are in S, then a,,! < j for every /3 < ao. Now choose 6 a limit point of S of cofinality 9, S E S (really SE&)

suffices). Then choose {jc: 5 < a} C S n S such that Min(5: q8(y) 2 j,} is strictly increasing; then J: - 6, ys = jg, fulfil the requirement because : S# U,€,SJ , hence is = qa MinS.7 (5) or is 2 S for 5 < 9, Y <a,*

6 - .

Case B. For some u, u < A < 2“, c<K = u 2 p<’. The proof is similar to [Sh 83, 2 . 7 1 . Let S, c {S < u+: cfS = a}

(i d u) be stationary, pairwise disjoint. For every SEU,,,S, choose an increasing continuous sequence qa of length 9 , S = uc<,q8(5). For any A E u let I, be defined as in Case A.


Let (4 : 6 < A} be a family of subsets of CT, any non-trivial Boolean combination of < K of them has power cr (they exist; see AP 1.5(2)). Now for any B c A let JB = &tBIA,, and it suffices to prove that if B, $4, f : JB, -+M(JBz), then there are xE, ye as required in Def. 2.2. We canchoosee(*)E&-Gandconsiderf r (IAcc., x {e(*)}).Letf((i, e ( * ) ) ) = 7i((. . . , (at ,a, q a ) , . . and again for some S G S,, S stationary, and

(1) 7t = 7*,a1 = a* for ~ E S , (2) { ( . . . ( a i , a , e t , a ) a at3 .a , = air.al - ai3.az - ai4.azl’

[et,, a, - ctz, up => ‘i,, a = et,,al - Ets,az = et4,az1, - -

(3) the truth values of e f , a = et,8, ” ; and the values of Min{t : q.,,$! b a,,p} are the same %,a = U i . 8

= vut, (t), “ a t , a E U , E A 6 t , o 4”, < (

for all iES,a < a*,B < a*, (4) if i, < i, are in S, then at,+ < i,, ( 5 ) i f f o r s o m e y * , i E S & a , , , # i & y < 9 = > r ] u l , . ( 6 ) ~ i v va t ,= (<)=

Now choose j ( i ) ~ & - u ~ < ~ * A ~ ~ for ~ E S [exists as e ( * ) ~ & - & . , C ~ , ~ E B , and the choice of {4:6 < A)]. Asj( i ) < cr w.1.o.g. j(i) =j(*) for every ~ E S . Now choose S a limit point of S; for appropriate { j t : t < a} G S , xt = (a,€(*)), ye = (je, e ( * ) ) we get the desired conclusion.

Case C. A is a strong limit of cofinality > K (or > K + if K is singular).

Similar to [Sh 83, 2.6, Case B]. We use (h ,g) such that for every 8, i < A if {j : g(i,j) # 0} # then i t is a set of successor ordinals, i is limit, the set has order type 9 x K , g(i, - ) is a one-to-one function over every interval of order type 8, its limit is a strong cardinal

7%” s,a (Y) < Y*.

Ai, hi < i < zA‘, Cf Ai = K .

FACT 2.10 : Let 6 < K < A be regular cardinals and let S+ = { S : S E A , (VSES) cfd = 9, and for every closed unbounded C E A for some 6 E C, cf 5 = K and S n 6 is a stationary subset o f t } .

Then any S E P can be partitioned to A subsets each in P.

Proof. For every ~ E S let qa be an increasing sequence of length 9 of ordinals < 6 whose limit is 8. Let for each a < 9, i < j

A:,, = { S E S : ~ d v,(a) <j}.

CH. XIII, $21 ON HAVINU MANY MODELS 633

We shall prove

SUBFACT 2.10A : For some a < 6 for every i < A for some j, i < j < A and A; . ,ES+.

This suffices as then we can define by induction o n j < A an ordinal i(j) such that j , <j * i(jl) < i(j) and forj-successor, A;(j-l),i(j) ES+. This is clearly possible and {A;(j),t(j+l):j < A} is a family of h pairwise disjoint subsets of S which are in S+. So we can get a partition as required.

Proof of the Subfact. If not, then for every a < 8 there is i, < A such that for no j, i, < j < A, there is a closed unbounded subset of A such that

[[EC; A cf[ = K] - [A; , fl [ is not a stationary subset of 51. 0.

Let i(*) = Ua<Bia < A, and

C* = {S < A : 6 is limit, S > i(*), and SE (7; for each a < 8, j < 8).

It is well known that C* is a closed unbounded subset of A. But S E S+, so there is [E C* such that cf [ = K and S n [ is a stationary subset of [. For each SES n 6 there is a6 < 6 such that qa(a,) > i(*). So S n [ = U,,,{SES n E:aa = a} and as 6 < K = cf[, for some p,

S , = {SES n [:a,= p} is a stationary subset of 6. Now the function 6 + q6(p) is regressive on S, (i.e. qa(p) < 8) so by a variant of the Fodor lemma, for somej < 6,

S, = { S E s l : r l , ( p ) <j>

is a stationary subset of 5. So by the definition of a6, p,&, S, for every SEE&, i, < i(*) < qa(a,) = q,(p) <j, hence SEA{*:,. So Is, E AtP,.

But [EC*, hence [E C&, hence A C j n 6 is not a stationary subset of 6, but Is, E Atp, and S, is a stationary subset of 6 , a contradiction.

Remark. We would like to have in 1.1(2) that IE(A, T) = 2A for every A > No. We think that i t is true, but having spent considerable time on proving “ I @ , T) = 2A for T unsuperstable, A 2 IT1 + N1” in all cases we have not found it challenging to do this again.


XIII.3. On the Morley conjecture

DEFINITION 3.1 : We say that Ml is an (N, a)-component of M if Ml C M , N U a C M,, tp,(M,, N U a) is almost orthogonal toNand plI is a maximal set under the previous conditions.

CLAIM 3.2: If {MI,&} is independent over &,A& E M1,& CM&M primary overM, U 4, GEM,, andM* isan (4, a)-component ofMl, then M* is an (4, a)-component of M provided that every regular stationary type not orthogonal to tp(M,,M*) is not orthogonal to M,,.


We now work in the context of Section 2 of Chapter XI. The following lemma shows that under suitable circumstances, we can “blow up a model M ” by increasing a suitable dimension, without losing control.

LEMMA 3.3 [The axioms used in XI, 2.81: We assume T is superstable without the dop and A satisjies (*) of XI, 2.3(3). Suppose N C *MI, {@( : i < /3} independent over N (not necessarily maximal !) tp(it,,N) regular, llyll < A , A < a < p, for i < a,@,dlP and 4 an (N,af)-component of M 1 ; for i 2 a (but </3) tp,(&,M’ U U,,,A$) does not fork over N , the 4 ’ s are pairwise isomorphic over N , and M2 is T- primary over M1 U Uf<& Suppose further that all the models are T- saturated. Then for every bEM’, tp(6,N) regular, and (N, 6)-component 4 of M2, there is an (N, 6) component Ml of M1 isomorphic to 4 over N U 6 provided that A+ \< /3.

LEMMA 3.3A : (1) Really for some automorphism of M2, F r (N U 6) = the identity, F(&) EM’.

( 2 ) Moreover, i f N U 6 c M c M’ n M,, F(M,) an (N, 6)-component of M’ and M2, then w.1.o.g. F rM = the identity provided that A+ < a.

Proof. We should note:

FACT 3.4: If a 2 A+ A c M2 of power < A , then there is an automorphism F of M2, mapping A into M1 such that F r (M’ n A ) = the identity. [Because, letting = N , N$> = &, (q : 7 E { ( i ) : i < a} U

CH. XIII, $31 ON THE MORLEY CONJECTURE 635

{( )}) is a (T, E *, A)-decomposition inside P, hence we can extend it to (q: Y E J ' ) , a (T, E *, A)-decomposition of LIP with 1 1 ~ 1 1 < A, w.1.o.g. (i)# J' when a < i < p. As "& is a (N,aJ-component of@, ( i )<qrlEJ'*( i )=qandeasi ly ( q : q ~ P ) , w h e r e J 2 z f J 1 U {(i); a < i < p} is a (T, s *, A)-decompositzon of W . W.1.o.g. A = WI, M prime over uveJq, J E J 2 , I JI < A, and the rest should be clear using the uniqueness of the T-prime, T-minimal model over A).]

FACT 3.4A: If A EM2 has power < 1011, B c A n P and IB( < IaI, then there is an automorphism F of M2 mapping A into M' such that F (B U N ) = the identity.

Proof. Same as the proof of 3.4.

FACT 3.4B : I f N U 6 c B, A EM^, B EN, IAl< la1 and tp,(B, N U 6) is almost orthogonal to N , then for some automorphism F of M2, F maps A into M' knd F r (N U B ) = the identity.

Remark. Instead of 6 finite, 6 of length < A suffices.

Proof. As we can replace (N, : i < a) by (N, : i E a\w) for any subset w of a of cardinality < )al, w.1.o.g. 6does not depend over (N, u,,J,), hence over (N, u,,BN,). Now we can find (x : v E J') be as there, such that: B E N*, N* T-prime, T-atomic over u{x: V E J3} , where J3 = {v:v = ( ) or v ( i ) 2 a (hence 2 p)}. The rest is the same.

Remark 3.4C. In 3.4A, 3.4B, a 2 w suffices.

Continuation of the proof of 3.3. Let (q, ~q €I2) be a (T, E *, A)- decomposition o f 4 with 11q11 < A, and N U b c q). Let If = {q€12: l (n) < n}. We can define by induction on n , q , a i , F , for SEE such that

(1) (3, (2) F, is an automorphism of M2 mapping

(3) For TE,?,-,

: q €I",) is a (T, E *, A)-decomposition inside M , onto 3, F,(G) = a:,

Fvrk s F, for k < l (q ) , and Fo r (N U 6) = the identity. (i) E I ~ } is independent over 3, each

tp(aqny,) ,q) is regular orthogonal to N,- if q # () and to N otherwise.

For n = 0 : Let FO be an automorphism of M', F o r (N u 6) = the identity, and Fo maps P into M' (exists by 3.4A). We let q) = F < > ( q ) ) (and q) are meaningless).

: q

(>


For n = m + 1 : We deal with a specific r] EL, of length m, {F,(qn(r)) : q ^ ( i ) ~ 1 , } is as required in (3); however, F,(qn(r)) does not necessarily belong to W . Let T be regular and we deal with I = {F,(atn(f)): r ] "(i) €4, tp(F,(q-(i)q) not orthogonal to r} . For each IZE I ,p = t p ( a , q ) is a regular type orthogonal to N [as N U 6 E q> E*T E*%, for every F E ~ , t p ( c , q ) is almost orthogonal to N (as tp,(A?JV U 6) is) hence is orthogonal to N (by Ax(C2))I. But tp*(U,,,,,y,W) does not fork over N, hence it is orthogonal to t p ( a , q ) . So clearly, tp (a ,W) forks o v e r q , and it does not fork over some F E W .

We can (using 3.4A) define by induction on k < w , dl EW realizing t p ( a , q U_ F U u,,,d,), so for some 1 {&: k < Z} is independent over q and dr (hence a) depends on them. [Otherwise, {Z,: k < w } is an indiscernible set based on t p ( a , q ) , but tp(a,q U C) = Av({d,: k < w} , q U q, a contradiction.] We can conclude that each ~ Z E I depends on J = J = (662111 : t p ( 6 , q ) is regular not orthogonal to r} over q.

Now it suffices to find for each @,-(,)EI an automorphism F' of W , Ff r q = the identity, F'(G,,n(f)) EW and {F*(a,-(f)) : a q n ( , ) ~ I ) is a maximal subset of J independent over q. [As then (by 3.4B) for some automorphism F: of W , F: r (q U F'(a,n(t>)) = the identity and F: maps F*(Fq(qaCf ) ) ) into M1, and let F,n(f) = F: o P OF,.]

If 14 < A, there is an automorphism F of@, F /'q = the identity, F maps I into M' (hence into J), hence the maximality of {a:-(,): r ] "(i) E P} implies the maximality of {F(G) : B E I ) , hence F' = F is as required.

If 14 > A , we still know that J has a maximal subset f' independent over?, and that 14 = PI. We then define by induction on 5 < 14, countable q E I-uC<tI$ q c UC<[q, 3 E P - - U ~ < & ,

doesnotforkoverq U U Ji U f i U q a n d t p G , I U dC<& U 3 ) d o e s not fork over q u u$ u q u $ and 141 = implies 8 U f i E$]. Now for each i let Fi be an automorphism of W , F : r (q U 8 u $) = the identity, F i ( q u c) E W exists by 3.4B with 6, q U 8 U $, q U Ji U $ U q U here corresponding to b, B, A there). Lastly, for a V m c r ) ~ I let F' be Fi when a ,n -< f )~q .

If T is trivial (e.g. when it has depth > 0) the proof is easier. Note : We can find a one-to-one function h from I onto& s.t. {a, h(a)} is not independent over q ; now use 3.4 for A = A u U h(n) for each a€ I .

$ = UC<& such that I = U& P = u*JL tp*(q, u <& u P u q) d KO and [ F E ~ n

CH. XIII , $31 ON THE MORLEY CONJECTURE 637

As was said above, this suffices to define F,-(,) (for 7 " ( i ) € 1 2 ) . So we have carried the definition of (F, : 7 €1,). As (3 : 7 €I2) is a

non-forking tree of models, F zf u {F, : 7 €1,) is an elementary map from Utsp$, we can extend F to an elementary mapping F+ from M , into M', thus finishing.

One point is still missing: Why is M , = Range(F+) an (N,6)- component of M' (and of W ) ? The reason is that for each ~ € 1 ~ : (*), {F,-<i)(q-(i)) : qn( i ) €I2} is a maximal independent subset of J, = (a~M2: tp(&,N,,) is regular not orthogonal to Nt- if 7 # ( ), to N if 7 = ( )} (or use the proof of 3.3A(1)).

Proof of 3.3A(I) . We can find a (T, G * , A)-decomposition (q, C$ : 7 E 13) ofM2 such that I2 = (7 : (0)- 7 EP}, and for 7 E 12, I Z ~ = I Z : ~ ~ , , ~ = N;"o)-,,q) = N. Now easily F+ U U{idN3:q~I3, ~ ( 0 ) > 0} is an elementary mapping and (by XI, $2) W is !I"-prime over its domain, so it can be extended to an embedding F* of M2 to itself (extending F+). But we can check that ( F ( y ) , ( F ( ~ Z ; ) : ~ E P ) is a (T, G *, A)-decomposition o f W (see ( * )7 above). As T does not have dop,

M2 is T-minimal over UVErJF*(T) hence the mapping is onto.

Proof of 3.3A(2). It is similar, but choosing a (T, c* ,A)- decomposition (P, I Z ~ : 7 €1') of M , we take care that M is T-prime, T-atomic over d q s r q , I E I2 close under initial segments (see XI, $2) ; when we choose F, add' the condition : (4) if ~€1: n I , then F , is the identity. Note that in the induction step, for each 7 €1; n I our commitment

on F(ct-(,)) ='IZ,yO when yn ( ~ ) E I is not an obstacle. In the end, instead of extending u,srzFt to an elementary

embedding of M , into M', we note that also UlrsrPF,idlCI is also an elementary mapping and M , is T-prime over its domain, and extend it to an embedding F+ of M , into M,. Then continue as in the proof of 3.3A(1).

LEMMA 3.5 [The axioms used in XI , 2.81: Assume T is superstable without the dop. Suppose M* is a model of T of power p > A Ef A(T, ~ * ) , 2 p > 2A, and M* is T-saturated and (*) fails but (**) holds where

(*I there are x A and a T-saturated N G *M*, ai EM* (i < f ) , {a, :i < x+} independent over N, & an (N, ai)-component of M*, the & pairwise isomorphic over N , 11&11 < x, IlNll < A ;

638 FOR THOMAS THE DOUBTER [CH. XIII , $ 3

(**) there are N Eq s M * , & Z N , IlNll d A, 11&11 d A (i < A+), {q: i < A+} independent over N and the 8 ’ s are pairwise isomorphic over N .

Then for every p( 1) 2 p, T has (at least) 2’ pairwise non-isomorphic T-saturated models of power p(1).

Proof. Let N , q ( i < A+) exemplify (**). Let p( * ) = Min{9:2’ = P}, hence A < p( *) d p. Let (3, : r] €1) be a (T, E *)-decomposition of M* (so [&I[ < A ) and w.1.o.g. N E q), and for each r] { i : r ] - ( i ) EI) is an ordinal aI(q). We concentrate on the case p(* ) is not strong limit (as for T countable we use only the case Dp(T) < o , p < a@). If p ( * ) is regular, then for some ~ € 1 , for p( * ) ordinals i , r ] - ( i ) ~ 1 , but {v: r] -(i) 4 v} has power <p( * ) . If there is such r ] , w.1.o.g. r] = (), and (V i < p ( * ) ) [ ( i ) ~ I A I{v: ( i ) Q V E I ) ~ < p(* ) ] . If there is no such r ] , then necessarily p( *) is singular, and let p( *) = xb, : i < cfp( *)} and w.1.o.g. pi is increasing, each pi regular, 2’6 (strictly) increasing and 2’t > 2A. For each i we can find r ] , ~ 1 such that 7, is 4 -

incomparable with q, for j < i, and for pi ordinals j, r] (j) €1, I{v: r] -(j) Q v}l < p,. W.1.o.g. r] , = ( i ) for i < cfp(*), and for j < p,: ( i , j ) ~ I , I[v: (i,j) Q ~€41 < p,. If cfp(*) < A we can again assume w.1.o.g. that al(( )) 2 p( * ) and for i < p( * ) , [{v: ( i ) Q v ~ r > l < p(*) .

So we have exactly one of the following cases. Case A. For i <,u(*), ( i ) ~ 1 and I { v ~ I : r ] ~ ( i ) Q v}I < p( * ) and

even, I { v ~ 1 : v(0) d i}l < p( * ) . Case B. Not Case A, p ( * ) > cfp(*) > A , for i < cfp(*):(i)EI,

( V j < p , ) [ ( i , j ) € 1 A I { u : ( i , j ) ~ U E I ) ~ < p i ] and p = xb,:i <cfp(*)}, pi, 2’i strictly increasing.

As {&: i < A+} is independent over N , N E NO for some i, < A+, {q: i,, d i < A+} is independent over (N,No), and w.1.o.g. i, = 0. Now in Case A, w.1.o.g. {q : i < A+} is independent over (N ,N U Ur<’(*) a(,,!. [If p( * ) > A+ is regular, for some u < p( * ) , {a(;c): a (for i < p( * ) ) , a(,) as @(,I(O)+,) for i < a (and make the corresponding renaming for q ,av ) . If p(* ) > A+ is singular, the proof is similar. If p( * ) = A+, define by induction on i < A+, p ( i ) < A+, y ( i ) < A+ such that tp,(q(,),N() U u,&,(,)) U U,<,q(,)) does not fork over N and

then rename a(,,(,)) as a(,),A&,(,)) as N(,,, and N,,,, as 8, etc.] tP*(q,(,,)),q) u U,<&,)) u U,<rq(,)) does not fork over 4) and

N,) u Ul<ct’(*)a(,)) (remember cfp(*) > A ) . Similarly in Case B, w.1.o.g. {q:i < A+} is independent over (N,

CH. XIII , $31 ON THE MORLEY CONJECTURE 639

Now we define M , 4 , a, (for i < p(*)) . If Case A, we let M =No, a, = a(,, and 4 be T-prime over u {q : (i) Q u} €1. If Case B, we let M c M * be T-prime over uitcfr(*)A&,, for i < p(*) let j(i) < cfp(*) be minimal such that pj(c, > i, Cii = lZ<i,j(i)> and @ be T-prime over M U U{q: (j(i),i) 4 ~€0. Clearly, IlMll, 11&11 are <p(*) , and really llMll Q h if Case A, IlMll = h+cfp(*), otherwise. Note that for y < p(*), & < y 11411 is <p(*) .No te : {at:i < /A(*)} G M * is independent over M , Mi an (M, a,)-component of M*.

