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Wolfgang Rautenberg
Berlin
An introduction tomathematical logic
Textbook
Typeset and layout: The author
Version from December 2004
V
Foreword to the English editionDRAFT
The friendly reception of the German-language edition of this book has made the
decision easier to prepare the first english version, a revised translation of the 2nd
german edition. Although the general conception has not been changed, all details
have been worked over. Moreover, Chapter 7 has been expanded in order to discuss
at least some of the latest results in the area of self-reference. Section 7.1 on the
Derivability Conditions is now completely self-contained.
The book is aimed at students of mathematics, computer science, or linguistics,
as well as students of philosophy with a mathematical background because of the
general epistemological interest of Godel’s Incompleteness Theorems. For an abbre-
viated course on mathematical logic, combined for example with an introduction
to set theory, the material for the logic part is covered by the first three chapters
(about 100 pages), which also include a discussion of the axiom system ZFC. The
last sections of Chapter 3 are of a partly descriptive nature, providing a view towards
decision problems, automated theorem proving and further subjects.
On top of the material for a one-semester course, basic material for a lecture course
in logic for computer scientists is included in Chapter 4 on logic programming where
an effort has been made to capture at least some of the mathematically interesting
aspects of this discipline’s logical foundations. Computable functions are made
precise using PROLOG programs, and the undecidability of the existence problem
for successful resolutions is proved as simple as possible.
Chapter 5 concerns applications of mathematical logic for various methods of
model construction and contains enough material for an introductory course to
model theory. It presents in particular a proof of quantifier eliminability in the
first-order theory of real closed fields, one of the basic results in this area.
A special aspect of the book is the thorough treatment of Godel’s Incompleteness
Theorems. These are based on the representability of recursive predicates in for-
malized theories. Hence, Chapter 6 starts with the foundations of recursion theory
which are basic both for applications to decidability and undecidability problems
and to Godel’s theorems. 6.2 is devoted to so-called Godelization. A classification
of defining formulas for arithmetical predicates is introduced already in 6.3 in order
to elucidate the close relationship between logic and recursion theory as early as pos-
sible. Along these lines we obtain in 6.4 in one sweep Godel’s First Incompleteness
Theorem, the undecidability of the tautology problem from Church, and Tarski’s
result on the non-definability of truth. Decidability and undecidability is dealt with
in 6.5 and 6.6 and includes a sketch of the solution to Hilbert’s Tenth Problem.
VI Foreword
Chapter 7 is devoted exclusively to Godel’s Second Incompleteness Theorem and
some of its generalizations. Of particular interest thereby is the fact that questions
about self-referential arithmetical sentences are algorithmically decidable due to
Solovay’s Completeness Theorems.
This book is meant to be used not only to accompany lectures, but can be used
for independent and individual study. For this reason an index and a list of symbols
have been as carefully prepared as possible; for the large part of the exercises hints
are given in a special section. Apart from a sufficient training in logical (or math-
ematical) deduction, there are no special prerequisites; only in Chapter 5 a basic
knowledge of classical algebra might be useful. The very last portion of the book
assumes some acquaintance with models for axiomatic set theory. The demands on
the reader grow from Chapter 4 on. They can best be met by attempting first to
solve the exercises without recourse to the hints at the end of the book.
Remarks in small print refer occasionally to notions which are either undefined or
will be introduced later. Likewise such remarks direct one towards references in the
bibliography, which can necessarily represent only an incomplete selection.
In spite of the variety of topics, this book can only provide a selection of results.
It is no longer possible to compile an encyclopaedic textbook even for sub-areas
of mathematical logic, and in the selection of material one can at most accentuate
certain topics. This is above all the case for Chapters 4, 5, 6 and 7, which go a step
beyond the elementary. Philosophical and foundational problems of mathematics
are not systematically dealt with, but are nonetheless considered where thought
appropriate, in particular with regard to Godel’s Theorems. A particular concern
of this book was to portray simple things simply and to avoid over-correct notation
which may divert from the essentials.
The seven chapters of the book consist of numbered sections. A reference like
Theorem 5.4 is to mean Theorem 4 in Section 5 of a given chapter. In cross-
referencing from another chapter, the chapter number will be adjoined, for instance
Theorem 6.5.4 is Theorem 5.4 in Chapter 6.
For helpful criticism I thank numerous colleagues and students; the list of names
is too long to be given here. Particularly useful for Chapter 7 were the hints from
L. Beklemishev (Moscow) and W. Buchholz (Munich).
Here: Thanks to the publisher.
