AXIOMATIC SET THEORY

http://dx.doi.org/10.1090/pspum/013.2

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

VOLUME XIII, PART II

AXIOMATIC SET THEORY

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1974

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA

JULY 10-AUGUST 5, 1967

EDITED BY

THOMAS J. JECH

Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-6698

Library of Congress Cataloging in Publication Data

Symposium in Pure Mathematics, University of California, Los Angeles, 1967. Axiomatic set theory.

The papers in pt. 1 of the proceedings represent revised and generally more detailed versions of the lec­tures.

Pt. 2 edited by T. J. Jech. Includes bibliographical references. 1. Axiomatic set theory-Congresses. I. Scott,

Dana S. ed. II. Jech, Thomas J., ed. HI. Title. IV. Series. QA248.S95 1967 51l'.3 78-125172 ISBN 0-8218-0246-1 (v. 2)

AMS (MOS) subject classifications (1970). Primary 02K99; Secondary 04-00

Copyright © 1974 by the American Mathematical Society

Printed in the United States of America

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission

of the publishers.

DDE

CONTENTS

Foreword vii Current problems in descriptive set theory 1

J. W. ADDISON

Predicatively reducible systems of set theory 11 SOLOMON FEFERMAN

Elementary embeddings of models of set-theory and certain subtheories 3 3 HAIM GAIFMAN

Set-theoretic functions for elementary syntax 103 R. O. GANDY

Second-order cardinal characterizability 127 STEPHEN J. GARLAND

The consistency of partial set theory without extensionality 147 P. C. GlLMORE

On the existence of certain cofinal subsets of "(D 155 STEPHEN H. HECHLER

Measurable cardinals and the GCH 175 RONALD BJORN JENSEN

The order extension principle 179 A. R. D. MATHIAS

"Embedding classical type theory in 'intuitionistic' type theory." A correction 185

JOHN MYHILL

Remarks on reflection principles, large cardinals, and elementary embeddings 189

W. N. REINHARDT

Axiomatizing set theory 207 DANA SCOTT

Author Index 215 Subject Index 217 Lectures delivered during the Institute 219

FOREWORD

This volume is the second (and last) part of the Proceedings of the Summer Institute on Axiomatic Set Theory held at U.C.L.A., July 10—August 5, 1967.

Many of the lectures delivered during the Institute have been published in the first volume of these PROCEEDINGS, edited by Dana S. Scott. Although we were unable to obtain all the remaining manuscripts, this volume contains most of them. A small number of the contributions was meanwhile published elsewhere; the complete list of lectures is provided at the end of this volume.

For several reasons, the publication of this volume was slightly delayed. I wish to thank the authors of the papers for their patience.

THOMAS J. JECH

Author Index

Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book.

Ackermann, W., 192, 205 Hadamard, Jaques, 1 Addison, J. W., 1, 10, 128, 146 Hanf, W. P., 71, 101, 145, 146, 205 Aleksandrov, Pavel S., 5 Hausdorff, Felix, 2, 3, 8, 155, 173 Asser, G., 135, 146 Hechler, Stephen H., 155

Hinman, Peter G., 7

Baire, Rene, 1, 2, 3 Barnes, Robert F., Jr., 4 Barwise, J., 103, 126 Bennett, J. H., 122, 126, 135, 146 Benson, Guy M., 5 Bernays, P., 195, 205 Borel, Emile, 1, 2, 3, 4, 8, 9 Brouwer, Luitzen Egbertus Jan, 4

Cantor, Georg, 2 Cohen, Paul J., 10, 100, 144, 146, 155, 159, 161,

165, 173 Coppleston, F., 198, 205

Devlin, K. J., 104, 126

Feferman, Solomon, 11, 11, 12, 16, 17, 20, 21, 30, 32,32, 116, 125, 126

Fitch, F. B., 149, 153 Fraenkel, Abraham A., 8, 9 Friedman, H., 32, 32

Gaifman, Haim, 33, 36, 37, 42, 46, 95, 96, 100, 101

Gandy, R. O., 30, 31, 32, 103, 126 Garland, Stephen J., 127 Gilmore, P. C , 147, 147, 150, 153 Godel, K., 10, 111, 126, 186, 187, 188 Grzegorczyk, A., 104, 126