Now we define by induction on y < p( *) an ordinal i ( y ) and a set J , G J = {aeM* : tp(a,M) is regular} such that (letting x = IlMll) def

(1 ) w < ( M + X ) + ? (2) a,(,,, JIII < llMll + h + IyI for every J; E J; independent

(3) a,(y) does not depend on up,,,.& U B over M , where B =

(4) Ja is a minimal subset of J such that :

over p, def

U,<A4.

(a) {%(pJ u U,<& GJa, and (b) if a’, J, and there is an automorphism F of M* over M ,

F(a’) = a”, then a’ eJa iff IZ’’E$. We shall show we can carry the induction.

Remark 3.5A. Note that there is an automorphism F of M* over M,F(a’) = &” iff there are M’, an (M,a’)-component of M*, and M“, an (M, &)-component ofM* and an isomorphism F fromM’ ontoM”. F rM = the identity and F(d) = a”. [Why? The implication * is trivial; so let us prove -=. We can find a , w , $(i < a) such that {w : i < a} is independent overM,M G *% G *M*,Mprime over u,,,w,

an (M, $)-component ofM*,Mh = M’, and if {a’, a”} is independent, MI = W . If {a’, a”} is independent, let F be a mapping of u,,,q on itself, F rMl = the identity if i > 1 , F rMo = F’, P rWl = (P’)-l. Clearly, F is well defined, elementary, hence can be extended to an automorphism of M*. If {a’,a’’} is not independent, define P by F rx = the identity for i > 0, F rMh = F’, and when tp(a’,M) is regular we finish easily and this is the case which interests us. (For the general case, decompose M’ and prove by induction on w(a,M).).] Hence, by T(*): for every CZEJ and J‘ E (a‘€ J ; for some automorphism F of M* over M , F(a) = a‘} if J’ is independent over M , then WI d x. Let us do the induction step.

As {E@:P < (171 +x)’} is an independent subset of J over M and < llMll +x’lPI, there is no problem to choose i ( y ) for which ( l ) , (3)


hold. Then we can choose JB satisfying (4) to which at(,,, belongs. The other half of (2) holds as (*) fails and 3.5A. Now for i < A+ let MP(*)+* be T-prime over M U 4 and EM*. Clearly, MP(*)+* (i < A+) are pairwise isomorphic over M. We define &@(*)+A+ < i < p(* ) +p( 1)) such that % is isomorphic to MP(*, over M and tp(&,M* U u,<&!,) does not fork over M. So {&: i < p(*)+p(l)} is independent over M, 11&11 < ,u( *), & # M, each & (and M) is T- saturated ; and M E *& (as (N?, a? : T,I €1) is a (T, E *)-decomposition of M*,NO E *M*, etc.). For each S E p( * ) let MO, be T-prime over u,,,A?& U Ui<AMP(.)+r, Mo, E M * , and then let Mk be T-prime over W, U Ui<p(l)MP(*)+A+6. Clearly, each M i is T-saturated and has power P(1 ) -

It suffices to prove {S:Mk ZW,(~,} is + ( * ) ~ ~ " ~ ~ (as p ( * ) ~ ~ " ~ ~ < 2"') because p( *) is not a strong limit and ( V 8 < p( *))2" < 2'9. Let Me, be T-prime over Mo, U ut<P(,)MP(.)+A++i,Mes E M,. Clearly, if Ws z Ms(o), then alsow, z Me,. By VIII, 1.2 it suffices to prove that IS(O)\S(l)l > x = L M ~ ~ ( ~ ) , M ~ ~ ( ~ ) are not isomorphic over M. So for some y , ~ E S ( O ) , y#S( l ) , andF an isomorphism fromMe,(,, ontoMes(l) overM, we have : F(a(i(y))) belongs to the model Mo,. So F(i&(y)) is an (M,F(at(y)))- component of P,,,,. By 3.3A(l), w.1.o.g. F maps &,,, into Mos(l), hence F ( 4 ( , ) ) is an (M, F(a<*(,,>))-cornponent of hence of MO,,,,. Now, using Exercise XI, 3.4(2), F(&(,)) iS an (M, F(a((,,))-component of M*. As #'(ai(,)) there are y(1) < y(2 ) < . . . < y(k) < ,u(*), such that F(ai(,)) depends on U B overM and {y(l), . . . , y ( k ) } is minimal with this property. By (4) above y > y(k) (and k = 0) are impossible and by the choice of y , y # y ( k ) , so y < y(k). Now a*(,) does not depend on . . . , @tt(y(k-l)} U B over M. Hence %y(k)) depends on {a l (y ( l ) ) , * * * , ai(y(k-1)) } U {F(at(,)) U B over M. But

E&(~), and ai,,,,e4 (check the definition of 4). By V, 1.14 depends on U . . . U &(R- l ) U 4 U B E ufl,,(k)JB UB over M, a contradiction. Hence we finish.

. . . ,

Remark 3.5A. (1) In the case p( * ) a strong limit, use S E p(* ) of cardinality llMll+.

(2) If N c *M* (i.e. this is added to the assumption (**)), then it was enough to have 4 for i < A.

LEMMA 3.6 [The axioms used in IX, 2.81 : Assume T i s superstable without the dop. Suppose A = A(T, ~ * ) , 2 " > 2A,p > 21TI,p(1) > p. ThenI(p,T,T) <1(p(l),T,T)xI{(X,Xl):A < x < x 1 <p)I.

CH. XIII, $31 ON THE MORLEY CONJECTURE 64 1

Remark 3.6A. (1 ) I ( p , T, T) is the number of T-saturated models of T of power p, up to isomorphism.

(2) " p > gTl" is needed only to get that (**) of 3.5 holds. ( 3 ) For simplicity we assume that the demand in Definition IX,

2.3(3) holds for every A' 2 h (as this is true in the cases we shall use 3.6).

Proof. As p > 2ITI, every model of T of power p satisfies (**) of 3.5. If some T-saturated model of T of power p fails to satisfy (*) of 3.5, then by 3.5, I (p ( l ) , T, T) 2 2', but trivially 2' 2 I ( p , T) 2 I ( p , T, T), so we finish. So we can assume that each T-saturated model of T of power p satisfies (*). In (*), we can assume that &,@* are defined for i < x1 (for some xl, x < x1 < p) , such that ((4, a,) : i < xl} is maximal (under the conditions there). It suffices to prove that for each pair (x, xl), the number of T-saturated models of T of power p for which we can get { (4, @*) : i < xl} as above (up to isomorphism) is < I @ ( l ) , T, T). So let (x,xl) be fixed, so h < x < x1 < p.

: i < xl)) : a < a( *)} be a maximal list such that M",Ma are not isomorphic for each a # p, each Mu a T-saturated model of T of power p, N", (N:, a%: i < xl) is as above (in 3.5(*)). It suffices to prove that la( *)I < I ( p ( l ) , T, T).

We can define for each u, for x1 < i < p( 1) a model N; isomorphic over N" to N;, such that tp(N;,Mu U u ,,N;) does not fork over N". Lastly, let M; be T-prime over Mu U d,,,,,,N:. Clearly, it is enough to prove that for a </3 < a(*), M;,M$ are not isomorphic. For notational simplicity we let u = 0 , p = 1 so we assume from F is an isomorphism from M"* onto M i . Clearly, if 4 is an automorphism of M i (for I = 0, l ) , then we can replace F by Fl o F OF,. So, for example, w.1.o.g. there are Mk E *&?;,Mi T-prime over M1 U u {q : j < p+p} such that F maps M", onto M i .

By the Lowenheim-Skolem argument there are, for 1 = 0 , l fi G

W' C p ( 1 ) , /&I = IW'-Kl = x and models Po such that: N:) is T- saturated, IIPol = x, PO c *Mi,* c Po for i~ K, {q: i * c M i T-primary over Po U UiEwi# such that F maps ?j* onto q j * . By Fact 3.4, w.1.o.g. No cM' and even P(>* cM', and by its proof we can preserve even the last sentence. Let 1; = { b ~ M k : tp(b,PO) is regular}. Clearly, F maps 1; onto 1:. As in the proof of 3.3 it suffices to find 4 c 1; n M' independent over Po, maximal, and for each

Let {(M", Nu, (N:,

def -

642 FOR THOMAS THE DOUBTER [CH. XIII, $ 3

6 ~ 4 an (Po, 6)-componentMfsofM', and a one-to-one function h from 4 onto I , , such that for 6 ~ 4 , F r q , can be extended to an isomorphism from lMos onto %-. By 3.3 it suffices to find such Mfs which are (flo, 6)-components of M i ; hence it is enough that some isomorphism Fs from M"* onto P* extend F rN& and maps 6 to h(6) . Let c = ( 6 ~ M i : tp(6,Po)) be regular not orthogonal to t p ( 4 , P ) for some (every) i ~ p ( l ) \ W } . Clearly, tp(#,iV), tp(F(@), F(N1)) are not orthogonal [as every indiscernible set EMfc orthogonal to tp(a& iV) has dimension +I, hence F maps

Now when we restrict ourselves to If\c (1 = 0 , l ) we can as in the proof of 3.3 define appropriately 4 n (If-c) E M and h r ( 4 n (I; -c)). So we can concentrate on c(l= 0 , l ) . Let

1; = ( 6 ~ M ~ : b realizes tp(@,P&) for i $ W * and there is an (Po, 6)-component ofMi isomorphic to the T-prime model over N: u PO (for i$ W') over PO}.

onto c.

Clearly, I; c I p and let

c = ( 6 ~ q : 6 depends on over Po}.

Now F maps onto [for this it suffices to prove that for every 6~ I;, F(6) depends on I; over ?> and also that every F E I; depends on F(I; ) overq ) . By the automorphismsMi have overP<) (see proof of 3.4) i t suffices to prove this for at least one ~ E I ; , E E I ; . But the choice of @(>*, W' ensure this.]

Again by the use of P(>*, it is clear how to define 4 n If c M' and hr(If" n If) . Lastly, we let 1; E 1; n M' be maximal subset, independent over (PO,f10 U uc) and we can deal with them too.

THEOREM 3.7 (Morley conjecture): Let T be a countable complete jirst-order theory. Then for A > p 2 No, I ( A , T ) 2 I (p , T ) except when A > ,u = No, T is complete, N,-categorical, not K,-categorical.

Remark. For not necessarily complete T, the conclusion holds except when A > p = No, every completion of T is N,-categofical (or have finite models only), there are only finitely many completions of T and a t least one such completion is not No-categorical (but have infinite models).

PToof. If T is not superstable, or superstable but with the dop or with the otop or is deep, then by XII, 6.1 we know that for A >

KoI(h, T) = 2A but 2A 2 2’ 2 I ( p , T), hence we finish. So we assume T is superstable without the dop without the otop and is shallow. We concentrate on non-KO-stable T. Let po = Min{X: 2K > 22n0} (hence (po > 2’0)). By IX, 1.20 for p < po, I (h , T) 2 Min{22no,2A} 2 Min{2p, 2’) 2 I ( p , T). So we assume p 2 po, and apply 3.5, 3.6 for (T, E *) = (Ti,, si0). We see that together they give the conclusion for h > p 2 po, provided that

(*I As T is KO-unstable, I (h , T) 2 Min{22xo,22”} 2 No. If T is not

unidimensional, I(A, T) 2 I{x: x < p}I easily, hence by cardinal arithmetic (*) holds. We deal with unidimensional T in detail in 4.15.

What about H,-stable T? For p = No, if T is categorical in No, this is trivial, otherwise by IX, 2.1 I(h, T) 2 KO for every A , so the only non-trivial case is I(Ko, T) > No; but then by [SHM 841 I (h , T) 2 2’0 = I ( KO, T). For 2’ > 2’0, apply 3.5,3.6,3.68(2) for (T, c *) = (Tio, G ~ ) , and again it suffices to prove

l ( h , T ) 2 No+ N X , XJ : 2’0 < x < x1 < A.

( * ) I W , T ) 2 ~ o + { I ( x , x 1 ) : ~ 0 x+x1 GPu>l. If T is multidimensional, this holds by Ex.X, 6.1; if T is not

multidimensional, use IX, 2.4. We are left with the case 2” = 2’0. We can assume T not

multidimensional (by IX, 2.4)). If h 2 K, use Ex.XI, 3.1 (which gives the exact number for h and a corresponding upper bound for p) . If h < N,, Dp(T) > 2, by X, 6.2, I(h, T) 2 2’0 = 2’ 2 I ( p , T). If A < K,, Dp(T) d 2, repeat the analysis of [SHM 841 considering all regular types and get either I @ , T) = No < I(h, T) or I ( p , T) = 2’0 < I(h, T).]

XIII.4. I(N,,T) for a large enough

(Hyp) T is superstable.

DEFINITION 4.1: (1) For any pair (No,Nl) of models of T such that No sN1 we define a cardinal SNDA(No,Nl) (=special number of dimensions) to be the number of models NESE,(N,,N’) up to isomorphism over N,, where SEA(No,N,) is the class of N satisfying


(*I (4, b)bsN, ~f (N, b)bENo, and for some B E N tp(B,N,) is a regular type of depth zero orthogonal to No, and tp,(N,N, U a) is almost orthogonal to 4.

(2) If ti exemplifies NESE~(N,,N,) we shall write also (N, a) €SEA

(3), We write NESE,(N,), (N,a)eSEA(N1) when (*) above holds omitting No (and the orthogonality of tp(a,N,) to No). If h = ITI+, we omit it.

Remark. (1) Clearly, if h increases, SEA(No,Nl) decreases. (2) For A = 0, (4, b)bsN, ~f (N, b )bsN, becomes 4 EN. (3) We may consider =: and then write SND:,SE,”. (4) Why are we restricting ourselves to “Dp(tp(&,N,)) = O ” ?

(4,’w.

Because by 4.5, X, 7.2 the other cases are not so interesting.

LEMMA 4.2: (1) I ~ N E S & ~ ( N , ) , then N is a minimal extension ofN,. (2) N* E SGO(Nl), N, is Ff-saturated, then N* is Ff-saturated. (3) If (N, a) ESE~(N,) , then N is F;l,-constructible over N, U a and

for every EN, 6$N, tp,(N,& U 6) as almost orthogonal to N , and Dp(6,q) = 0. If T is countable N is even F;,-constructible over 4 U a.

(4) If N, E ;toN, B E N , tp,(N,N, U a) is almost orthogonal to N, then N is F;,-atomic over N, U a.

Remark 4 .2~4 . If T is countable, we can replace S&, by SE, in ( l ) , (2) (e.g. in (1) use N** S M , Fko-constructible over M, U N*) . We can omit countability in (3) by 111.

Proof. (1) Like XII, 4.17. Suppose N is not minimal over N,, so there is N*,N, EN* C N,N, # N* # N. Let h > llNll+ IT1 be regular, M, be an F:-saturated extension ofN,, such that (M,,N) is independent over &. Now tp,(N,N, U a) is almost orthogonal to &, hence N is Ff- constructible over M, U a (remember h > llNll). Let M be Ff-primary over M, U N , hence also over M, U a. As Dp(a,N,) = 0 every non- algebraic PEU,,,S~(M) is not orthogonal to M,, hence for every FEN, F#N*, tp(F,N*) is not orthogonal to N,. Remember N, s ; t0N, hence by XI, 3.1 1 for some such Ftp(c,N*) does not fork over N,, and choose ~ E N * , ~ $ N . As M is F,”-primary over M, U a, tp(a,M,) is regular by V, 3.3, M is Ff-primary over M, U 6. By 111,4.22 it is also F:,-atomic over M, U 6, hence tp(c,M, U 6) is F&-isolated. But tp(F,M, U N*) does not fork over M,, hence over N,. So by IV, 4.3 tp(i?,N,) is Fff,-isolated. But this contradicts N, FEN, F$N,.

CH. XIII, $41 I(&, T) FOR U LARGE ENOUGH 645

( 2 ) Let p > I(N*II + ITI+, (N*, a) ES&~(N,) and let N be F,"-primary over p*l, so we assume N # N* and essentially get a contradiction. Let M, be an Fz-saturated extension of N,, tp,(Ml,N) does not fork over Nl. Easily N* is Fi-constructible over H, U c, N is F,"- constructible overM, U N*, and there is an Fz-primary modelM over Ml U N , hence over Ml U a. From this we see that for every F E N , F$N*, tp(F,N*) is not orthogonal to N , , hence by XI, 3.11 for some such FEN, F#N* and tp(c,N*) does not fork over Nl. As N is F,"- primary over N*, tp(F,N*) is F,"-isolated, and as Nl is F:-saturated, we get a contradiction to IV, 4.3.

and is Fio-constructible over Nl U @. We assume A # N and get a contradiction. Let p ,Ml ,M be as in the proof of 4 .2 (2 ) . By 4.2(2) M is Fio-saturated ; over 4 U A there is an Fio-primary model W , hence w.1.o.g. W E M . Clearly, MES&~(M,) , hence by 4.2(1) W = M . But M2 is Fin-atomic over M, U A . Hence for every F E N , tp(c,Ml u A ) is Fio-isolated. This is a contradiction to A # N (and the maximality of

So we have proved "N is Fio-constructible over N, U a". If T is countable, we can find a maximal A G N including Nl U @ and FAo- constructible over & U a. We can find W E M which is FAo- constructible overM, U a, hence F&o-atomic over A (see XII , 1.2) and continue as above.

The second conclusion to 4.2(3) can be similarly proved using V, 3.3. (4) Similar to the above.

( 3 ) Let A be a maximal subset of N , which includes N , U

A) *

CONCLUSION 4.3 : (1 ) Suppose T has the ( < co, 2)-existence property. If NESENo(N1), M, s Nl,M, s M , and M c N , {M,N,} independent over M,,M # M,, then N i s primary atomic and minimal over M U N,.

( 2 ) If also HESEo(Ml), N atomic over M U &, then NESE,(N,), if in addition M E S & (M,), then NES&~(N,) .

( 3 ) If in addiGon MESE,(M,), N primary over M U N, , then NESE, (4).

Proof. ( I ) By the hypothesis there is N* primary and atomic over M U 4, so w.1.o.g. N* s N. But by 4.2(1)N* = N, soNis primary and atomic over M U N,. Again by 4.2(1) N is minimal over M U N,.

(21, ( 3 ) Easy.


LEMMA 4.4 : Suppose T has the ( < a, 2)-existence property but not the dop. Suppose, further, & EN" c N ( l = 1,2),N, Epl,,+N, and ( N " , ~ ) E SE(N,) ( for 1 = 1,2). If {a1, a"} i s not independent over N, or even N, 5 A c N, tp(d,A) does not fork over N, (1 = 1,2) but {al,iZ2} i s not independent over A, then W , P are isomorphic over N,.

Proof. Clearly, tp,(A,N") does not fork over N,. By XI, $ 2 there is M c N,N, E A E M , tp*(M,W) does not fork over & and N is primary over M U W . Clearly, tp(a'2,M) does not fork over N, (by V, 1.14): otherwise as tp($,A U a") G tp(al,M U a") forks over&, also tp(al, M) forks over N,, a contradiction. Hence tp,(P,M) does not fork over N,, and by XI, $2, M Gb,+N,N is primary overM U N2 too. Let h be large enough regular cardinal so that T, N , M , N,, N1, P E H(h). Let 2l be an elementary submodel of @(A) , E) of power IT1 which includes { i : i < ITI} U {ITI ,N,M,N, ,N1,P,~1,a2} . Clearly by 4.3,N" is primary (and atomic) over N, U (N" n 2l) (for 1 = 1,2). AlsoN n 2l is primary over (M n 2l) U (N1 n a): this is proved as follows: as there is a construction (M U N,, (a,: i < a ) ) of N over M U N,, there is such a construction in %. Clearly, ( ( M n 2l) u (W n a), ( a t : i c a n 2l)) is a construction of N n 2l. Clearly, {M n 2l,W n %} is independent over& n %,and& n 2 l i M n %,N1 n %<N.

So N n 2l is primary over (M n %) u (Nl n 2l). Similarly N n 2l is primary over (M n 2l) u (P n %). Now there is an elementary mappingffromM n 2l into N, which is

theidentityoverN, n %(asN, Eb,+N, (1%11 < ITI)andletM* = Rang f. We can extend f to f l by the identity on N1 n 2l (and it is still elementary, by the non-forking) and then extend it to an elementary mapping g2 from N n into N1 [as N n % is primary over (N1.u M ) n 2l]. We know that M'%*Rang(g,) is atomic over M* U (N" n a), and easily M* U (N" n 2l) ct& U (N" n %), hence tp*(Rang(g,),N,) does not fork over M*. Clearly, {M1,N,} is independent over M*, and by 4.3 N" is primary over M1 U N,. As g2 o q;l is an isomorphism from M1 into M2, over M*, i t can be extended to an isomorphism from N1 into N2 over N,. By 4.2 it is onto P, so W , N2 are isomorphic over N,.

LEMMA 4.5: Suppose (N", d ) ESE(N,) (1 = 1,2) and tp(d,N,), tp(lEz, N,) are not orthogonal, and they are not trivial. Then W,N2 are isomorphic over N,.

Proof. By 4.2(3), w.1.o.g. @' = (b ' ) , and for every c2~N"-N, , R[tp(c',

CH. XIII, $41 I(Nal T) FOR LARGE ENOUGH 647

N , ) , L , 001 2 R[tp(b2,N,),L, 001. We also can find v , ( x ; 6')~ tp(b',N,), such that a, g e R [ ~ , ( x , F ~ ) , L , 001 = R[tp(b',N,), L, co]. W.1.o.g. a, d a,.

Next notice that tp(bl,iQ, tp(b2,N,) are not weakly orthogonal. This is because we can define by induction on i < wb:, b ,2~N, , such that b: realizes stp(bl,a' U c2 U u,,,b,l U u,,,b,2) and b,2 realizes stp(b2,a1 U F2 U UjGib; U u,,,b,2). Now apply, V, 2.7.

Now note that by XI, 1.9 there is b' EP such that !=ql[b', all and b'+Nll hence a2 < R[tp(b',N,),L, 003 d R[vl(x, cl) ,L, co] = all so necessarily a1 = a2 and w.1.o.g. c1 = iY,q1 = v2, so let F = a 2 , p = q,.

W.1.o.g. tp(NI,P) does not fork over N,, and let N be primary and atomic over NI U N2 (exists by the (< co12)-existence property). By XI, $$2,3,N, sb,+N. By Pact XI, 1.4 (and X, 7.1, like XI, 1.8) there is b3 E N , I=v[b3, F] such that {bl , b3}, and {b2, b3} are independent over& but {b', b2, b3} is not. By XI, $2 (applied to cia) there i s P E Nsuch that N , U {b3} S I P and tp,(P,N, U {b3}) is almost orthogonal toN,. As N, E;,+N also N, Ebl+2v3. We want to apply 4.4 twice: first to prove that N I , P are isomorphic over Nl and then that P,P are isomorphic over N,. The only missing point is ' ' tp (b3 ,P) is regular". If this fails, then by XI 3.10 (formally, its proof') there is b4~p)(N, Q-N, , such that tp(b4,&) forks over a, henceR[tp(b4,&),L, co] < a,. There are J ~ E W , Z ' E P such that tp(b4,W U P)EFR,(& U d,) and so necessarily tp(&,N, U (2, U {b4}) forks over &. W.1.o.g. tp(d,^d,-(b4),N,) does not fork over El and we can easily find d E& realizing stp(d2, @ hence tp(d', F U dJ = tp(d2, C U d,) ; then find (see XI, 1.4) b 5 € N 1 such that i=v[b5,r] and tp(&,F U d' U {b5} ) forks over a, hence R[tp(b5, F U ?tl U d), L, co] < a,. So b5 contradicts the choice of a,.

DEFINITION 4.6: Assume T has finite depth. Let No E$,+N,, and we define

( 1 ) n(No,Nl) is the maximal n for which there are 4 ( 2 < I d n+ 1) such that for 12 1, (&, b)beNt- , Ebl+(&+l, b)bsNl- l l and there is a , ~ q+l-&l tp(a,,q) is regular orthogonal to and tp*(A$+,,l$ U a,) is almost orthogonal to 4.

(2) snd(No,N,) is the first K 2 1 such that for every 4 ( 2 < I < n(N,,N,)+ 1 ) as in ( l ) , SND(N,-,,N,) < K .

(3) k(No,Nl) is the maximal k such that k < n(No,N,) and there are q ( 2 < I < k + 1) as in (1) such that for every h < K there are 4(lc+1 < 16 n(No,N,)+l ) ) such that (4:s < I < n(No,N,)+ 1) is as in (1 ) and SND(N,-,,N,) 2 A.


(4) We define l(No,N,) as n(No,N,)-k(No,N,).

DEFINITION 4.7 : ( 1 ) We define n(N,), snd(N,), k(N,), l(N,) as in 4.6, just omitting the requirements connected with No.

(2) We define n(Tl) as the maximal n(Nl), we define snd(T) as the supremum of {snd(Nl) : n(Nl) = n(T)}. If this supremum is obtained, k ( T ) is the maximal

{k(N,) : n(N,) = n(Tl), snd(Nl) = snd(T))

and 1(T) = n ( T ) - k ( T ) . If the supremum is not obtained, k ( T ) = O,l(T) = n ( T ) .

MAIN THEOREM 4.8 : Suppose T is superstable without the dop but with the (a, 2)-existence property and of finite depth. Then for any Ha such that a > 2,IT',

](Ha, = a f i ( T ) [ c al(T)(~a+ IIA)]. A c s n d ( T )

This section up to 4.13 is devoted to the proof of this theorem. So we assume its hypothesis, and 4.11 and 4.12 implies it easily.

FACT 4.9: ( 1 ) I n Def. 4.6, 4.7 let No Gh,+Nl; if n(No,Nl) = 0, then snd(No,Nl),k(No,Nl), E(No, Nl) are not defined, if n(No,N,) > 0, then snd(No, N , ) , k(N, ,N,) , l(N,,N,) are defined, 0 < k(N,,N,) < n(N,,N,). I f n(Nl) = 0, snd(No), k(No), l(No) are not deJined; i fn(Nl) > 0, then they are defined and 0 < k(Nl) < n(Nl) . Lastly, n(T) = Dp(T) 2 0.

(2) 1 < snd(N,,N,) < (21T1)+, 1 d snd(N,) < (21Tl)+ [but i f N is Ff- saturated or just for some N,, Nl Gb,+N,, Nl # N,, then 2 < snd(Nl)] and 2 d snd(T) < (21Tl)+, if n(N,,N,) = 1 , then snd(N,,N,) = SND(No,Nl)+.

(3) I n Def. 4.6, 4.7 we can restrict ourselves to models of power <2ITI, without changing the results ( for those models and for T) . Of course in 4.6, 4.6(2) we should assume llNIII < 2ITI. (4) If N , c$ ,+Nl ,aENl , tp*(N,,&, U a) is almost orthogonal to N,,

then n(No,Nl) d Dp(No,Nl). (If No,Nl are Ff-saturated, equality holds.)

LEMMA 4.10: (1 ) Suppose No &b,+N,, llN,II < 2IT1, n(No,N,) = Dp(No, N , ) 2 1 and Ha > 2ITI. Then the number of models in K = qfi,, N , ) = { N : Nl G ;(,,+N, IlNll = K,, and tp,(N,Nl) orthogonal toNo} up to isomorphism over Nl is at most 3Dp(N,,N,) (1" + a 1 2lT') .

CH. XIII, $41 l ( N a , T) FOR 01 LARQE ENOUQH 649

( 2 ) If in addition n(N,,N,) = Dp(N,,N,), then the number is at most

' k ( N 0 . N,) [ A<snd(No. N , ) a,(No,N,)(("+al"+2s.)]. o r A - 1

Prooj. We prove this by induction on n = Dp(No,Nl) (simultaneously for (1) and ( 2 ) ) . We concentrate on the proof of ( 2 ) ((1) is easier). Let N E K . If n(No,N,) = 1, then k(No,Nl) = 0, Z(No,Nl) = 1 by XI, 2.6(2) , 2.8 there are (P, & ' ) E S E ( N ~ , ~ ) for i < a, such that {P: i < a} is independent over 4, tp(a',N,) is regular orthogonal to N,, tp,(P, N , U a') is almost orthogonal to N1,& G P and N is primary over ufP. So N is determined by the function giving for each isomorphism type of NESE(N,,N,) over N , how many times it appears among the q ' s . This function has domain of power SND(No,N,) and range {h:O < h < Nu} which has power la+wl. As by 4.9(3) snd(No, 4) = SND(N0,4)+, we finish.

Next let n(No,N,) = 2. By XI, 2.6(2) , 2.8 there are (P ,@)ESE(N, ,N , ) for i < a such that

{P : i < a} is independent over Nl, tp(d,N,) is regular orthogonal to 4, tp,(P,N, U a) is almost orthogonal to Nl,N, GP,~+P and N is primary over ul0P. Clearly Dp(&,N,) < 1 and w.1.o.g. Dp(a*,N,) = 1 iff i < p. The number of isomorphism types of P over Nl (/3 < i < a ) is < < I T ' (see 4.3(1)) and the number of times each appears

belongs to {x : 0 < x < Nu}, so the number of possible isomorphism types of UBcf<aM over 4 is at most

I" + a(21T'.

Now let i < /3 and apply XI, 2.16 for (TLo, Ga) (SON, C q) C , N , ) . There are <2ITI possible isomorphism types for hence < a,( la + " I A ( ( ) + 21T9 possible isomorphism types of P over N , , and h(i) = SND(N,,Pz) for suitable P2. So the number of isomorphism types of u, ,&" over 4 is the number of functions to {x : 0 < x < No} from the set of isomorphism types of P over N,, which we have bound above, a bound 3 I { x : O < x < No}l, hence the number is

A < snd(N,, N 1 )

Together with the bound on the number of possible isomorphism types of uBgl<aP we get the required bound, when k(&,N,) = 1. If


k(N,,N,) = 0, then SUPr<ah< < snd(N,,N,) and we get XA<snd(N,,,N,)

3,(lu+olA + 2ITI) implying the condition. For n > 2 choose N", a' as above and w.1.o.g. Dp(a*,N,) = n- 1 iff

i < b. Choose P2 cP,l+N?,I( U c N?,, llN?211 < 2ITI. The number of possible isomorphism types ofP2,/3 < i < 01, is < 3,(1fl) and for each the number of possible isomorphism types of N? over it is <3n-1(Io+u121T') (by the induction hypothesis and 4.9(2)). The number of possible isomorphism types of M2 over Nl is < 3,(1q) and the number of possible isomorphism types of N" over N?, for i < a is

The rest is easy, too.

LEMMA 4.11 : (1) Suppose llN, 11 < 2ITI, N, 2 2ITI. Then the number of models in

up to isomorphism over N, is at most 3Dp(T,((~+a121T'), when n(Nl) = Dp(T) is ,<

K = W,N, =;.,,+N, IlNll = N,)

&WI)[ A < s n d ( N , ) c 3,iN~)(lo+al"f21T1)].

'(N,,T) < > k ( T ) [ c 31(T)(IW+a1A+2'TJ) *

A < snd(T) 1 (2)

Proof. Just like 4.10.

Proof. Straightforward (remembering X, $6, Exercise XI, 3.2 (we shall not use in (2) the " +No")).

EXAMPLE 4.13 : For every k , 1 < o and cardinal K 2 No there is a totally transcendental theory T, IT1 = K such that for N, 2 K+ + H a

' ( N a , T ) = 31 [zr ~ ~ ( I ~ I A I ] *

CH. XIII, $41 I(&, T) FOR a LARGE ENOUGH 65 1

Proof. There is a theory q such that 1q1 = A,I(N,, q) = )aIA (just A pairwise disjoint infinite monadic predicates). So it is easy to find Tf,Z(N,, Ti) = l l(lal)A). By disjoint sums of I{x:x < K } J such theories we find Tk,,, IT",,I < U,,<,p++No, I(NQ, Tk,, = lk(lalA). Now it is easy to define T:: as required.

DISCUSSION 4.14: (1) So in 4.12(1) equality holds when T (is superstable and) with ( < 00,2)-existence without dop otop and of finite depth, and a 2 /I+2lTI.

So for T countable a 2 2'0, we can compute I(&, T) from few invariants of T. But what can snd( T) be ? So we are stuck with the continuum hypothesis type problem, i.e. whether snd(T) > N, * snd(T) = (2'0)'. Of course, if 2'0 = K, this holds, and if we start with V!= G.C.H. and add K , (cfK > No) generic reals, still snd(T) > N, * snd(T) = (2'0)+. If we translate our problem to descriptive set theory we get as a sufficient condition:

Assume there is a set A of reals IAJ > No, and (V'Torl.. .E A( [A, ro#&(ro, r l , . . .)] where 4 is an analytic function. Is there such A, IAl = 2'0?

THEOREM 4.15: For every countable theory exactly one of the following occurs :

(i) I(NQ, T) = 1 for every a > 0, (ii) I&, T ) = a,, ( i . e . Min{2'*, 1,)) for every a > 0, (iii) I(NQ, T) 2 la+ 11, for every a.

Proof. By XII, 6.1 we can assume that T is superstable, without the dop and otop, and shallow. If T is totally transcendental, then if T is unidimensional by IX, 1.8(2), Case (i) holds, and if T is not unidimensional by IX, 2.1, Case (iii) holds. If T is not unidimensional (not necessarily totally transcendental), Case (iii) holds by V, 2.10(2) (and Def. V, 2.2).

So we can assume that T is not No-stable, is unidimensional, hence has depth 1. By IX, 1.20 I(&, T) 2 Min{2'#, a,} for a > 0. Now if snd(T) > 2, by 4.12 I ( N u , T) 2 la1 when a 2 2'0, hence together with the previous sentence implies (iii). However, if snd(T) < 2, it is 2, and by 4.11 I@,, T) < 22'T' = 1, so (ii) necessarily holds. [As an alternative to the use of 4.12 we can note that if there is (for T) a regular non-trivial type, then every regular type is not orthogonal to it (as T is unidimensional), hence is non-trivial (by X, 7.3(4)), so 4.5


is always SND(N,) < 1, and it is easy to prove (ii). On the other hand, if every regular type is trivial the computation of the dimension becomes easy (see X, $7) . ]

A personal remark. This theorem has a personal significance for me. In 1969, just after reading Morley’s paper, I conjectured it. As this has been, in a sense, the first step toward this book, i t seems appropriate to close the book with an affirmation of the conjecture.

APPENDIX

A.O. Introduction

This is a technical chapter containing the combinatorial theorems needed in the book. Of some self-interest may be 1.6, in which we find when a family of sets haa independent subfamilies; 2.4 which is a weakened version of Halperin-Lauchli’s theorem, proved by a different proof; and 2.6 which generalizes the ErdosRado theorem to trees. In Section 1 we introduce filter and ultrafilters, with some basic

lemmas (e.g., every non-trivial ultrafilter can be extended to an ultrafilter). Then we deal with the filter D(a) (generated by the closed unbounded subsets of a) end prove that we c&11 usually split a stationary set to of a stationary subsets. We prove some theorems saying that families of sets have some nicely looking subfamilies, and the existence of families of 2” functions from h to h (or subsets of A) which are very independent. In Section 2 we deal with the partition calculus: we prove the

theorems of Ramsey, a weakening of Halperin and Lauchli, Erdos- Rado and a generalization to trees; and that if IS1 > A, 5 E B =- lf(z)l < A, there is an independent B* c B, IS*[ = 181.

Section 3 deals with K-contradictory orders and connected antisymmetric relations.

A.l. Filters, stationary sets and families of sets

DEFINITION 1.1: (1) D is a filter over I, if D is a non-empty family of subsets of I and

(i) A, B E D implies A n BED, (ii) i f A c D , A c B r I t h e n B E D . (2) The filter is trivial if 0 E D, we sometimes “forget” to my our

filter is non-trivial. 653

654 APPENDIX lap., § 1

( 3 ) D is an ultrafilter over I, if D is a non-trivial filter over I, and for

(4) Let S be a family of subsets of I, the filter [S] generated by the

(6) The filter D over I is principal iffor some A , D = { B c I : A s B}. (6) The filter D is A-complete if A, E D(i < a < A) impliea

every A G I, A E D or I - A E D.

family S is {A: A E I , and n,<,, B, c A for some B, E Is).

n,<= 8, E D.

THEOREM 1.1 : (1) For any family E of subsets of I , [ E ] is a filter Over I . (2) The filter [El is non-trivd iff every finite intersectbn of members

( 3 ) Any non-trivialjElter can be extended to an UltrajElter. (4) Any filter ie *,-complete. (6 ) An dtra$lter D Over I i s ~ r u t i p d ifl for 8 m t E I , {t} E D, 80

of 8 ie non-empty.

D = {A E I : ~ E A } .

Proof. Part (3) follows by Zorn’s Lemma. The other parte are eaey ,and well known.

DEFINITION 1.2 : ( 1 ) A set S is unbounded under a if for every /3 < a, there is y E Lg, p 5 y < a. (We omit “under a” when it is clear.)

(2) A subset S of a is closed, if for every limit 6 < a when 8 is unbounded under 6, 6 E S .

( 3 ) D(a) is the filter generated by the family of closed unbounded subsets of a.

THEOREM 1.2: If cf a > No, then D(a) is a (cf a)-mplete non-trivial JiZter Over a.

Proof. As any unbounded subset of a is not empty; it suffices to prove that if A, (i X,, Un<a, p,, < a and as each A, is closed and unbounded under

and if p,, is defined, define inductively p,, (i < /3) so that = P,,, pi+‘ E A,, u,<, < p,, < a. This is possible aa each A, is unbounded under a, and p < cf a. Clearly pn+l =

vn, P n + J n 4 z 0 for i < 8. [(a, &+I) = {Y: ptt < Y < A+l).I

uncm pn, unea, p,, E n,<# A,, hence nfCb A, is unbounded. b t p 0 = y

a < a will be suitable. For simplicity, we shall now concentrate on D(A) for regular A.