Jech, T., 183, 183, 198, 205 Jensen, Roland Bjorn, 103, 104, 106, 108, 109,

126, 175, 179, 183 Johsson, B., 179, 183

Kalmar, Laszlo, 4, 8, 9 Kantorovic, Leonid, 6 Karp, C , 103, 108, 126 Keisler, H. J., 68, 84, 96, 100, 101 Kleene, Stephen C , 4, 137, 138, 146 Kolmogorov, Andrei N., 7, 8 Kreisel, G„ 17, 21, 30, 31, 32, 32, 125, 126 Kripke, S., 30, 32 Kunen, K., 37, 38, 65, 71, 72, 76, 92, 95, 96,

101, 131, 144, 146, 200, 201, 205

La Vallee Poussin, Charles J. de, 2 Lavrent'ev, Mihail A., 3 Lebesgue, Henri, 1, 3 Levy, A., 103, 108, 126, 141, 146, 192, 194, 196,

205 Livenson, E. M., 6 Luzin, Nikolai N., 3, 4, 5, 6, 8 Lyndon, Roger C , 148, 149, 150, 153

Magidor, M., 86, 88, 94, 101 Martin, Donald Anthony, 10 Mathias, A. R. D., 175, 179 Montague, R. M., 143, 146

215

216 AUTHOR INDEX

Moschovakis, Yiannis N., 6, 10, 126, 128, 146 Mostowski, Andrzej, 181, 183 Myhill, John, 185, 185

Pincus, D., 183, 183 Platek, R., 30, 32 Powell, W. C , 35, 91, 94, 101, 198, 205

Reinhardt, W. N., 35, 78, 86, 94, 101, 189, 192, 198, 200, 205

Ritchie, R. W., 121, 126 Rogers, Hartley, Jr., 2 Rowbottom, F., 36, 101

Scholz, H., 135, 146 Scott, Dana S., 13, 71, 95, 101, 145, 146, 200,

205, 207 Selivanovskil, E. A., 5 Shoenfield, J. R., 71, 74, 101, 132, 136, 137, 146,

198, 205

Sierpinski, Waclaw, 155, 173 Silver, J. H., 9, 36, 72, 101, 145, 146, 198, 205 Smullyan, R. M., 104, 122, 126 Sochor, A., 183, 183 Solovay, Robert M., 35, 78, 86, 94, 101, 155,

173, 189 Suslin, Mihail Ya., 5, 6, 7, 8 Suzuki, Y., 137, 146

Tarski, Alfred, 68, 96, 100, 101 Tennenbaum, Stanley, 155, 173

Urysohn, P. S., 5

Vaught, R. L., 42, 101, 205

Zermelo, Ernst, 8, 9 Zykov, A. A., 130, 146

Subject Index

AJ, 128 a-scale, 155 Alternating sum, 3 Antitone, 3 Approximations of embeddings (y-approxima-

tions, < ̂ -approximations), 34, 78-83, 86, 87

Axiom(s) Vv la (Ra exists Av e Ra], 33, 54 of constructibility, 9 of definable determinateness, 10 of extendibility, 199 of infinity, 9, 189, 199 of measurable cardinals, 9 recognizing, 204-205

Basic closure, 109 Basic functions, 105 Basic numerals, 118 Bilateral, 3 Blowing up small structures, 72-75 Borel hierarchy, 2

Cardinals in models of ZC+, 95 Cardinals, see Large cardinals Cartesian product, 105, 111 Category (of structures), 40, 41 Characterizable cardinals, 128 Classes (as distinguished from sets), 37, 50-54 Classical descriptive set theory, 2 Cofinal embedding, 33, 34, 36, 54-60, 79, 90, 91

y and < y-cofinality, 34, 35, 78-84 Consistency results, 144, 147 Constructible closure, 111 Construction principle, 3 Coreduction property, 7