AP.9 § 11 FILTERS AND STATIONARY SETS 655

DEFINITION 1.3: A subset A Of A is 8 ~ ~ ~ ? ’ 9 if A - A $ D(A).

THEOREM 1.3: (1) If A is a stationary subset of A, A regular > No, f a function from A into A, 0 # u ~ A - + f ( u ) < a, then for some /3 {y E A : f ( y ) = p} i8 StatiOWWy.

(2) If A is a Stationary subset of A, A regular > KO, then A can be partitioned into A pairwise disjoint, statitmury subsets, pmvided tirat for some p < A, (Vu E A)[cf a 5 p].

(3) If K < A are regular, then {u < A: of a = K } is a stationary subset of A.

Remark. The additional condition in (2) involving p is not, in fact, necessary.

Proof. (1) Suppose not, then for every f? < A there is s# E D( A) such that y €88 * f ( y ) # p. h t R = {(y, p ) : y €86) and M = (A, < , f , R) and Ma the submodel with universe a. By VII, 1.4 S = {u < A: Ma < E D(A) hence A is not disjoint to S.

So let S E S n A, then clearly for every p < S,S# is unbounded under A hence under S (using that M C (Vz)(3y)[z < y A R(y,p)] and

(2) For every u E A choose an inorewing sequence &(i ) (i < of a 5 p) whose limit is a. Let S(P, i) = {u E A: pa(i) = p} (p < A, i < p), so clearly 8, # pa, i < p =- S& i) n S(&, i ) = 0. If for some i < p, {p c A: S(P, i) is stationary} has cardinality A, clearly we finish. Otherwise for every i < p there is p[i] < A such that for /I 2 p[ i ] , S(@, i) is not stationary, so let it be disjoint to some closed unbounded S*(/I, i). Clearly p* = supler p [ i ] c A as p < A. Let

< M) hence 6 E S ~ . But f ( 6 ) < 6, contradiction.

R, = {(u, i, &(i)): a E A, i < cf a},

& = {(& i, y>: y ES*@ i), /3 2 P*, i < p},

M = (A, < , R,, R,, p*) and Ma the submodel of M with universe u. E D(A), so there is S > p*,

6 ES n A. As in the proof of (1) it is easy to show that for i < p, p 2 p*, p < 6, 6 ES*(B, i), hence &(i) # p. But a8 p* < 6, and the sequence pa({) (i < cf 8) converges to 6, we easily get a contradiction.

By VII, 1.4 S = {u < A: Ma <

(3) We leave it to the reader.

THEOREM 1.4: (1) If A is regular and x < A =- x<” < A, and < p f o r t € W,where (W( = A y t 7 m f o r s m W G W , IW’( = h a n d s , for a n y s # t ~ W * , S ~ n S , = S .

656 APPENDIX k . 9 8 1

(2) Moremer i f St = {a;: i < a(t)} we can ae%unce tlrat t E W' * a(t) = aQandforsome U c a * ; i E U , t E W ' * 4 = d a n d 8 = { s ' : i ~ U ) a n d 8:(1, = 8&lo e+ 8&4) = 8{(2) for t ( I), t (2) E w', and 8!(1) = 8&a), i # j E w' * t ( l ) , t(2) E u. Proof. W.1.o.g. W = A, 8, E A and let K = p when p is regular, and K = p+ otherwise. Clearly K < A and K is regular. For any a < A, cf a = K, clearly S, n a is a bounded subset of a, so let h(a) < a be a bound of it. By 1.3(3) {a < A: of a = K } is stationary, and by 1.3(1) on some stationary W1 c {a < A: cf a = K } h haa constant value 8. As ISl<fi < A, by 1.3(1) there are a stationary Wa c W1, and 8 c 8 such that for every a E Was 8, n a = 8, n 8 = 8. We can easily find a stationary W' c Wa such that a < y E W' implies 8, c y. Clearly W , 8 satisfy our requirements.

THEOREM 1.5: (1) I f ASK = A, t h there ie a family '3 of 2" functiOne from A into A, such that for any d ie t inc t f ,EB (i < a < K ) and any ordinals y, < A (i < a), (5 < A: for every i < a,f ,(5) = y;} # 0.

(2) If A<' = A there are subsets S; (i < 2 9 of A, d that for any disjoint nun-empty U, V c A, IUl + I Vl < K , the eet UieUS, - UtevS,

Proof. (1) Let {(At, (q: 5 < a;), (j!: 5 < a;)): i < A} be an enumeration of all triples (A, (0,: 5 < a), (j,: 5 < a)) such that:

(2) The same proof, eesentidy.

has caraimiity A.

(i) A is a subset of A, IA1 < K.

(ii) 0, is a subset of A and 5( 1) # 5(2) implies Cc,l, # Cc(a,. (iii) a < K and j, < A.

(Clearly the number of triples is A, aa A<" = A.) Now for every set B c A we define a function f B : A + A. We define f & ) aa follows: if

If Batn ( 5 < a < K ) are distinct and j, < A ( 5 < a ) we can find A c A, IAI < K so that B , n A m distinot; ao for some i, A, = A, a; = a, C: = B,,, n A, j: = j,, so fBaCc,(i) = j,. Hence ( f B : B c A} is a family satisfying our requirements. .

(2) Immediate by (l), for if B = (f,: i < 2"}, let St = {a < A:

THEOREM 1.6: (1) I f S ie an infinite family of subsets of I , then 2oe can jEndt, ,EI,A,ES(n < w)suchthat

B n A; = Cis f B ( i ) = j:. Otherwise f B ( i ) = 0.

ft(4 = 01.

(i) t, E A, iff n = m, or (ii) t , E A , i , n # m,or

f l . 9 8 11 FILTERS AND STATIONARY SETS 657

(iii) t, E A , iff n c m, or (iv) t, E A , iff n 2 m. (2) For every n < o there ie k < o such that i f IS1 2 k, S a family of

8et8 then we canjhd t,, A, (i < n) a% above.

Proof. (1) We define by induction on n, t,, A,, S, such that: (a) So = S, #,+, c S,, S, is infinite. (p ) Si = {A E S,: t, E A } and Si = {A E S,: t, $ A} am non-empty.

This is easy as we can always find a suitable t,, since S, is infinite, and then it is easy to choose A, and S, + By Ramsey’s theorem for a 2-place function with range of cardinality 4 (see 2.1) the conclusion is easy.

( y ) A n ~ S i o A n $ S ; o S n + l # SieSn+l = S : .

(2) Essentially the same proof.

DEFINITION 1.4: (1) Ded A is the first cardinal p , such that no tree with A nodes has 2 p branches (a tree is a partially ordered set T such that for every x E T {y E T : y < x} is well-ordered. A branch is a maximal linearly ordered subset).

(2) Ded, A is the first regular cardinal which is 2 Ded A.

THEOREM 1.7: (1) Swppo~e S i8 a family of subset8 of I , I injnite, IS1 2 DedJIl. Then there are, for every n, element8 to, . . . , tn-l of I , 8wh that for every w c n there is A, ES such that: ti E A, e i E w for i c n.

(2) Suppose S ie a family of subset8 of I , I is jnite, IS1 > Zisn ( y ). Then there are element8 to, . . . , t, - , E I 8wh that for every w c n there is A,,, ES such thd t, E A, oi E w.

Procf. (1) Let A = 111, p = Ded, A, and w.1.o.g. I = A, and J c I, IJI < A implies I{A n J: A ~ L 9 ) 1 < p. Let for a s A S: = { (A n a, a): A EL^). So clearly IS:[ < p for a < A. Let (A , a) 1 /3 = (A n /3, min{a, 18)). On Uarh St we define the following partial order: (A l , a l ) 5 <A,, .a) 8 a1 s aa, A, = A, n al. Clearly under this order Uarh Sg is a tree. Let for a < A

658 APPENDIX

1s: - sy = u ( t E s g : 8 zs t } 8ESb) - s: I a<A

< p

EXERCISE 1.1: Show that if p < Ded, A, III = A, then there is a family S of subsets of I such that the conclusion of 1.7(1) fails even for n = 2.

EXERCISE 1.2: Show that the bound in 1.7(2) is the best possible.

PROBLEM 1.3: Can we replace Ded,lIl by DedlIl in 1.7(1)?

EXERCISE 1.4: Prove x < Ded A iff there is an ordered set J, I J I 2 x , and a dense subset I E J, IIl zs A. (Hint: For * use a lexicographic order on the tree plus its branches. For -= the nodes will be intervals of I.)

LEMMA 1.8: (1)Suppose 8, i8 TegUlar , < N, for t E W, I Wl = NB+, then for 8ome W' G W, lw'l = NB+, and V , IVl < N, and for every distinct s # t E W', S, n S, E V . If W E (6 < K,,, : cf S 2 Ka} is stationary, we can have W' stationary too.

AP., § 21 FILTERS AND STATIONARY SETS 659

(2) Imtead "8, is regular" we can demand lStl < cf X, for t E W . If W = gB+ny 8, G HE+,,, we can a 8 8 U W y ES, - V impJim y 2 a f o r Y E W'.

Proof. (1) Similar to 1.4: we can assume for some p lStl = p for every t ; W = h,Sa c h.Thenforsomestationary W1 G {a < &+n:Cfa = p+}, and y < N,, for every a E W1, S, n a c y ; and S, c a for 4' < a. Now we prove by downward induction on 1 5 n that there is V , c y , IVl < HE+,, and I{a E W': S, n y c V,}l = a,,,. For Z = n Vn = y ; for 1 = 0 we get our conclusion, and if V,+l is defined, let V,+, = Ui Vl, 'IVlI < HE+,, Vj increasing (i < KB+l); one of the Vj is as required on Vl .

(2) Similarly.

A.2. Partition theorems

THEOREM 2.1 (Ramsey's Theorem): (1) For any infinite ordered set I , and n-place function f from I , with range of cardinality < No there is an infinite set J c I a h thud if I , Z E "J, 3, 8 increasing, then f ( I ) = f ( 8 ) .

( 2 ) For any n, k, 1 < w there is m = m,(n, k, 1) such that if I is an ordered set of cardinality r m , and f an n-place function from I with range of cardinality sZ, then for some J c I , I J I = k; I , Z E nJ, I , Z increasing =- f(3) = f ( t ) .

Proof. (1) For simplicity we assume I = w . We prove the assertion by induction on n. For n = 0 or n = 1 there is nothing to prove, so assume we have proved for n, and we shall prove for n + 1.

We now define by induction on k c w , natural numbers Z(k) and infinite sets 8, c w such that

(i) Z(k) < Z(k + 1 ) s 8, C Sk-1, (ii) for any m E sk, Z(k) < my (iii) for any m,, . . ., m,,-l c k, m ESk

f V(m0), . - . , J(mn-l)y W ) ) = f(l(mo), - . - 3 4mn-1), m).

We choose Z(0) = 0, So = {i: 0 < i c w}, and if Z(k), SI, are defined, let E , be the equivalence relation on Sk defined by: iE,j o for any mo,. . ., mn-1 s k, f(Z(m0),.. ., Z(mn-l), i) = f(+,), . . ., J(mn-l) , j ) . clearly 8, has only finitely many equivalence classes, so at least one of them is inhite; choose Z(k + 1) from such a class and let = {i: i E S ~ , Z(k + 1) < i, iE,$(k + 1)).

660 APPENDIX [-., § 2

Now define on I' = {Z(k): k c w } a functionf',f'(Z(k,), . . . , t(k,,-l)) =

Now by the induction hypothesis (applied to I* , f ') we get our con-

(2) We leave it to the reader.

f (Z(ko), . . . , Z(k,,_,), Z(E)) for all large enough k.

clusion easily.

THEOREM 2.2: For any n, k < w we can define an increasing sequence Z(i) (i < w ) w h that the, following h o h : Suppose f is an n - p h fandim from 1(:)2 into k, then t7we are h, : I r 2 --t l(:)22, rn c n, and k* c k Buch that

(i) hm(q) has tenstla 4J(q))* (ii) q 4 u * hm(7) Q It&). (iii) hm is one-to-one. (iv) I f 9 , ~ ~ 2 ( m c n)aredistinct,thenk* = f(ho(qo), ...,h,,-l(q,,-l)).

Proof. Let Z(0) = 0 and if Z(i) is defined, we want Z(i + 1) to be such that:

(*I If IStl 2 Z(i + 1) for i E I, 111 = n2":), and g is an n-place function from UaerSa into k then there are distinct a:, ui E 8, (a E I) and an n-place function g' from I into k such that g(a;'& . . . , a:(&) = g'(a(l), . . . , a(n)) for any distinct a ( l ) , . . . , a ( n ) ~ I and anyj(l),...,j(n)c{O, 1).

We can do this, for let Zr(0) = k, Z:(a + 1) be 2 to the power n[lr(a)n21(i)]n, and 21(1+1)-1(') 2 E;"(n29. Now for any suitable I, Xu, g let I = n2":) for simplicity, and now define by induction on a c n2'") sets W , and elements a:, a; such that:

(i) W, E L. I".: a s j < n2l(:)} and W,+l E W,, (ii) I w, n s,i = . ..: * -2lW - a) for j i ,-:, (iii) a: # af ES, n W,, (iv) if b o y . . . , b,,-a E {uj: y c 2, p c a} u U {S,: a c j c n2"{)} and

m 5 n - 1, then

g ( ~ o , . . . Y ~ r n - l , u ~ , b ~ , . . . ~ b , , - a ) = g(boy...,bm-l,af,bm,.. .,bn-d.

If we have defined for fi c a, we first choose Wa to satisfy (i) and (ii) and then we can eaaily satisfy (iii) and (iv). Clearly a!, ui me as required in (*).

Now we prove the theorem by induction on i. For i = 0 there is nothing to prove. Suppose we have proved for i and we shall prove for

A P . 9 5 21 PARTITION THEOREMS 66 1

i + 1. We apply (*) for I = l(O2 x n, f (or more exactly, some suitable extension of it) and B<,,,> = {v: v E l({+l)2, r) Q Y) , and get'distinct p&,> E B<,,,), a = 0, 1 and an n-place function f' from into k, so

Now we use the induction hypothesis on i for f ' and get suitable h; (m < n), k*. Now define h, so that for r) E i22, hm(r)) = h&(r)) and for

a(n-1) thatf @%b).o>, &&,1>, * - * 9 P<r(n-l).n-l> = f'(do), ~ ( 1 ) s - - * 9 q(n - 1))-

r ) E '2, hn(r)Ya)) = P7v.m>.

THEOREM 2.3: Suppose n, k < w and f is an n-place function from O>2 into k. Then we can jind functions h:, hz from 2 (for rn < n) and k* < k and natural numbers lo( i ) , P(i) such that

2 into

(i) lo(i) , P(i) increase with i, (ii) h:(r)) 7m Zength lo(l(q)), hi(r)) h.9 length l l( l(r))); h: and h i are

one-to-one, (iii) r ) Q v iflhi(r)) Q h%4, (iv) h i m -4 %(r)), (v) for any i and q0, . . . , E 9 , distinct,

k* = f(h:(r)o), - * - 9 hLl(7S-l)).

Proof of 2.3. Define l ( i ) as in 2.2; and apply 2.2 to get for each i < w ,

m < n, functions hm,{ (m < n) and kr < k. For every j the number of possible (h,,* '+2: m < n ) is finite, hence

by Konig's lemma there is an increasing sequence i (a ) < w (a < w ) and k* so that h,,{(") t j 2 2 = hm,f(j) t j r 2 for a 2 j and k&) = k*. Now for r) ~ ' 2 let r)' E *('I2 be defined by r)'[a] = r)[a], a < j, r)'[a] = 0,

j I; a < i(j) and let %(.I) = h,,dq'), h h ) = h,,~o,(r)) and clearly they prove 2.3.

THEOREM 2.4: Suppose f , is an n(m)-place function from O > 2 into a jinite set. Then there are functions ho, h1 : ">2 +- O>2 such that:

(i) Z(r)) = l(v) implies l(ho(r))) = l(ho(v)), l(hl(r))) = l(hl(v)); ho and hl are one-to-one.

(ii) 7 Q Y iifs hl(v) Q hl(v),

(iv) i f m I i I min{a, ,!I}, ql E "2, vz E 82 (for l < n(m)), vl t i = (iii) hl(r)) 4 ho(r)),

r i, ~ ( 1 ) z ~ ( 2 ) r i z vlca, r i , , thn

fm(ho(70), * - 2 ho(7n(m)-1)) = fm(ho(vo), * * 9 ho(vn(m)-l))-

Proof. Immediate by repeated uae of 2.3.

662 APPENDIX [M., § 2

DEFINITION 2.1 : A + (K); if for every n-place function f from h into p there are 8 5 A, 181 = K and a < p such that for any increming sequence a E n8, f (a) = a.

THEOREM 2.6: an@)+ -.(A+):+'.

Proof. By induction on n. For n = 0 it is easy, so suppose we have proved it for n, and we shall prove it for n + 1. Let x = p = an@); and define an increasing sequence pi (i < p') of ordinals < x + such that (*) if 8 c pi, lLSl 5 p, then for every y < x+ there is anordinaly' < ~i+lsuchthatforanyao, . . . , s n - , ~ 8 f ( s 0 ,..., 8n-1,y) = f(sO,. . . ,an-l ,y ') a n d y ' E B e y E 8 . Thisiseasilydone byinduc- tion, by cardinality considerations; let p* = Uiep+ pi. Now define by induction on i < &+, yi < /3f+l, such that for any so,. . ., an-1 E

{y j : j < i},

f (80, - - - 9 sn-19 ri) = f (80, - - - 9 an-l ,p*h j < i * rj Z: Yi.

Define an n-place function f ' on p+ :

f ' ( i ( o ) , . **ni(n - 1)) =f(3/i(o),...,ri(n-l),B*).

By the induction hypothesis there is 8' c p+, 18'1 = A+ so that for any i(0) i (n - 1 ) ~ 8 ' , f'(i(O),...,i(n - 1)) = ao, for some fixed ao. Now S = {yi: i E 8') proves the theorem.

THEOREM 2.6: For every n, m < w there is k = k(n ,m) < w (and k(n, 1) = 0) such that whenever h = 5&)+ the follolving ho&: I f f is an m-place function from n2 X into x (or any set of cardinality 5 x), then there is I s such that:

(i), ( ) E I and i f q E I n n'A, then l{a: a < A, f ( a ) E I } ! = x+, (ii), if qo, . . . , qm- 1, yo, . . . , vmW1 E I and (qo- - -) - (vo, . . .) (see

V I I , Definitions 2.3 and 3.1), then f (qo, . . . ) = f (yo, . . . ). Remark. We do not try to get the best k(n, m).

Proof. We fist prove for m = 1 that we can choose k(n, 1) = 0, and then prove by induction on n. Clearly we can assume n, m > 0.

Case 1. m = 1. Let k = 0, so A = x+. We define, by induction on j 5 n, for any q E n - f h a set I, E {v: V E " ~ A , q Q v or q = v} and ordinals an(q), . . . , an- j (q ) < x such that:

(i) q E I, and if v E I, n " 'A, then l{a c A: v-(a) E I,,}] = x+,

A P . 9 5 21 PARTITION THEOREMS 663

(ii) foranyvEI,,n'A,n - j 5 i 4 n,f(v) = q(r)).

For j = 0, r ) €"A, I,, = {r)}. Suppose we have defined I , for any 1 7 ~ " - j + l h , and let r ) ~ ~ - f A . For any P < A, the sequence (a,,(r)^(j?)), . . . , t~,,-,+~(r)~@))) is well defined, and there are only 5 2 = x such sequences, hence there is a sequence (a,,, . . . , a,,-,+,) which we get for A = x + pa. Let a,,(r)) = a,,,. . .,t~,,-~+,(r)) = a,,-,+l

and Qn-,(r)) = f (4 and

I , = (7) " u { In^: (%(r)-(P)), * - * 9 % - j + l ( f ) >

= (a,,, * * ., %-,+l)}* Clearly this satisfies our demands.

Case 2. n = 1. This follows by 2.6, taking k(n, a) = m - 1.

Case 3. Suppose we have proved for n, m, and we shall prove for n + 1, m. Let p = &,a+,,,+,(x), k(n + l , m ) = k(n,m) + m2 + m + 2. Define

an m-place function g on n2A:

g(r),, . . . , r),,,-,) = {(h, Po, . . . , P,,,-,, a): h a function, w -c my h: w + my

Pi < p (1 < m) and a = f (v,, . . . , v,,- 1) where

1 E w * Vl = r)h(l)-(Pi), 1 $ w =+ Vl = q3.

Clearly the range of g haa cardinality I 2'. By the hypothesis of Cam (3) there is a set I , c IZ A which satisfies:

(i)' ( ) E I, and r ] E I, n " > A implies I{a < A: f ( a ) E I,}! = (2")+, (ii)' if +j, i j E I,, q - i j , then g( i j ) = g(ij).

We cas find I, c I, such that (i)" ( ) € I , , and r ) €1, n implies I{a < A : f ( a ) E I , ) ~ = x +

(define I, n 'A by induction on i). In view of (ii)' it suffices to find for each r ) E " A n I, a set I,, c

{r)^(a): a < p}, iI,l = x+ such that:

(*I If ij,q are similar sequences of length m from I* = U{I,,:~EI,n~A}uI,andij[I] I n = $1 rn thf (7 j ) =

f (6). For this it suffices to prove Theorem 2.7 below.

THEOREM 2.7: 8wppme IX,l = amnc,,,+l,(x)+ for i < x ( X i well ordered by 5 ) and f k an mplace fundion f r m Uiex X i into a set of cardinality

664 APPENDIX LAP., $ 2

S X . Then there are sets Y , E Xi, I YiI = X + 8w:h that: if s(Z), t(1) E Ti(,, (1 < m) a d i( l l ) = i(Za) [8(11) < a($,) o t(ll) < t(l,)] then

Proof. We define for n 4 m , i < x sets XT such that (1) X! = x 1.3 xn+l i c - xn i , IXFl=~m(rn-n+l ) (~ )+ , (2) if 8(1), t(1) EX;,), 2 < m , I{i(l): i(1) r i(lo)}l 4 n, and

then f (8(0), . . . , s(m - 1 ) ) = f (t(O), . . . , t (m - 1)). We define Xr by induction on n. For n = 0, Xp = Xi and clearly

(1) and (2) hold. Suppose we have defined for n, and we shall define XT+l by induction on i. Suppose we have defined X;+l for j < i; ohoose @ c X:, IS:[ = m for i < a < X .

By 2.6 we ccm easily find XP+l c X:, IX;L+lI = arn(rn-n)(X)+ such that (2) holds for n + 1, i (lo) = i when i (1) 4 i * s(l), t(1) E Xzlf and i ( l ) > i =. s(l), t(1) E&). By the induction hypothesis on n clearly (2) holds. Now let Y , = Xf end clearly the Y,'; rn as required.

THEOREM 2.8: Suppme IS1 > A, S is the h d n o f f , and for x €8, If (x)l < A ( f ( x ) is a set). Then there ie S* €8 , IS*! = 181 8w:h thut s # t E S* implie8 e $ f (t).

Proof. Cwe 1. IS1 = p is regular. W.1.o.g. S = p; x E S =. f ( x ) s p and define Si by induction on i s A, so that Si is a maximal subset of p - a f [where 4 = sup Uj,, 4, a? = sup U { f (B ) u v}: B < a:)] which satisfies 8 # t E st s- 8 $ f (t).

If for some i, ISigrl = p we finish. Otherwise, i 4 A * a? < p, and notice that j < i, s E p - a? implies f (8) n S , # 0. Now clearly the 8,'s &re pairwise disjoint and for any 8 E p - a!; f (8) n Si # 0 for i < A, hence If ( a ) ] 2 A, contradiction.

Cme 2. IS1 = p is singular. Let p = Z,,,p(i), K = ofp, p( i ) , i < K , is a strictly increasing sequence, A, K < p(0). W.1.o.g. we can assume that S = p, end x € 8 e- f ( x ) G 8; end a < /3 =. /3 $ f (a ) (as we can replace 8 by any S' c S , IS'I = p) and /3 # y ~ ( p ( i ) , ~ ( i ) + ) = {a: p( i ) < a < p( i )+) implies B $ f ( y ) (by Case 1). Now we define by

f i . 9 8 21 PARTITION THEOREMS 665

induction on t; < x = max{K+, A + } sets 8: c (p(i),p(i)+) (for i < K)

such that (i) a €8: impliesf(a) is disjoint to U,<, 8{, (ii) a E Sl, /3 E Ueec 8: implies /3 < a, (iii) 1S:I = p(i) or It$! < p(i) and St is a maximal subset of

(p(i), p(i)+) satisfying (i) and (ii). Clearly for any t; < x, ct # /3 E

u,<%t$ =- a$f(/3) hence if lUf<xS~l = p we finish. Otherwiae, for every t; < x there isj([) < K such that IUf<lsl$l < p(j(t;)). As x is regular and > K there is j0 < K such that IUl = x where U = {t; < x:j(t;) 5 jO}; so for all t ; ~ U, l8f'l < p(jo). There is a such that p( jo ) , sup Uc6uS{o < a < p( jo )+ so by (iii) for all t; E U, f ( a ) n U,<,o#{ p 0 hence lf(a)I 2 x > A, contradiction.

EXERCISE 2.1: Suppose I c 0 2 A , and v 4 7 , 7 E I =- v E I and for r ) E = > A n I define A,, = l{i < A: r)^(i) E I)I.

(1) I f a = n < w,f:I+XandcfA,, > Xforeveryr]En>AnIthen there are J c I and al, I 5 n such that ( ) E J and E J n " > A implies h, = I{i < A: r)^(i) E J}I and 7 E J impliesf(7) = aI(,,).

( 2 ) If a = w and for every 7 E I , cf X, > Xn, then there are J E I and aI IM above.

EXERCISE 2.2: Reprove 2.7 as follows: Msume IX,l = A = D,,,(x+)+, rn > 1, X , = h x {i} (for i < p) and f 5 x+ and define

and use 2.6; and get suitable S E A, 181 = x+ w.1.o.g. 8 = x+ and let Y, = (xi, ~ ( i + 1)) (or xi, x(i + 1)-ordinal products).

EXERCISE 2.3: Reprove 2.6 similarly, assuming h = p = IRangefl, 2' 5 x+ using

g(ao, - - * 9 amn - 1 ) =

= {(c, (i:, * - i m - 1 iP,o)), - - * (ic-l, * * * 9 I ( m - 1 ) ) ) :

c = f ( ( q 8 , . . .), . . . , (a,;-1,. . .)), Z(k) < n, i: < mn}

get 8 s A, IS( = x+ from 2.6, w.1.o.g. S = x+ and let I = {(ao,. . . , a I ) : 1 5 n, and for each k 5 1

Ak+lak < ak+l < Ak+lak + A and 0 < a. < A}.

666 APPENDIX [AP., § 3

A.3. Vsrious results

LEMMA3.1:IfK < A 5 x,xBinguZarandforeveryp < xAD(~,A,A,K) holde, then AD(x, A, A, K) hide (see Definition VII, 1.11).

Proof. Clearly if AD(p, A, A, K ) holds, 8 is a set of cardinality A, there is a (A, K)-family of p subsets of 8. Let x = z,<,,o p,, p, < x ; and let 8' be a (A , K)-family of subsets of A, 8' = {A,: i < po}. As lA,l = A, there is a (A, K)-fadY, 8,, of p1 subsets of A, for i < po. Clearly (J,<uo8, is a (A, K)-family of x subsets of A.

EXERCI8E 3.1: Prove A D ( A , A, A, 1) and natural implications.

LEMMA 3.2: Suppo8e AD(x, A, p, K ) holds. (1) Then An 2 x; 80 x = 2A, (2) If h > 2", N, = min{A,: A: 2 A} and A s Nu, then for 8ome /3,

2 K implies An = 2".

A 2 N, = /3 > N, > 2%.

Proof. (1) Let 8 = {A,: i < x } be a (p, +family of subsets of A. Choose B, c A,, lB,l = K ; so i # j + B, # B,, hence.AK = I{B c A: IBI = K } I 2 l{B,: i < x}l = x.

(2) Immediate by VII, 1.9 (A , K , p correspond to N,, p, x). On K-contradictory orders and K-skeleton like sequences see Defini-

tions VIII, 3.1 and 3.2 (VIII, Section 3 is the only place they are used).

THEOREM 3.3: For every A = A + K + , K regular, there are 2A pai&e K-cOntradkt("y Order8 Of Ca?'di?U&ty A.

We prove 3.3 by a series of claims.

DEFINITION 3.1: The orders I, J will be called strongly K-contradictory if they have cofinalities 2 K and there are no orders I,, J , with cofinality 2 K such that there is a model M with an anti-symmetric relation < and K-SkeletOn like sequences (a8: 8 E I , + I*) and (b,: 8 E J , + J*) such that:

(1) For every t E J* and 8' E I , there is 8 E I , , 8, < 8, M C a, < b,. ( 2 ) For every ~ E I * and 8 l ~ J , there is ~ E J , , 8, < 8, M Cb, < a,.

(I* is I with order the inverse of the order of I.)

Remark. By the definition of K-skeleton like sequences (Definition VIII, 3.1) (1) implies that for every t E J * there are 8 O E I , , 8, E I* such that

dp., !I 31 VARIOUS RESULTS 667

I , + I* k 80 I; 8 s 81 implies M C a, < b,. Similarly for (2). We shall use this many times.

CLAIM 3.4: Any t ~ 8 ~ ~ ~ ~ y K - C O d ~ a d i c t 0 p . y Order8 are K-COd?'&i&Wy.

Proof. Immediate.

QUELZTION 3.2: Is the converse true?

CLAIM 3.6: If I, J have distinct cojkl i t im >K, then I , J are strongly K - d r d i d 0 p . y .

Proof. W.1.o.g. c f J = p > A = cfI , A 2 K . Suppose I, J are not strongly K-contradictory, and we shall get a contradiction. By Definition 3.1, there are M, I,, J,, (as: 8 E I , + I * ) , (b,: t E J , + J * ) satisfying the conditions mentioned there. Let (44: a < A) be an increasing unbounded sequence in I, and for each a < h choose t ( a ) E J and t'(a) E J , such that J , + J * C t'(a) I; t I; t(a) implies b, < a,(,). As J has cofinality > A, there is a bound t ( * ) E J to {t(a): a < A}.

So for every a < A, J 1 + J * C t'(a) I; t ( * ) I; t(a), hence b,(.) < a,(,). This contradicts (1) from Definition 3.1 (by the remark to it).

CLAIM 3.6: If A i8 a regular cardinal > K , K regular, I = Z a e h I:, J = I,<,, J,* and {a < A: cf a 2 K and I,, J , are 8trongly K-contradictory} # 0 mod D(A) ( A = 0 mod D(A) ifSA - A E D(A), A G A), then I , J are ~trongly ~-COdrdictory.

Proof. Suppose there are a model N, orders I , , J 1 and sequences (a,: 8 E I1 + I*), (b,: t E J 1 + J *) satisfying the conditions from Definition 3.1, and we shall get a contradiction. As I,, J , are non-empty, we can choose 8, = 8(a) E I,, t, = t(a) E J,.

Now we define ordinals a, = a(i) for i < h such that: (i) j < i < A implies a, < a, < A, (ii) for a limit ordinal 6, ad = sup{a,: i < a}, (iii) if 8 E I*, I' + I* C 8 I; 8 ~ , + 1 ) then N C a, < btca(,),, (iv) if t E J * , J 1 + J * C t I; tact+,) then M C b, < a,,,,,,. Let us define by induotion:

Case 1. i = 0; then a. = 0.

Case 2. i = S is limit; then a, = Ufi, a, (it exists as i < A and h is regular).

668 APPENDIX bP.9 6 3

Case 3. a, is defined and we s h d define a,+1.

By the remark to Definition 3.1, there is 8l E I* such that 8 E I*, I' + I* C 8 s 8l implies M C a, < be,,,,,. Similarly, there is t1 E J* such that t E J* , J l + J* C t 5 t l implies bt < aNMt,,,,. Let a,+1 be the

Clearly {a,: i < A, i limit} is a closed unbounded subset of A, so it ED(A). Hence by the hypothesis there is a limit 8 < A such that

tc-contradictory.

J {tN0: i < 8). We define the model N such that IN1 = 1611, x N = {(a, b): (b, a) E <") and N = (INI, <N). Now the sequences (a,: 8 E I: + ( I + ) * ) @ , : t E J r + ( J + ) * ) in the model N , and the orders I + , J +, I : , JZ satisfy the conditions mentioned in Definition

contradiction.

firetordinal > ~ s u c h t h a t I 1 + I* C8cr(t+1) < d , J 1 + J* CtCr(i+l) < t l .

O f f a d 2 K (80 Cledy Cf 6 = Cf a d 2 K ) and lcr(d),Jcr(d) &l'0 Stroll&

h t US define I + = In(d), J + = JMb), It = I r {8at): i < a}, JZ =

3.1, hence I + = Icr(d), J + = J,(d) &l'0 not SkOIlglJ' K - O O ~ t ~ a d i O ~ ~ ,

Proof. By 1.3(3) A = {a: a < A, cf a = K } is a stationary subset of A, and by 1.3(2) there are pairwise disjoint, stationary A, c A (i < A). For any set W c h and a < A let I,,, be K if aEUBewAs and K +

otherwise; and let I, = C,<AI$,,. By claim 3.6 K , K + strongly K-contradictory. If W, U are distinct subsets of A, for some y, y E W e

{a: of a 2 K ; I,,,, I,,, are strongly K - c o n t r a d i d r y }

includes A,, hence by claim 3.6, I,, I, are strongly K-contradictory. Our conclusion follows immediately.

Y4USso

CLAIM 3.8: If A i8 eingular and > K , K regular, then tirere ia a family of 2A pairwi8e 8 t r q l y ~-cont~acEic t~ry d r 8 of cardinality A.

Proof. Let A = zQcy A, where p = cf A, K < A,, h, incredg, A, regular and choose a regular x , p + K+ I x < A, and let A, (i < p) be disjoint stationary subsets of {a: a < x , cf a = K}. Let K, be a family of 2% pairwise strongly K-contradictory orders of cardinality A,. For any f E n,<,, K , (i.e., a function with domain p, f (a) E K,) and fl < x

D., o 31 VABIOUS RESULTS 669

let I,,,, be f (a) when /3 E A,, and K when /3 # UaeP A,. Let I , = ZBex IF,8; m in 3.7 we o m prove (by 3.5, 3.6) that I,, I , me strongly tc-contradiatory when f # g E n,<,, 9,. As II,l = h and In,<,, 9,l = nu<,, 2% = 2 A we Gnish.

Proof of T h e m 3.3. Immediate by Claims 3.7, 3.8, and 3.4.

DEFINITION 3.2: ( 1 ) An m-place relation R over a set 8 is wnneoted [antisymmetric] iff for every distinct .so,. E S there is a permutation a of m suoh that R(8,(0), . . . , & ( , , , - I ) ) holds [does not hold].

(2) If R is an m-plaoe relation over 8, u a permutation over 8, then

R'' = ((80, - . . Y h - 1 ) : <~o(o)Y - * * Y %(m-i)) E R}.

LEMMA 3.9: ( 1 ) 8-e R ie an mplace, connected and antiqpnmetric relation over w, and w b a A,-m-indiecernible sequence, A , =

which is a Boolean com- {.Nxo, * * 9 %I} .

Then tkre i.9 a relation R*(xo, . . . , xgm- bination of imhnaa of R, such thd when i c k # 1

R*(mi,mi + 1 ,...,mi + m - l ,mk,mk + 1 ,... ,mk + m - 1 ,

m2,mi + 1, . . . , m i + m - 1 )

holds iff k c 1 . (2) 8uppo8e R ie an mplace, connected and antieyntmetriC reldtion

Over m = (0 , . , . , m - 1) . Then for 8ome n < m - 1 and permWion a

R'(O,. . . , n - 1, n, n + 1 , n + 2 , . . ., m - 1)

of m

i ,R'(O,. . ., n - 1 , n + 1 , n, n + 2,. . ., m - 1).

Proof. (1) Let R* be

A {B(Xh(o), . . . , Xh(m- l ) ) t : h & fUM3tiOIl from m hlto %,

h one-to-one, and

t = 0 o ' t # 1 * R(h(O), . . . , W(m - 1)) holds),

and let

P ( i , k , l ) = R*(im,im + l , . . . , i m + m - l , k m y . . - s h - - m )

(eo P ia a three-place relation over w).

670 APPENDIX P., 8 3

By the R-m-indiscernibility of w , 0 < k < 1 implies P(0, k, I ) holds. Suppose it holds for 0 < 1 < k, and we shall get a contradiction. By the R-m-indiscernibility of w we can assume 1 = 1, k = 2.

P ( 0 , 1,. . ., m - 1)

Let u be a permutation of m, n < m - 1, then

ifF R'(O,l ,..., n , m + n + 1 , 2 m + n + 2 , ..., 2 m + m - 1 )

iff R'(O ,..., n,2m + n + l , m + n + 2, ..., m + m - 1)

ifF Ro(0 ,..., n - l , m + n , 2 m + n + 1,

(as w ie R-m-indiscernible)

(as P(0, 2, 1) holds)

m + n + 2, ..., m + m - 1) (as w is R-m-indiscernible)

2m + n + 2, ..., 2m + m - 1) (as P(0,2, 1) holds)

(as w is R-m-indiscernible) . Hence it suffices to prove (2). (2) Let nu = min{i 5 m: u(i) # i or i = m}, assume there are no

such u and n; and we shall prove by downward induction on nu that u EC = (0': P'(o,. . . , m - 1) = R(0, . . . , m - 1)). Clearly our assumption means u €2, n < m - 1 implies u(n, n + 1) EC ((n, n + 1) is the permutation interchanging n and n + 1). If nu = m this is trivial, nu = m - 1 is impossible, and nu = m - 2 follows by the assumption.

Suppose we have proved for n + 1, and nu = n, and let u(i) = n (so clearly n nu, hence uo €27. But clearly u = uo(n, n + 1) (n + l , n + 2) . . . ( i - 1,i)soclemlyuEZ.

iff P ( 0 ,..., n - 1 , 2 m + n , m + n + 1,

iff R'(O ,..., n - l , n + 1,n,n + 2 ,..., m - 1)

EXERCISE 3.3: Suppose 111 2 2m, R an m-place connected and antisymmetric relation over a, < l, < , orders on I, and (I, < l), (I, < I) are R-m-indiscernible sequences.