S-separated, 8 Decomposition of embeddings, 34, 82, 83 Sm, 136

A0 predicates, 105, 106, 108, 114-117 A0-separation, 12 Definability, 36, 39

general concept, 41, 42 Definable, 1 Defining schema, 39, 40, 41, 42 Descriptive set theory, 1 Detail, 4 OJ,128 Direct limit, 36, 41, 46, 47 Duals, 3

Embeddings of intuitionistic type theory, 185 ordinals and the first ordinal move by, 34, 35,

67, 75, 85-87 critical ordinals of, 67 see also: Approximations of embeddings, Co-

final embeddings, and "Local" conditions Exhaustion principle, 5 VJ, 128 Extendible, 192, 197, 199, 202 Extendible cardinals, 35, 94, 197, 199

various concepts of extendibility, complete extendibility, 86, 87, 94, 95

Extensionality axiom, 147 Extension operators, 36, 39-42, 95-99

Iterations of—see Iterations of extension operators

First separation principles, 7 First separation property, 7 Forcing, 156 Functor, 41, 45, 49 Fusion, 5

General recursive functions, 2 Generalized continuum hypothesis, 131 Godel embedding, 185

217

218 SUBJECT INDEX

Hausdorff hierarchy, 3 Hierarchies, 2,128 Higher-order definability, 143 Hyperarithmetic comprehension rule, 16

Inclusion principle, 5 Indexing of functions, 123 Indiscernibles, 49, 50, 68, 71 Inner quantifier, 10 Intuitionistic type theory, 185 Invariant for e-extensions relative to S, 17 Irreducible cover, 201, 202 Iterations of extension operators, 36-38, 43-54,

60-62, 64-78

Kalmar hierarchy, 4

A-extendible, 94,199 Large cardinal(s), 145

definitions of, via elementary embeddings, 35, 85-88, 189, 198

properties, 193 in iterated extensions, 75-78 see also: Measurable cardinals, Extendible

cardinals and Supercompactness Liftings of embeddings, 35, 36, 88-93 Limit ultrapowers, 34, 83-85 "Local*' conditions on elementary embeddings,

35, 86-88 Luzin hierarchy, 6

Measurable cardinal(s), 36, 68, 74, 75,145 Models of set theory, 144

Natural extension operator, see Extension operators

O-classes, 198, 199, 200, 201 Operation (A), 5 Operator, see Extension operators Operator R, 7 Ordinal functions Km(fi) and the ordinal r0» 12 Ordinal sufficiency rule, 13 Outer quantifier, 10

Partial set, 147 Persistent for 8-extensions relative to S, 17 Il-formula, 13 Predicative predicate, 109

see also: Simple predicate Predicative set theories, 115

Predicatively reducible systems, 11 Projections, 5 Projective hierarchy, 6 Provably definite relative to 5, 18 Provably 2 H II formula, 13 Pushing up ordinals, 37, 68, 69, 72

R-sets, 7 Ramsey cardinals, 36, 145 Random reals, 155 Recursive function theory, 2 Reduction property, 7 Reflection principles, 190 Representation structures (which are ordinally

coded well-founded trees in OJ), 12 Rudimentary predicates, 104,121 Rudimentary functions, 104

S-admissible set, 30 Scales, 155 Self-extension(s), 58

see also: Extension operator Separated unions, 9 Separation Urprinciple, 8 X H 11-separation rule, 14 X-formula, 13 ^-reflection rule, 12 Simple predicate, 104 Spectrum, 128 Spectrum problem, 135 Strongly compact, 202 Strong separation principle, 7 Subtheories of set theory

Z+ , 33, 54 Z\ 55 f̂inite* 55, 59

ZC+, 33, 78 Supercompactness, 35, 86, 87, 202 Suslin hierarchy, 5 Syntax, elementary, functions and predicates of,