Then (1) or (2) holds: (1) I = I , u I,, (Il, I , disjoint); and < l, < a are identical on each

1, (1 = 1, 2) and 81 E I , implies 81 < 82, 8I < a 81. (2) I = I , u I , U I , (the II's pairwise disjoint); < l, < a are identical

on I, , 11, U 131 S m - 2 and 81 E I , implies 81 <' 8, <' 83, 81 < a 82

< 83.

AP., 3 31 VARIOUS RESULTS 671

DEFINITION 3.3: A dependency relation on a set W , is a relation R between members of Wand subsets of W (x depends on w) satisfying the following conditions (where w is called independent if, for no x E w, does x depend on w - {x}) :

(0) x depends on {x}. ( 1 ) Exchange principle : if {x$ : i < a} is not independent, then for

some p < a, xp depends on {xi: i < p}. (2) Finite character: x depends on w iff x depends on some finite

u E w (so we have monotonicity: if x depends on u,u E w, then x depends on w).

(3) (Weak) transitivity : if w, u are independent, x depends on u, and every y ~ u depends on w, then x depends on w.

DEFINITION 3.4 : A nice dependency relation on W is defined similarly, strengthening (3) to

(3)’ Full transitivity: if x depends on u, and every y ~ u depends on w (where u,v E W ) , then x depends on w.

LEMMA 3.10: For a dependency relation on W, any w E W , any two maximal independent subsets of w have the same power. Also, every independent subset of w can be extended to a maximal independent subset of w.

Proof. The second sentence follows from the finite character (2). For the first, suppose uo c w is a maximal independent subset of w, and u c w is an independent subset of w. It suffices to prove luol 2 IuI so assume this fails. If JuI is infinite, for every XEU there is a finite w, E uo on which x depends. By cardinality consideration for some finite uh G uo, u’ = {x: w, = ui} is infinite, hence Iu’J > luJ and clearly every x E u‘ depends on ui, So uh U u’, uh, u’ satisfy the assumptions on w,uo,u and uh is finite. So w.1.o.g. uo is finite. We choose a counterexample with minimal Iuo-uI. Let uo = {xi:i = 1 , n} with uo n u = {xi:i = l , m } . Choose x,~u-u, (exists as luol < lul). So {xi:i Q m} is independent (being G u), but {xt:i < n} is not (as X ~ E U

depends on uo = {xi : i = 1, n} and x,(i Q n) are distinct as xo 4 uo). So let I , (0, ..., m} c I E (0, ..., n} be maximal s.t. {x(:i~I} is independent and w.1.o.g. I = (0, ..., k} so m < k < n.

Clearly, ui = {xi:icI) is independent, and every member xi of uo depends on it (if ~ E I by (0) and monotonicity, if i$I, by the exchange principle for (xo, xl, . . . , xk, xi} x, depends on ui). Now for

def

672 APPENDIX LAP., 8 3

every x E u, x depends on u,, by assumption and every y E u,, depends on u;. As u,,,u; are independent, x depends on u;. So u;,u satisfy the assumptions, but lui - uI = I{i : m < i < k}l < I{i : m < i B n}l = Iu,,-uI as k < n and u;,u form a counterexample (as IuJ < luJ < lu,,l). This is a contradiction to the choice of uo,u.

HISTORICBL REMARKS

Unfortunately, though this will surely be the most carefully read part of the book, the author had become allergic to writing by the time he reached this section. So he apologizes in advance for any inaccuracy. Theorems and notions not credited are due to the author and/or are trivial. Chapters 11, 111, VII and VI I I were distributed in Spring, ’74; most of Chapter V in the Fall. It is still valuable to look at [Sh 711, Section 0.

I. 1

The theorems are classical, for credits see, e.g. [CK 731 (in 1.3 the use of VIII, 4.7 can be easily avoided by constructing the expansion by approximations of cardinality < A).

1.2

The notion “T totally transcendental” ( # No-stable, by our terminology, as totally transcendental here doesn’t imply countable), was suggested by Morley [Mo 661, “T stable in A”, by Rowbottom [Ro 641, “stable”, “superstable” by [Sh 691 and [Sh 69a] “indiscernible sequence” by EhrenfeuchtMostowski [EM 661, “indiscernible set over A”, by Morley [Mo 661. Lemmas 2.1, 2.12, 2.13 and 2.9 are trivial, 2.4 was deduced by Ehrenfeucht-Mostowski [EM 661, from Ramsey theorem [Ra 291. Theorem 2.8 proved through 2.6, 6 and 7 is from Shelah [Sh 72bl. Morley proved 2.8 for models of countable No-stable theories. His proof was by choosing a type p E Sm(A) with minimal rank a = R(p, L, No), realized by > A member of I, choose inductively a, E I realizing p such that R[tp(Ei, A u UjK1 si,] = a, and prove it is an indiscernible sequence like 2.6. See 11, 2.17, 8 and 9 for a similar proof. Notice that for stable T, there is no corresponding to minimal. Rowbottom has a weaker unpublished result (a direct generalization of

673

674 EISTOIUCAL REMARKS

[Mo 651). In [Sh 691 and [Sh 701 there were approximations to this. See 111, 4.23 for another way to prove. Theorems 2.10 and 2.11 are from [Sh 72bl. (Erdos and Makkai [ErM 661 is the case A = KO of 2.11 for the parallel combinatorial theorem.)

II.1

Morley [Mo 651, influenced by the Cantor-Bendixon rank, defines Rm(p, L, KO) and Mltl(p, L, H,) for complete p (in different notation). He proves for this version the parallels of 1.1, 1.2, 1.4(3), 1.6, 1.9(1), 1.10(1) and (2). In [Sh 691 and [Sh 69a] Rm(p,d, 2) was defined for complete A-types, the interesting case waa d Gnite or singleton whom research was continued in [Sh 711, where the completeness was dropped; P ( p , L, m) wa8 used in [Sh 7OaJ. In [Sh 72a] P ( p , L, Ho) for p not necessarily a d-type, and [Sh 71d] introduce Rm(p, A, A). The “(A, n)- indiscernibility ” was introduced in [Sh 711.

Exercises 1.6 and 1.6 are due to L. S f d and A. Hinkis, resp., in their Maater thesis.

II.2

Everything is from [Sh 711. Concerning the stability spectrum (2.13) see 111, Section 3. Baldwin [Bl 70a] proved independently, a weaker version of 2.12(1) (for T totally transcendental A a definable set). 2.17, 8 and 9 generalize the proof of Morley [Mo 651 on the existence of indiscernible sets. Concerning 2.20, Harnik and Ressayre [HR] independently prove that if T stable in A, 2” > A, I indiscernible over A, then for every si there is J s I, I J I < A, I - J is indiscernible over A u si, and Shelah [Sh 701 proved there is J G I, I J I < K(T) , such that I - J is indiscernible over A u ii u U J (see 111, Section 3 for definition, but K ( T ) is always sp, and sometimes <p) and Shelah [Sh 711 proved 2.20 which is the finite version.

Morley [Mo 651 proved that if there is an indiscernible sequence which is not an indiscernible set, T is not KO-stable, exhibiting in the proof order (see 2.16).

II.3

Morley [Mo 651 proved that for countable K,-stable T, sup,Rm(p,L,No) c ol; and Lachlan [La 711 solving a question from [Mo 651 remove the “ No-stable” from the hypothesis. Theorem 3.1 is a slight generalization. Morley [Mo 651 proves 3.3 and 3.2 in different terminology. D“(p, A, A)

HISTOBICAL REMARKS 675

was defined in [Sh 711, Section 6, some of its properties and its oon- nection to superatability 3.9 and 3.14 were established.

More on ranks, see 111, Section 4, V, Section 7, and Baldwin and Blnss [BB 741.

II.4

The finite cover property was suggested by Keisler [Ke 671 (for his order, see VI), and prove the property (E) implies it (Ehrenfeucht property from [Eh 67I-there is a formula q(z,, . . . , z,,) defining on an infinite set a connected antisymmetric relation; (E) is related to but weaker than the order property); so this is related to 4.2.

Most of the section is from [Sh 711, but 4.16 (if some q(Z; j j ) has the strict order property, then some ~(z; y) has the strict order property) conjectured in [Sh 711, is of Lachlan [La 76a], and 4.4 was added lately.

III

Sections 1, 2, 3 and 6 m e from [Sh 71a] (with some additions; 6.17, the proof of 2.8), some parts me exposed in [Sh 76c] with different proofs.

m i - a The stability spectrum theorem evolves as follows. Morley P o 661

proves that if T is countable, stable in H, then it is stable in every A. Successively and independently Rowbottom [Ro 641 (using GCH) and Resseyre [Re 691 prove that if T is stable in A, h < A h then T is stable in every p 2 A, and Shelah [Sh 68a] [Sh 691 proves that if T is stable in one A, it is stable in every h = XITI, and if T is stable in one A, h IT1 waa proved by Harnik [He 761, seeing a proof of VI, 6.3 (then 111, 3.10 was essentially included in it) and then 3.11 for cf h > IT1 was clear.

m.4

A result b d on this was announced in [Sh 711, p. 276-6. Concerning 4.6(3), Lachlan [La 721, codrming a conjecture of L. Blum, proved Mltl[tp(G, 1iWI), L, H,] = 1. In Werent terms, some of this, mainly the symmetry lemma 4.13, was developed later and independently by Lascar [La 731, [La 761, who concentrates on superstable T and complete types over models (see also on V, 7.12).

676 HISTORICAL REMARKS

III.5

Theorem 5.18 was proved independently by Lachlan [La 741 (using a version of 2.8) and Shelah (it is immediate from [Sh 71a], Lemmas 38, 40(p.106,108),asnoticedsomewhatlater).Theorem 6.14isfrom [Ke 71al.

III.7

Theorems 7.4 and 7.2 were proven for countable T by Keisler m e 761 (so K = No) partially confirming a conjecture 4E, p. 330 [Sh 711 (see (4) there). He proved a version of 7.3 by induction.

IV. 1,2

Vaught [Va 611 investigates Fi,-prime models over 0, Morley uses F&,-prime model (mainly for No-stable T), Ressayre [Re 691 and Shelah [Sh 691, [Sh 69a] use, independently, Fi-prime, isolated. In [Sh 701 Fi-prime, isolated, were used. About the same time and independently Lachlan [La 721 (ad hoc) and Shelah (see [Sh 711 7) p. 275) use FL,. Fragments of lemmas here were parts of proofs. In Rowbottom [Ro 641, Ressayre [Re 691 and Shelah [Sh 691, [Sh 69a], 2.18(1) appeared for Fi, 2A > p, T stable in p.

Morley [Mo 651 proved 2.18(1) for FL,, T &stable.

1V.Q

For the appropriate specific F’s, 3.1, 3.10, 3.12 and 3.17 appear in Morley [Mo 651, Ressayre [Re 691, Shelah [Sh 691, [Sh 69aI; 3.2 appear in [Mo 651 and [Re 691; 3.3-3.8, 3.11, 3.16, 3.18 were not traced exactly by the author; 3.9 for Fi is due to Ressayre (unpublished) and 3.15 is essentially from [Sh 72al.

Iv.4

in [Sh 711. Most results appeared in [Sh 71d] and [Sh 72a], and were announced

Iv.5

Theorems 5.1 and 5.2 for A = M, axe from Vaught [Va 611, and generally from Hmnik and Ressayre [HR 711. For 5.3 see credits in [CK 731, 5.4 is of Vaught [Va 611. Theorem 5.6 was announced in [Sh 74bl. Theorem 5.13 (without maximality) is due to Silver, answering a question from [Mo 651. Theorem 5.17 for countable T, is due to Grilliot [Gr 721, and the main case was announced in [Sh 761.

HISTORICAL R E W S 677

v. 1

Minimal formulas were introduced by Marsh 661, who proved the existence of dimension for it. Harnik and Ressayre [HR 711 generalize this to minimal types, weakly minimal formulas were introduced in [Sh 741.

v.2 The part of Theorem 2.10 asserting the equivalence of (l), (3), and

(4),, is from [Sh 701, for regular A. Ha& [Ha 761 proved, independently of this, the following weaker result (GCH): every T satisfies exactly one of the following; (i) for every p > A 2 ITI, A regular, T has a maximally A-saturated model of cardinality 2p, (ii) there is A, such that every A,-saturated model of T of cardinality > piT' is p + -saturated. H a d [Ha 75a] and the author independently observe that the regularity is not needed. Most of the theorem was proved in [Sh 71dl.

v.3 Theorems 3.11,3.19, and Exercises 3.7-3.16 onward were added after

the author saw Lascar [La 761, and was motivated by problems there.

v.4 Exercise 4.4 was asked in the first version and was answered inde-

pendently by Rosenhl and Shelah on the one hand and Lascar on the other hand (using algebraically closed fields).

V.6 On n-cardinals theorems generally see [CK 731. In [Sh 691 (see [Sh 711)

the 2-cardinal theorem for stable theories was proved (6.14(2)) and then the n-cardinal theorem was announced in [Sh 72dl. Later Forrest [Po 7x1 proved those theorems. In [Sh 691 models of stable theories with the Chang quantifier 0"" were also discussed, and the transfer theorem proved (starting from a regular cardinal).

Lachlan [La 721 proved a much stronger theorem: 6.14(1) (if T countable and stable, P ( M ) G N 4 M, then there is M,, M < M,, M # M,, P ( M , ) = P ( M ) ) . Baldwin [Bl 751 gives another proof. Harnik [Ha 76a] generalizes this to A-compact models, with P replaced by a type.

Theorem 6.16 was proved by Lascar [La 761 for T No-stable, A regular.

678 RISTORIOAL REMARKS

v.7

Gaifman [Ga 731 proved 7.4(2) (without stability assumptions). Baldwin [Bl 731, proves sup, Rm(p, L, KO) < w for T K,-categorical; essentially he proved 7.9 for cp(z, 6) minimal; he solved by this one of the problems from Morley [Mo 651. Erimbetov [Er 751 and the author proved 7.11 (2) independently.

The rank L ( p ) and Theorem 1.12 on it are due to Lascar [Ls 761.

VI.1

Ultraproducts were introduced by bog and revived in Frayne, Morel and Scott W S 621. See [CK 731 for references.

vI.2

For most results see [CK 731 for references, Theorems 2.6 (and 2.7, 2.8), 2.9 and 2.10 are due to Shelah, and announced in [Sh 75b] (a weaker form of 2.6 was announced in [Sh 71gl). Exercise 2.10 for ultrafilters was asked in Keisler [Ke 67a], proved in [Sh 7Obl. Koppel- berg [Kp 751 proved this for filters (which is the exercise) by another proof and in the exercise we hint how the proof in [Sh 70b] works for this too.

VI.3

Keisler [Ke 641 proved the existence of a good N,-incomplete ultrafilter over A when A+ = 2*, Kunen [Ku 721 eliminates this hypothesis, and we represent this result in 3.1-3.4; 3.5 is Keisler’s theorem on isomorphic ultrapower (see [Ke 64a]), 3.6 is of Frayne, Morel and Scott [FMR 621; 3.7 and 3.8 is of Keisler [Ke 651. Exercise 3.1-3.5 represent [Sh 71c] (with a slight improvement), Exercises 3.6 and 3.7 are from [Sh 72~1, and Exercise 3.11(2) was announced in [Sh 701. Exercises 3.17, 3.18 and 3.19 are of Benda [Be 721. For uniform filters and Exercises 3.20-3.23 see [CK 731. Exercise 3.26(1) and Lemma 3.9 for M,, = PMn is from Ellentuck and Rucker [ER 721. The results of the rest of the section were announced in [Sh 76bl. Keisler [Ke 67a] shows (assuming GCH) that for “almost all sets 8 of successor cardinals, 5 A + , for some ultrafilter D over A, S = w u {n n,/D: nf c w } ; he uses products of ultrafilters, and the operation from Exercise 3.20. On Boolean ultrapower (Definition 3.8, Exercise 3.32) see Mansfield [Mn 711. Theorem 3.12 answers a question of Keisler (see [CK 731).

HISTORICAL REMARKS 679

vI.4

Most of the results in Sections 4, 5 and 6 appear in [Sh 721, and announced in [Sh 711. Keisler’s order, and 4.1, 4.2, 4.6 and 4.6 are of Keisler [Ke 671. Independently Keisler and the author prove 4.7 for the Tor&, the theory of infinite linear order. On limit ultrapower (Definition 4.2, Exercise 4.10) see Keisler [Ke 631. Concerning 4.3, Keisler [Ke 671 gives a similar theorem with a much stronger condition on T. By Benda [Be 721 the condition could be weakened to: T is @-maximal if for some rp(x, y) for every n, and S s {w: w c n},

w s n

vI.5

Conjecture 6.1 was made in [Sh 721, later and independently Eklof asks the same question (in a little different terminology). Keisler and Prikry [KP 741 prove a positive answer for good ultrafilters; see also Exercise 6.10, 5.11 and 6.12 (which is the result of [Kp 741).

Concerning 6.8, Keisler [Ke 671 proved that @-minimal theories are not @-maximal, and does not have the f.c.p. Exercise 6.8 is of Keisler [Ke 631, and Exercise 6.7 was proved by Morley independently of Theorem 6.3.

vI.6

Ultralimits (Definition 6.1) are from Kochen [KO 611 and Keisler [Ke 631, 6.1 should have appeared there; on p-descending complete filters see [CK 731; 6.3 appear in [Sh 721, but it was not traced.

M. 1

The main results of the section were announced in [Sh 7181. On 1.1-1.4 see [CK 731 for references. On AD (see Definition 1.11) see Baumgartner [Ba 701 [Ba 761.

Exercise 1.7 was not traced.

M . 2

Ehrenfeucht-Mostowski [EM 661 introduce the models (in our notation) E H ( I ) , I an order, and prove their existence, i.e., 2.3. We give here a generalization. Ehrenfeucht [Eh 671 proves a variant of 2.4.

680 HISTORIOAL REMARKS

[EM 661 proved E M ( I ) realized few types, and Morley [Mo 661 proved that for well ordered I, the model is stable in appropriate cardinalities (i.e., 2.7(2) and 2.9(1)).

M.a Theorem 3.6 is from [Sh 741. Theorem 3.7 was first proved assuming

there is a measurable cardinal. The proof is hinted in Exercise 3.3. (The main consequence was announced in [Sh 7281 with measurable, and in [Sh 731 without measurable.) M.4

The main result waa announced in [Sh 731.

vII.5 Most references are well known (starting with Morley’s omitting

type theorem). Now 5.5(8) seems new; p(A) = a,(,, was by Barwise and Kunen [BK 711 (and see Shelah [Sh 72dl). vm. 1

1.1 is trivial, 1.2(1) is essentially due to Ehrenfeucht [Eh 681; it seemed 1.3 waa first mentioned in [Sh 711. Concerning 1.7(2), Keisler [Ke 701 proved that when ID(T)I = 2N0 there are 2h models for which the sets of types realized in distinct models are distinct. In [Sh 7281 (with mertsurable) and [Sh 731 (without) 1.7(2) was announced.

Concerning 1.8, the simplest example (T = Th(M), M = (w2, E,, F,, . . . ), qE,w o q n = w n end (P,(q))[k] = q[k] iff n # k) was proved by Baumgartner and Laver.

vm.2,a Ehrenfeucht [Eh 671 proved that if T has property (E) (which implies

unstability) A = 2’ then it has a t least two non-isomorphic models in A. Scott replaces A = 2’ by h = px > p. In [Sh 69a] it is proved: if T is unstable, lTll = K, < Kp, then TIT) 2 IP-al and similar results for unsuperstable T). In [Sh 71b], 3.1 and 3.2 for K = KO, A > I TI are proved (using only Case 1 and a proof similar to 3.1) and in [Sh 731 the whole result (3.3, 3.1) was announced.

In [Sh 74a] the proofs are sketched and they are discussed also in the introduction of [Sh 7601.

vm.4 In 4.1 the fact that T is stable in p, when 12’1 s p < A, waa proved

by Morley for IT1 = X,, T = T I , and it waa observed that the proof

HISTORICAL REMARKS 68 1

is given generally by Keieler [Ke 711, Cudnovski [Cu 701 (for countable), Rowbottom [Ro 641, Ressayre [Re 691 and Shelah [Sh 681 [Sh 69a] (independently). Keisler [Ke 671 proved for countable T, T does not have the f.c.p.

(3) is essentially proved in Keisler [Ke 661; and (2) * (3) was proved, independently by Cud- novskii [Cu 701, Keisler [Ke 711 and Shelah [Sh 681 [Sh 691 [Sh 69al. The same holds for 4.4. Most cases of 4.7 which are proved here me from [Sh 701. Keisler [Ke 661 proved that 4.3(1) (hence (2), (3)) holds whenp = 8, > ]TI.

In Theorem 4.3, for countable T, (1)

IX. 1

Theorem 1.4 answered a question of Sabbagh [Sb 761 who gave modules as a counterexample (so its theory is stable but not superstable, see Exercise 1.2 for another example). Theorem 1.6 is due independently to Engeler [En 691, Ryll-Nardzewski [Ry 591 and Svenonius [Sv 691. Theorem 1.8( 1)-(4) and 1.9 is Morley categoricity theorem confirming Ws' conjecture from [Mo 661, and (5) was added in Morley [Mo 671. Lemma 1.10 and many others are essentially from [Sh 741. Concerning 1.11(1) Baldwin [Bl 731 proved for K,-categorical countable T that sup, Rm(p, L, KO) < w and also 1.1(2) in this context.

Theorem 1.16, first part, answers a question in [Mo 651. Rowbottom [Ro 641, Ressayre [Re 691 and Shelah [Sh 68a] [Sh 691 [Sh 69a] were successive and independent approximations. Rowbottom proves (GCH) that categorioity in A. > xo = mink: xHo > x 2 ITI}, implies T is categorical in each A z Ao. Ressayre eliminates GCH, and proves it also for ITI+ < A, < x,, and mostly for A, = ]TI+ , and shows T is categorical in some A < > ( p i ) + , Shelah shows it for A, > IT/, A. # x,, and that T is categorical in some A < p(lT1). The final answer is [Sh 741. The second part of 1.16 answers a question of Keisler and so is 1.16(i); they were announced in [Sh 70~1, [Sh 71fl. 1.16(ii) too answers a question of Keisler, and appears in [Sh 701.

On countable T categorical in X, see also Baldwin [Bl 721 [Bl 72a], Dickmann [Di 731 and Makowski [Ma 741; on uncountable T categorical in I TI + see also Andler [An 761.

Concerning 1.19, it was asked in Morley [Mo 661, Keisler [Ke 71a] proves it when No < 12'1 < 2#0, IT1 regular, and Shelah [Sh 70e] when IT!". = IT].

Theorem 1.20 was announced in [Sh 711.

682 HISTORICAL REMAEtKS

Ix.2

In Theorem 2.2(1) one inequality ( 5 KO) is of Morley [Mo 671, the other inequality and 2.2(2) and (3) a m of Baldwin and Lachlan [BL 711, thus answering a question from [Mo 661. Theorem 2.1(1) is of Lachlan [La 731, Lascar [Ls 761 gave an alternative proof, and here we give another (but in Lachlan’s proof we can weaken a little the supersfability demand). Morley [Mo 701 proves I(No, To) > N, I(No, To) = P o

(thus partially confirming Vaught conjecture). The first part of 2.1(2) is from [Sh 701 and confirms a conjecture of Harnik, the second, for T countable is of Lachlan [La 761. Theorem 2.3 was essentially announced in [Sh 711, and much more than 2.4 in [Sh 741 and [Sh 75~1. Latter and independently Lachlan proves 2.4 (for countable totally transcendental theories). Exercise 2.2 is from Harnik and Ressayre [HR 711.

X

This is based on [Sh 82a], [Sh 82b] (and we thank the Israel Journal of Mathematics for permission to use part of it). It was done in 1972 (with the parallel for KO-stable T), and announced, for example, in [Sh 741, and distributed in 1979 and 1980, respectively. The main change here is in writing down fully the proofs for the IE case.

It was planned in this series to compute I (h ,T) for T totally transcendental (the complication being as in Example XIII, 4.13), hence the trivial cases of computing I(h, T) for h large enough were not written down. (See Ex. XI, 3.1.) However, Saffe computed and proved the hard cases for T totally transcendental.

XI

This chapter, as well as Chapters XI1 and XIII, were written in 1982, circulated in September 1983. It continues on the appropriate parts of [Sh 82a], [Sh 861 (but is more general).

XI1

Done and announced in early 1982.

XI11

Completed, done and announced in the Fall of 1982.

HISTORIOAL REMARKS 683

A. 1

See [CK 731 for references for 1.1-1.6; Theorem 1.7 appears in [Sh 711 (in different notation) (the h i t e case was completed together with Perles) and Sauer [Su 721 proved later and independently 1.7(2) (the finite case).

A.2

Theorem 1.1 is of Ramsey [Ra 291; Theorems 2.2, 2.3 and 2.4 form a weak version of the Halperin and Lauchli [HL 601 theorem (more exactly, a variant of it which Laver proved, and Pincus showed inessential changes in their proof gives); 2.5 is the Erdos-Rado Theorem (see [EHR 651; 2.6 waa mentioned in [Sh 711; 2.7 is from Erdos, Hajnal and R d o P H R 663. Theorem 2.8 is of Hajnal [Hj 611.

A.3

For 3.1 see [Sh 71b] (inside the proof') and [Ba 761, for 3.2 and 3.3 see [Sh 71bl. Theorem 3.9 is essentially from Morley P o . 651.

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[B17Oa]

[B172]

[B172a]

[B173]

[Bl73a]

[B176]

[BL 711

[Bush 891

D. Andler, Serni-minimcrl theories and categoricity, J. Symb. Logic 40 (1976) 419440. J. T. Baldwin and S. Shelah, Classification of theories by second order quantifiers, Notre Dame J. of Formal Logic 26 (1985) 22S303. J. E. Baumgartner, Results and independence proofs in combh- torial set theory, Ph.D. Thesis, University of Celifornie, 1970. J. E. Baumgartner, Almost-diejoint sew, the dense eet problem and partition calculus, A n d of Math. Lo& 9 (1976) 401. J. E. Baumgaxtner, A new clsss of order types, A n d of Math. Logic 9 (1976) 187. J. T. Baldwin and A. Blsss, Skolemization and mtegoricity, Noticee

J. T. Baldwin and A. Bless, An axiomatic approach to rank in model theory, A n d Math. Logic 7 (1974) 296-324. M. Ben&, On reduced produote and 6ltem, An& of Mdh. Log.ic 4

J. Berwise and K. Kunen, Had numbem for fragments of L,.,, I m l J. Mdh. 10 (1971) 308-320. J. T. Baldwin, Countable theories categorical in uncountable power, Ph.D. Thesis, Simon Fraser Univemity, 1970. J. T. Baldwin, A note on debbi l i ty in totally transcendentel theories, N o t h A.M.S. 17 (1970, Nov.) 1087. J. T. Baldwin, Almost strongly minimal theories, I, J. Symb. Log.ic 87 (1972) 481493. J. T. Baldwin, Almost strongly minimcrl theories. II. J. Symb. Logic

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J. T. Baldwin, The number of automorphisms of a model of an N,- categorical theory, Fund. Math. (1)88 (1973), 1-6. J. T. Baldwin, Conservetive extensions end the two cardinel theorem for stable theories, Fund. Math. 88 (1976) 7-9. J. T. Baldwin end A. H. h h h , On strongly . 4 sets, J. Symb. Logk 86 (1971) 79-96. S. Shelah and S. Buechler, On the existence of regular types, Annals of Pure and Applied Logic 45 (1989) 277-308.

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684

REBERENUES 685

[CHL 861

[CK 731

[Cu 701

[Di 731

[Eh 671

[Eh 681

[EM 561

[En 691

[ ~ r 751

[ER 721

[ErM 661

[EHR 661

[Fl 711

[Po 7x1 [FMS 621

[Ge 731

[Ga 761

[Gr 721

[GS 731

[GSh 831

[Ha 711

[He 761

[He 76a]

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[He 731

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[Sh 76b]

[Sh 781

[Sh 78a]

[Sh 791

[Sh 79a]

[Sh 801

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[Sh 821

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[Sh 831

[Sh 83a]

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[Sh 85c]

[Sh 85d]

[Sh 85e]

[Sh 861

[Sh 86a]

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[Sh 89a] [SHM 841

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Logic 34 (1987) 291-310.

93-100.

INDEX OF DEFINITIONS AND ABBREVIATIONS*

wl (algebraic closure)

a d m i s s i b 1 e

AD el (after last) algebraic

alge braically

fdmost closed

mtisymmetric a t (atomic) atomic

atP Av (average)

Ax (axiom)

automorphism

based

baaic bi-set

acl(A)

(1) Fa-admissible (2) F-*admissible (3) F-**admissible A D ( A , x , p, K )

al(4 (1) algebraic formula

[type], d algebraic over A

(2) K-algebraic A is algebraically closed

formula [type] is almost over A

( A , &)-compact (1) B is F-atomic over A (2) atomic type (3) A is T-atomic over B atp (8, J , 4 , atp @ , I )

(2) Av(D,A) (1) Avd(1, A ) , Av(I, A )

(1) Ax(I)-Ax(XII)

(2) Ax(Al)-(E2) f ie an automorphism of M

(1) Z is based on A (2) I is based on p (3) a is based on W

bi-set function

111, Definition 6.1(4), p. 130

IV, Definition 4.2, p. 192 IV, Definition 4.3, p. 197 IV, Exercise 4.6, p. 202 VII, Definition 1.11(2), p. 410 IV, Definition 1.2(5), pp. 155-156 111, Definition 6.1(2), p. 130

V, Definition 7.1, p. 305 III, Definition 6.1(4), p. 130

111, Definition 2.1(1), (2), p. 94

I, Definition 2.5, p. 11 VI, Definition 1.5, p. 328 IV, Definition 1.5, p. 157 VI, Definition 1.4, p. 327 XI, Definition 2.2(2), p. 562 VII, Definition 2.2, p. 412 111, Definition 1.5, p. 89 VII, Definition 4.1(1), p. 426 IV, pp. 152-153, and Table 1 ,

XI,-pp. 562-565 I, Definition 1.4, p. 5

111, Definitionl.8, p. 90 111, Definition 4.3, p. 118 VI, Definition 3.7(3), p. 358 see Horn VII, Definition 1.8, p. 409

p. 169

* Notice that for e.g., Av(Z, A), P ( p , A, A), strongly A-homogeneous, you should look at Av, D, homogeneous, reap. So look at the main ward, ignoring strongly, explicitly, etc., but semi-, uni-, multi-, bi-, are considered part of the word. Note that the same word may have different meaning depending on the text (e.g., regular cardinal and regular type; Av(Z, A) and Av(D, I)), whereas some variations are only shortening (e.g. P ( p , A, A), D(p, A, A ) ) . So we number the distinct meanings. Some too well-known notions were omitted.

69 1

692 INDEX O F DEFINITIONS AND ABBREVIATIONS

Boolean Boolaan ultrapower N(m)/D VI, Definition 3.8, p. 369 ultrapower

C

cnat (conjunction of atomic)

cer (or Card)

cb (CenoniCBl

b-4 cc (chain

cdt (contradictory type)

of (cofinality) cl (closure)

cntegoricel

condition)

closed compaot

complete

component connected

coneervative contradictory constructible construction contradictory

CP (combinetorial principle)

CR (complete -1

Ctp (canonical type)

Cer K

(4) Ff-compact model (1) D is a A-complete filter

(2) complete model (3) A-descending complete

ultrefilter (4) complete type (6) complete theory M , is a

explicitly contredictory F-constructible F-construction (1) x-contradictory (2) strongly

K-contradictory (3) n-contradictory (CPA. x.p. x. e)

CR@, A )

V, Notation, p. 289 V, p. 290 VI, Definition 1.6, p. 328

V, Notation, p. 289 sea Introduction III, Theorem 6.10, p. 134

VI, Definition 3.7(3), p. 358

111, Definition 7.2, p. 141

Notation, p. xxix 111, Definition 6.1(6), p. 130 V, Definition 4.4(6), pp. 277-278 IV, Definition 1.2(2), pp. 155-15 I, Definition 1.2(2), (3). p. 3 VI, Definition 1.6, p. 328 VII, Definition 1.4(2), p. 402

end MI, Definition 1.9, p. 410

IV, Definition 2.1(3), p. 157 VI, Definition 1.3(6), p. 326

Appendix, Definition 1.1 (6),

VI, Definition 5.1, pp. 382-383 VI, Definition 6.2, pp, 391-392

see Notation see Notation XIII, Definition 3.1, I, Definition 2.5, p. lf 634

p. 653-654

Appendix, Definition 3.2( l ) , P. 669

VII, Definition 2.1(2), p. 41 1 11, Definition 1.1(3)(i), p. 21 IV, Definition 1.3, p. 156 IT, Definition 1.2(1), p. 155 VIII, Definition 3.2, p. 464 Appendix, Definition 3.1,

see n-inconsistent XIII, Definition 2.2, p. 628

p. 666

11, Definition 3.4(1), p. 66

111, Theorem 6.10, p. 134

INDEX OF DEFINlTIONS AND ABBREVIATIONS 693

D (degree)

dcl (definable

DC decomposition

closure)

Ded

Dedekind deep definable, defined

definable closure define

definitional extension

degree depend

depth

didip (discontinuity dimensional Property 1

divides df (definition) dim (dimension)

dcl(A)

DC(I), DCW(I), DCW) (1) (N,,, a,: q E I ) is a

(T, &*)- decomposition

decomposition

decomposition inside M

decomposition of M

(2) u (T, C*, A ) -

(3) u (T, E*)-

(4) u (T, G*)-

(1) Ded h

(2) Ded, h

(A , p)-Dedekind cut T is deep p is definable over A,

p is ($, A)-dehble , p is $-defined

d is defined by a formula CtYPel

(1) a formula depends on an equivalence relation

(2) E depends on 0 mod Y (3) d depends on I (1) the depth of (N, N', a) (2) the depth of T T has the didip

11, Definition 3.2, p. 42 V, Definition7.2(1),pp.30&307 111, Definition 5.5, p. 127 Appendix, Definition 1.2,

111, Definition 6.1(3), p. 130

VII, Definition 1.10, p. 410

XI, Definition 2.4(1), p. 565

p. 654

XI, Definition 2.4(2), p. 565 XI, Definition 2.5(1), p. 566


Appendix, Definition 1.4,


VII, Definition 1.10, p. 410 X, Definition 4.2(2), p. 528 11, Definition 2.1, p. 31

p. 657

p. 490

see do1 111, Definition f3.1(1), p. 130

111, Theorem 5.14, p. 128

see D 111, Definition 2.1(1), p. 94

111, Definition 5.3, p. 124 111, Definition 4.4, p. 118 X, Definition 4.1, pp. 527-528 X, Definition 4.2(1), p. 528 X, Definition 2.6, p. 520



111, Definition 4.5(1), (2), (4), dim(p, B, A) p. 119

(3) dim(Z, A, M ) , dh(1, A ) 111, Definition 3.3, p. 106

694 INDEX OY DEFINITIONS AND ABBREVIA'FIONS

Dom (domain)

(dimensional order property)

d0P

DP

ds (descending sequences)

E

EC ECN (equivalence

class number) Elementary

E M

eq (equivalences classes)

equivalent

existence

extension property

family

f.c.p. FE (finitmy

filter equivalences)

Dom P T has the dop

(1) E+ (2) E of a representation EC(To. F) ECN(Eo, @)

(1) Elementary submodel (2) (M, N)-elementary

EM'(1, a), EM'(I,N),

(1) @Q, Tea, pea, MeQ

mapping

EM1(I), EM(I, @), etc.

(2) e q ( 4 (1) p is equivalent to q (2) J, Z are equivalent (3) elementarily equivalent (4) equivalent

representations (1) (A, 2)-existence property (2) T hes the (A, n)-

(3) T has the true existence property

( A , n)-existence Property

existence property (4) strong (X,, n)-

( A , K)-family

FEm(A)

(1) filter over I

(2) trivial filter

(3) principal filter

X, Definition 2.1, p. 512

X, Definition 4.1, p. 527 X, Definition 4.2(1), p. 528 X, Definition 4.3(2), p. 528 X, Definition 4.3(3), p. 528 X, Definition 4.4A, p. 529 11, Definition 3.3, p. 44

111, Definition 6.5, p. 133 X, Definition 5.8, p. 536 VII, Definition 5.1(1), p. 432 111, Definition 5.2, p. 124

I, Definition 1.1, p. 2 I, Definition 1.3, p. 4

VII, Definition 2.6, p. 413; VII, Lemma 2.6, p. 415

111, Definition 6.2, p. 131 ; 111, Definition 6.3, p. 132

VI, Definition 4.2, p. 375 11, Definition 1.3, p. 23 111, Definition 1.6, p. 89 see Notation x, Definition 5.17B, p. 547

XII, Definition 4.2, pp. 60&609 XII, Definition 5.2, p. 616

XII, Definition 5.4(1), p. 616

XII, Definition 5.4(2), p. 616


VII, Definition 1.11(1), p.

11, Definition 4.1, p. 62 111, Notation, p. 94

Appendix, Definition 1.1( I),

Appendix, Definition 1.1(2),

Appendix, Definition 1.1(6),

410

p. 653

p. 486

pp. 653-654

INDEX OR' DEFINITIONS AND ABBREVIATIONS 695

finite cover

FI (finite

fork free

Property

intersection)

freedom full function

U gap U E GI good

h H homogeneous

Horn

I

IE

incomplete inconsistent ind

independence property

F W , FI,('3)

p forks over A S [(S, A)] is free [ (T, , 2')-

free] [free in (T,, T)] (T,, T) has (p, A)-freedom a full model ( A , a)-function

QiwlJ, Ml)

GEi(M, N) UIi(iV, M ) (1 ) good [K-goodl

the main gap theorem

[h-good] model (2) good [A-good] filter;

A-good Boolean algebra

(3) A is a good set

h(9, v) H ( I ) (1) homogeneous,

(2) strongly

(3) A-model homogeneous (1) Horn formula

(sentence) (2) basic Horn formula

K-hOmOg0neOUS

K- homogeneous

888 f.c.p.