103-104,122,125

Theory of definability, 1 Transfinite

induction rule, 16 recursion rule, 16

Undefinability, 2 Uniformization problem, 1

Well-foundedness, 37,62-64, 73-75, 88-90,101

LECTURES DELIVERED DURING THE INSTITUTE

John Addison Current problems in descriptive set theory

Robert Bradford Undecidability of the theory of Dedekind cardinal addition

C. C. Chang Sets constructible using LKK

Paul Cohen Remarks on the foundation of set theory

John E. Doner and Alfred Tarski Extended arithmetic of ordinal numbers

Paul Erdos Finite and infinite combinatorial analysis

Paul Erdos Problems and results on combinatorial set theory

S. Feferman Predicative set theory

Paul Fjelstad Set theory as algebra

William Frascella Tactical configurations for infinite sets

Haim Gaifman Pushing up the measurable cardinal

Robin Gandy Recursive functions of sets

Stephen Garland Second-order cardinal characterizability

R. J. Gauntt Undefinability of cardinality

219

2 2 0 LECTURES DELIVERED DURING THE INSTITUTE

P. C. Gilmore Partial set theory

Petr Hajek Bibliography of the Prague seminar on foundations of set theory (1962-1967)

James Halpern The Boolean prime ideal theorem

Stephen Hechler Generalized scales on number-theoretic functions

Ronald Jensen Measurable cardinals and the GCH

Carol Karp Some interconnections between infinitary logic and set theory

H. Jerome Keisler End extensions of models of set theory

Saul Kripke On the application of Boolean-valued models to solutions of problems in Boolean algebra

Saul Kripke Transfinite recursion, constructive sets, and analogues of cardinals

Kenneth Kunen Indescribability and the continuum

F. William Lawvere Category-valued higher-order logic

Azriel Levy The independence of the axiom of choice, the generalized continuum hypothesis and the axiom of cons true tibility ofyL2 statements of set theory

Azriel Levy Independence proofs by the Cohen forcing method. I

Azriel Levy Independence proofs by the Cohen forcing method. II

Azriel Levy The sizes of the indescribable cardinals

Saunders Mac Lane Categorical alternatives to set theory

R. Mansfield The solution of one of Ulam's problems concerning analytic rectangles

A. R. D. Mathias The order-extension principle

Kenneth McAloon Cohen's method applied to questions about ordinal definability

LECTURES DELIVERED DURING THE INSTITUTE 221

Yiannis N. Moschovakis Predicative classes

Jan Mycielski On the axiom of determinateness

John Myhill Imbedding classical type theory in Hntuitionistic' type theory

John Myhill and Dana Scott Ordinal definability

K. Namba An axiom of strong infinity and the analytic hierarchy of ordinal numbers

Katuzi Ono On Russell type paradoxes and some related problems

Richard Platek Eliminating the continuum hypothesis

Lawrence Pozsgay Liberal intuitionism as a basis for set theory

Karel Prikry Models constructed using perfect sets

Hilary Putnam Degrees of unsolvability and constructive sets

William Reinhardt Conditions on natural models of set theory

Gerald Sacks Degrees of nonconstructibility

Dana Scott Axiomatizing set theory

Dana Scott Lectures on Boolean-valued models for set theory

J. R. Shoenfield Constructible sets

Jack Silver On the consistency of the GCH with the existence of a measurable cardinal

Jack Silver The independence ofKurepa's conjecture and two-cardinal conjectures in model theory

Robert Solovay Measurable cardinals and the axiom of determinateness

Robert Solovay Real-valued measurable cardinals

111 LECTURES DELIVERED DURING THE INSTITUTE

Robert Solovay Solution of a problem ofFodor and Hajnal

Robert Solovay IfZF + AC + MC is consistent, then ZF + MC + 2Ko is a real-valued meas­urable cardinal is consistent. A sketch of the proof

G. L. Sward Transfinite sequences of axiom systems for set theory

Gaisi Takeuti Hypotheses on power set

R. L. Vaught Axiomatizability by a schema

Martin Zuckerman Finite versions of the axiom of choice