111, Definition 1.4, p. 85 VIII, Definition 1.2(1), (2),

VIII, Definition 1.2(3), p. 445 VI, Definition 6.2, p. 383 11, Definition 3.3, p. 44

VI, Definition 1.6, p. 328 XII, Theorem 6.1, p. 620 VIII, Definition 2.1, pp. 459-460 W I , Definition 2.1, pp. 45S460 V, Definition 6.1, p. 294

VI, Definition 2.1(1), (2),

p. 445

p. 333; VI, Exercise 3.16, pp. 354-355

XII, Definition 3.2, p. 604

VII, Definition 3.1, p. 422 V, Notation, p. 289 I, Definition 1.6, (I), p. 6

I, Definition 1.6(2), p. 6

VIII, Remark, p. 472. VI, Definition 1.2, p. 326

VI, Definition 1.2, p. 326

VIII, Definition l .l(l), (3),

IX, Definition 2.1, pp. 459-460; X, Definition 1.8(1), p. 511 X, Definition 1.8(2), p. 511 VIII, Definition 1.1(2), (3),

X, Definition 1.8(2), p. 511 X, Definition 1.8(2), p. 511 VI, Definition 1.3(6), p. 326 11, Definition 3.2(3)(ii), p. 43 11, Theorem 4.8(2), p. 72 11, Exercise 4.6, p. 80 11, Definition 4.2, p. 69

p. 444;

p. 444

696 INDEX O F DEFINITIONS AND ABBREVIdTIONB

independence

indiscernible

inevitable

inp (independent partitions)

ird (independent orders)

isolated isomorphic

k

K

Keisler kind

Kr

L (hecar rank) I

large

(1) Z is independent over A , [over (B, 4 1 [over PI

(2) I7 is independent over @andmod!?'

(3) Y is independent mod D

K-independent (1) ( A , n) indiscernible (set

of sequences), A-n- i n b r n i b l e , n-in&- oernible, eta.

sequences, A-n-indiscernible, n-indiscernible

set

(4) (31, gn, D)-

(2) (A , n)-indiecernible

(3) maximal indiscernible

(4) absolutely indiscernible (6) (A, n)-indiscernible

indexed set, etc. ~ E S " ( A ) is

inevitable K c p ( T ) , 'CindT), K r E p ,

&d(T), K i r d ( T )

T)

is F-isolated over A M, N are isomorphic

(1) k(N0,Nl) (2.1 k(N1) (3) k(T) (1) K = (F, w, A, P ) (2) Kh (3) Km, TI, W A ) , W A ) ,

K(A), etc.

(7) K i Keisler order @, @ A

(1) Ba is the kind of d (2) (used only in X, $5) W ( A , T), K r y A ) , etc.

L ( P ) (1) W 0 , N l ) (2) W,) (3) v) J is a A-large

(< A-large) subtree

III, Definition 4.4(1), (21, p. 118

ELI, Definition 6.4, p. 125

VI, Definition 3.1, p. 345;

VI, D W t i o n 3.3, p. 350

I, Definition 2.4, p. 10




IV, Definition 6.1, p. 212

VII, Definition 2.4, p. 413 IV, Definition 5.9A, p. 210

III, Definition 7.3, p. 145


IV, Section 1, p. 153 I, Definition 1.4, p. 6

XIII, Definition 4.6(3), p. 647 XIII, Definition 4.7(1), p. 648 XIII, Definition 4.7(2), p. 648 V, Section 6, p. 289 V, Definition 6.2, p. 296 II, Definition 4.4(1), (3), p. 75

M, Definition 2.5, p. 413 X, Definition 4.1, pp. 527-528 X, Definition 4.3(1), p. 528 X, Definition 4.3(1), p. 528 VI, Definition 4.1, pp. 37&371 111, Definition 6.4, p. 132 X, Definition 5.16F, p. 545 11, Definition 4.4(2), (3), p. 75

V, Definition 7.6, p. 316 XIII, Definition 4.6(4), p. 648 XIII, Definition 4.7(2), p. 648 XIII, Definition 4.7(2), p. 648 X, Definition 5.8, p. 536

-

INDEX OF DEFINITIONS AND ABBREVIATIONS 697

lcf (lower cofinelity )

1gw limit ultrapower log (logarithm) low (lower weight)

MA (Martin's

minimal Axiom)

Mlt

mod (modulo) monotonic multidimeneional

multiplicative multiplicity

r )

narrow ND (number of

dimeneione)

omit

ord order property

orthogonal

h f ( K , D)

lgw(P). lgw(& A ) the limit ultrapower M$(D

low(p), low&), low(U, A),


V, Definition 3.6(1), p. 261 VI, Definition 4.2, p. 375

V, Definition 3.4, p. 259; log, m

low*(& A) , low&), l o w ~ ( 4 A )

V, Definition 4.4(4), pp. 277-278

olementery mapping (1) I a m x b l indis-

(2) F-n~&~nal

I, Definition 1.3, p. 4 111, Definition 3.2, p. 106

V, Remark, p. 289 See [ndS 701

cernible set over A in M

(1) M is F-minimal over A

formula

cernible eet]

(6) A is T-minimal .over B

(2 is 1 or 2)

(2) weekly minimal

(3) minimnl t p [indk-

(4) R-weaklyminimsl

UtYp, A, A), MltYp, A )

(1) multidimeneioml I (2) multidimeneioml T


V, Definition 1.3(3), p. 238

V, Definition 1.3(1), (2),

V, Definition 7.2(3), pp. 306-307 XI, Definition 2.2(6), p. 563

p. 238

II, Definition 1.2, p. 21

see depend VI, Definition 2.1(1), p. 333 V, Definition 5.2, p. 286 V, Definition 6.3, p. 286 VI, Definition 2.1(1), p. 333 eee Mlt

(1) n(E) (number of the

(2) W W 4 )

(3) n(NJ (4) n(TJ A-narrow ND(T)

111, Notation, p. 94

XIII, Definition 4.6(1), p. 647 XIII, Definition 4.7(1), p. 648 XIII, Definition 4.7(2), p. 648 X, Definition 5.16A, p. 544 IX, Theorem 2.3, p. 502

equivalence classes of E ) -

(1) M omits a type (2) M Strongly omits 8

Tord formula hae the order

property (order p) (1) weekly orthogonal

Notation VI, Definition 6.3, p. 392

II, Theorem 4.8(1), p. 72 11, Theorem 2.2(3), pp. 30-31

V, Definition l.l(l), p. 230

type P

types

698 INDEX OF DEFINITIONS AND ABBREVIATIONS

otop (omitting type order property)

perdel pertition

PC (pseudo- elementary)

power primery

prime

primitive

product

proper

qf (quantifier

qd (quantifier free)

depth)

R (r=w rank Ramsey Theorem

reduced power reduced product

(2) orthogonal types, V, Definition 1.1(2), (3), p.230 orthogonal indiscernible seh

(3) p is orthogonal to B (4) p is strongly-

orthogonal to 9 (5 ) p orthogonal to A (6) p is almost

orthogonal to A T has the otop

parallel types

PC(T1, TI, PC,(TI, TI

pertition of a Boolean algebra

set [model] is F-primary

(1) model [set] is F-prime

(2) M is prime over A

(3) A is T prime over B (1) set [model] is F-

primitive over A, or (F, p)-primitive

(2) M(A) is primitive over A i n K

(3) A is T-primitive (1) reduced product (2) product of filters

@ proper for (I, T)

over A

over A

for K

D, x Dl

(qf m)-type, compaot

qd,,; (qd,,, ml-type, compact model

model

V, Definition 4.4(2), p. 277 V, Definition 4.4(3), pp. 277-278

V, Definition 1.1(4), p. 230 X, Definition 1.5, p. 511

XII, Definition 4.1, p. 608

111, Definition 4.2, p. 117 VI, Definition 3.7, p. 358

VI, Definition 6.2, p. 383; VII , Definition 2.1(1), p. 411

see reduced power IV, Definition 1.4(1), p. 156

IV, Definition 1.4(3), p. 156


XI, Definition 2.2(5) , p. 563 IV, Definition 1.4(2), p. 156


XI, Definition 2.2(4), p. 562 V, Definition 1.2, p. 233 VI, Definition 3.6, p. 358





see R Appendix, Theorem 2.1,

VI, Definition 1.1(3), pp. 324-325 VI, Definition 1.1(2), p. 324

p. 659

INDEX OF DEFINITIONS AND ABBREVIATIONS 699

regular

(6) regular family of sets (7) D is A-regular [regular]

filter (8) A is a regular

cardinality regularize family of sets regularize

representation a filter

(1) (N7, uV: ~ € 1 ) is a

(2) an F-representation (3) E of the

(4) equivalent

(5) standard

representation

representation

representation

representation

(2) (F, p)-saturated (3) F-semi-saturated (4) F-aturated (6) (A, H I , H2)-saturated

(6) (A, &saturated model model

sct

SE

V, Definition 1.2, p. 233 V, Definition 3.1, p. 251 V, Definition 3.6, p. 265

V, Definition 4.4(1), p. 277 V, Definition 4.6(4), (6),

VI, Definition 1,3(1), p. 326 TI, Definition 1.3(3), (4),

see Notation

p. 278

p. 326


X, Definition 5.2, p. 534; redefined X, 5.17, p. 547

X, Definition 5.3, p. 534 X, Definition 5.8, p. 536

X, Definition 5.17B, p. 547

X, Definition 5.8(2), p. 536



I, Definition 2.1(2), (3),

p. 240

P. 9

I, Definition 1.2(1), (3),

IV, Definition l.l(l), p. 155 IV, Definition 1.1(3), p. 155 IV, Definition 1.1(4), p. 155 VII, Definition 1.4( l), p. 402; W, Definition 1.9, p. 410

VII, Definition 1.6, p. 403; see VII, Section 1

I, Definition 1.5, p. 6 XI, Definition 2.2(1), p. 562 111, Definition 7.6, p. 149

XIII, Definition 4.1(1), pp. 643-644 XIII, Definition 4.1(2), p. 644 XIII, Definition 4.1(3), p. 644 XIII, Definition 4.1(3), p. 644 VII, Definition 4.1(2), (3),

P. 3

p. 426

700 INDEX OF DEFINITIONS AND ABBREVLILTIONS

semi-minimel

semi-regular

semi-simple semi-weakly-

minimal set

shallow simple

skeleton Skolem

and

SM) (special number of dimensions)

split

srd (strict independent partitions)

stable

stable system

standard

stationarkat ion

stationary

(1) semi-minimal type (2) semi-minimal formula (3) strongly semi-minimal

(1) semi-regular type (2) P-semi-regular type

formula

a formula is semi-weakly-

set function H, pure set

T is shallow

minimal

function H

(1) B-simple type

(2) strongly 9-simple type

(1) stable theory (in A) , stable model (in A),

A-stable (2) ~(3, $)-stable formula

(3) A-atomically stable (1) stable system

(2) sp. etable system (3) true sp. stable

system the representation is

standard q is the stationarization of

(1) p is stationary over A (2) p is stationary

(in A)

p over A

V, Definition 4.1, pp. 267-269 V. Definition 4.3, pp. 276-277 V, Definition 4.3, pp. 276-277

V, Definition 4.1, p. 267 V, Definition 4.6(3),

V, Definition 4.6, p. 278 V, Definition 4.2, p. 274

VII, Dsfinition 1.3, p. 402

X, Def. 4.2(2), p. 528 V, Definition 4.6(1), (6),


VI, Definition 4.6, p. 377 VIII, Definition 3.1, p. 464 VII, Definition 1.1, p. 400

XIII, Definition 4.6(2), p. 647 XIII, Delinition 4.7(1), p. 648 XIII, Definition 4.7(2), p. 648 XIII, Definition 4.1(1), pp. 643-6

p. 278

p. 278

p. 278


111, Definition 1.2, p. 85 111, Definition 7.4, p. 145

I, Definition 2.2, p. 9 11, Theorem 2.13, p. 36

11, Theorem 2.2(1), p. 30

VII, Definition 2.8, p. 415 XII, Definition 2.1, p. 598 ;

Uefinition 5.1, p. 616 XII, Definition 5.1, p. 616 XII, Definition 5.3, p. 616

X, Definition 5.8(2), p. 536

111, Definition 4.2(2), p. 117

111, Definition 1.7, p. 90 111, Definition 4.1, p. 117 VII, Definition 4.1(6), p. 426

INDEX OF DEFINITIONS AND ABBREWATIONS 70 1

StP

strict order property

Rubmodel

superstable supported system

T Tarski-Vaught

Th (theory) tP (type)

T P TPC TPS TPX transcendental

tree

trival true W (Tarski-

Vaught) type

ugw ultrafilter

ultrapower ultraproduct UL (ultralimit) unidimensional uniform universal

unstable unsuperetable

(3) S c A is stationary

(4) p is stationary inside A

(5) p is stationary inside (B, A)

(1) stp(4 A ) (2) stp*(& A ) , stp*(d, A ) formula q(Z; 5) (theory T )

has the strict order Property

(1) elementary submodel (2) submodel superstable theory T

TPX"(a,M) (1) transcendental T (2) totally transcendental T (1) (5, Bl-tree (2) strong tree (3) uniform &tree a trival type true dimension 'J!VZ(O, A )

q-type, n - t m e , Fff-t-0

UgW(P), urn@. A )

W M , D, 4

uniform filter universal, A-universal,

(< A)-universel model negation of stable negation of superstable


XI, Definition l . l ( l ) , p. 557


111, Definition 2.1, (3), p. 94 111, Definition 2.1(3), p. 94 11, Definition 4.3, p. 69

p. 655

I, Definition 1.1, p. 2 Notation I, Definition 2.2(6), p. 9 VI, Definition 3.7(2), p. 358 me. stable system

VIII, Lemma 1.3, p. 445 XI, Definition 3.1(5), p. 573

Notation I, Definition 2.1(1), (3), p. 9

111, Definition 1.1, p. 85 VII, Definition 1.6, p. 406 see TPX aee TPX VII, Definition 1.7, p. 406 II, Definition 3.4(2), p. 66 11, Definition 3.1, p. 41 VII, Definition 3.1, p. 422 VII, Definition 3.2, p. 423 VI I , Definition 3.2, p. 423 X, Definition 7.1, pp. 55&551 III, Definition 3.3, p. 106 IX, Definition 1.1, p. 492

W , Definition 2.1(1), p. 157

V, Definition 3.5; p. 261 Appendix, Deihtion 1.1(3),

VI, Definition 1.1(3), pp. 324-325 VI, Definition 1.1(3), pp. 324-325 VI, Definition 8.1, p. 390 V, Definition 2.2. p. 247 VI, Definition 3.4, p. 355 I, Definition 1.8, p. 6

p. 653-654

702

UPW (upper weight)

w (weight)

w (weakly)

W

INDEX OF DEFINITIONS AND ABBREVIATIONS

UPW(P), UPW@, 4 V, Definition 3.3, p. 259

(1) W ( P ) l 4% A )

(2) %(P)9 wI?(4 A ) I I , J , P swq

(1) set of triples

V, Definition 3.2, p. 252; 888

V, Theorem 3.9, p. 254 V, Definition 7.2(2), pp. 306-307 V, Definition 2.1(1), (3),

V, Section 6, p. 289 p. 240

(2) WE winning strategy

witness n-witness

ZFC (Zermelo- Fraenkel with choice)

V, Claim 7.1, p. 306 VI, Definition 1.6, p. 328;

V, Definition 7.3, p. 309 VIII, Definition 2.1, p. 459

INDEX OF SPMBOLS

elementa of 6 function funetion, or see VII, Definition 3.1, p. 422 ordinals natural numbers tYP- elements of an ordered set

finite set variables

tmbeets of (?; filter equivdence relation or such a form&, or rarely 8 family of sets generating a filter functions function, or see beginning of V, Section 6 ordered sets or trees or index symbols (in Chapters I-IV and IX-XI11 they are interchanged with 1,J)

cless of modele, or see beginning of V, Section 6 first-order hguege models predicate or relation, usually unary predicates or relations

on I+, I- see X, Section 2

on Q$ see XIII, Definition 1.2 686 theory (usually fixed) set, or see beginning of V, Section 6 Sets

Greek k&%r8

a, B1 Y ordinah 6

2 ordinel 9 sequence, w d y of ordinals

limit ordinal; on S ( A ) , S (A , K ) see VII, Definition 5.1, p. 432 on 7- see X, Section 2

703

INDBX OB SYMBOLS

formul8 d i n s 1 ( d y infinite)

on K"(T), K(T); eee III, Definition 3.1, p. 102

on &(T) eee IV, Definition 2.4, p. 165 on At( T) aee IV, Definition 2.6, p. 165 on A(T) see IX, Conclusion 1.14(4), p. 488 on X(F), A,(F) 888 IV, p. 153 on X(K) see V, p. 289 on A*(p) eee 111, Definition 6.1, p. 124 on A( c *), A ( T ) , A(T, E *), A,(T, E *) see XI, 2.3

on p(A), p(A, K ) see VII, Definition 6.1, p. 432 on p(D) see VI, Definition 6.2, pp. 391-392 on p(F) see IV, Definition 1.1(2), p. 155 on p(K) me V, p. 289

csrdinel(uSuallyinfinite);

cerdinel ( d y infinite);

sequenoe, UBUBuy of ordinals Ordinel sequenoe, d y of ordinale permutation, or term, or eequenoe of ordinele term formule cerdinel, uBu&z1y infinite formule h t infinite ordinel am infinite ordinal

set offormuha; see II, Definition 3.3, p. 44; eee 11, Theorem 2.2(4), p.31 sets of formuh set of formulas or types

oths* -8

a infinite eequenoe of elements B Boolean algebra d b model B Seep. 7 F

oonetrudion; see rV, Seotion I

888 Iv, Sedion 1, p. 153; on F$ see IV, Definition 2.1, IV, Definition 2.2, IV, Definition 2.3, pp. 157-158, IV, Definition 2.6, IV, Definition 2.7, p. 168, VII, Definition 4.2, p. 428, XII, Definition 1 . 1 , p. 591, XII, Definition 1.6, p. 593 on Fgak see IV, Definition 4.1, p. 185 on F(T) see XI, 2.2 on F'j see XII, Definition 1 .1 on Fi see XII, Definition 1.6

Y f d y of funotions 1, J

9

sete of sequenoee from 6, usually of h e d length, mostly indkernible (in chapters I-V and IX-XI11 they are interchanged with I , J ) power set see X, Section 5, XI1 on %(8) see XII, 2.1

705 INDEX OF SYMBOL8

T t truth value (0, 1)

T ( A ) a family of ‘‘small types” over A , see XI, 2.1, on T f see XI, 2.2

othcrsynbob

existential quantifier;

empty eet elementary submodel; see I, Definition 1.1, p. 2;

equivalenoe relation;

produot

on 3( X see V, Section 6, p. 289

on 4; see V, Section 6, p. 289

on w D see VI, Definition l.l(l), p. 324

Bum

natural sum; see V, Definition 7.0, p. 316 Keisler order; see VI, Definition 4.1(2), p. 370 see VI, Definition 4.3, p. 375 initial segment; see Notation restriation; see Notation and VI, Notation, p. 331 infinity sign subseta (not necepnnrily proper)

satisfaation; BBB Notation see 11, Definition 1.3, p. 23 on B < AC see X, Definition 1.2

on E:,, Gz see XI, Definitions 3.4, 3.5

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