Berlin, December 2004,
W. Rautenberg
VII
Contents
Foreword V
Introduction XI
Notation XIV
1 Propositional logic 1
1.1 Boolean functions and formulas . . . . . . . . . . . . . . . . . . . . . 2
1.2 Semantic equivalence and normal forms . . . . . . . . . . . . . . . . . 9
1.3 Tautologies and logical consequence . . . . . . . . . . . . . . . . . . . 14
1.4 A complete calculus for . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Applications of the Compactness Theorem . . . . . . . . . . . . . . . 25
1.6 Hilbert calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Predicate logic 33
2.1 Mathematical structures . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Syntax of elementary languages . . . . . . . . . . . . . . . . . . . . . 43
2.3 Semantics of elementary languages . . . . . . . . . . . . . . . . . . . 49
2.4 General validity and logical equivalence . . . . . . . . . . . . . . . . . 58
2.5 Logical consequence and theories . . . . . . . . . . . . . . . . . . . . 62
2.6 Expansions of languages . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Godel’s Completeness Theorem 71
3.1 A calculus of natural deduction . . . . . . . . . . . . . . . . . . . . . 72
3.2 The completeness proof . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3 First applications – non-standard models . . . . . . . . . . . . . . . . 81
3.4 ZFC and Skolem’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5 Enumerability and decidability . . . . . . . . . . . . . . . . . . . . . . 92
VIII Contents
3.6 Complete Hilbert calculi . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7 First-order fragments and extensions . . . . . . . . . . . . . . . . . . 99
4 The foundations of logic programming 105
4.1 Term-models and Herbrand’s Theorem . . . . . . . . . . . . . . . . . 106
4.2 Propositional resolution . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Logic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Elements of model theory 131
5.1 Elementary extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Complete and κ-categorical theories . . . . . . . . . . . . . . . . . . . 137
5.3 Ehrenfeucht’s Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.4 Embedding and characterization theorems . . . . . . . . . . . . . . . 145
5.5 Model completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.7 Reduced products and ultraproducts . . . . . . . . . . . . . . . . . . 163
6 Incompleteness and undecidability 167
6.1 Recursive and primitive recursive functions . . . . . . . . . . . . . . . 169
6.2 Arithmetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3 Representability of arithmetical predicates . . . . . . . . . . . . . . . 182
6.4 The Representability Theorem . . . . . . . . . . . . . . . . . . . . . . 189
6.5 The Theorems of Godel, Tarski, Church . . . . . . . . . . . . . . . . 194
6.6 Transfer by interpretation . . . . . . . . . . . . . . . . . . . . . . . . 200
6.7 The arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 205
7 On the theory of self-reference 209
7.1 The Derivability Conditions . . . . . . . . . . . . . . . . . . . . . . . 210
7.2 The theorems of Godel and Lob . . . . . . . . . . . . . . . . . . . . . 217
7.3 The provability logic G . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.4 The modal treatment of self-reference . . . . . . . . . . . . . . . . . . 223
7.5 A bimodal provability logic for PA . . . . . . . . . . . . . . . . . . . . 226
7.6 Modal operators in ZFC . . . . . . . . . . . . . . . . . . . . . . . . . 228
Hints to the Exercises 231
Contents IX
Literature 241
Index 247
List of symbols 255
X
XI
IntroductionTraditional logic as a part of Philosophy is one of the oldest scientific disciplines
and goes back to the Stoics and to Aristotle 1). It is one of the roots of what
nowadays is called Philosophical logic. Mathematical logic, however, is a relatively
young discipline and arose from the endeavors of Peano, Frege, Russell and others to
create a logistic foundation for mathematics. It steadily developed during the 20th
century into a broad discipline with several subareas and numerous applications in
mathematics, informatics, linguistics and philosophy.
There are several english textbooks on mathematical logic written in the 2nd half
of the last century, among them very good ones. Our motive to add another one is
simple: it has proven its usefulness for german readers and it is more concise than
most other textbooks. Since the material is treated in a rather streamlined fashion,
it covers nonetheless the most important topics, and it contains some material on
self-reference not yet contained in other textbooks.
One of the features of modern logic is a clear distinction between object-language
and meta-language. The meta-language is normally a sort of a colloquial language
and is more or less the same for various formalized object languages, although it
differs from author to author and depends also on the audience the author has
in mind. Since this book concerns mathematical logic, the meta-language is here
ordinary English mixed up with some semi-formal elements which mostly have a
set-theory origin. The amount of set theory involved depends on one’s objectives.
For instance, general semantics and model theory use stronger set theoretical tools
than proof theory. But in the average it is little more assumed than the knowledge
of the set-theoretical terminology briefly described in a separate section Notation
following this introduction. Mathematicians are familiar with this terminology and
hence may skip most of it.
One of several goals of mathematical logic is the investigation of formalized object-
languages (their syntax and semantics) with mathematical tools. The way of argu-
ing about formal languages and theories is traditionally called the metatheory. An
important task of a metatheoretical (or metamathematical) analysis is to specify
procedures of logical inference by so-called logical calculi which operate with syntac-
tic rules only. Basic metatheoretical tools are in any case the naive natural numbers
and proofs by induction on these numbers. We will sometimes call them proofs by
metainduction, in particular when talking about formalized theories which them-
selves aim at speaking about natural numbers using certain formalized induction
principles. These must clearly be distinguished from metainduction.
1)The Aristotelian syllogisms are useful examples for inferences in an elementary language withunary predicate symbols. One of these serves as an example in Section 4.4 on Logic programming.
XII Introduction
The logical means of the metatheory are sometimes allowed or even explicitly re-
quired to be different from those of the object-language. But normally both the logic
of object languages and of the meta-language is classical, two-valued logic. There are
good reasons to argue that classical first-order logic is the logic of common sense.
This is not only a convenient convention for mathematics. Computer scientists,
linguists, philosophers, physicists and other scientists need to agree on a common
logical basis in the process of communication, and this basis is essentially classical
first order logic. It therefore is the central subject of our investigation.
It should be noticed that logic used in the sciences differs essentially from logic
used in everyday language where logic is understood as the task or the art of saying
what follows from what. In everyday language, nearly every utterance depends on
the context. In most cases logical relations are only alluded to and seldom explicitly
expressed. Even a basic assumption of two-valued logic mostly fails, namely that
a proposition is either true or false. For instance, look at the proposition “Money
ensures privileges” whose truth value clearly depends on the circumstances under
which it is uttered. Problems of this type are not treated in this book. To some
extent, many-valued logic and Kripke-semantics can help to clarify the situation
but these belong rather to linguistics or perhaps to a special domain of philosoph-
ical logic. This does not exclude that intrinsic mathematical methods have to be
developed and applied in analyzing and solving such problems.
The language of this book is similar to the colloquial language common to all
mathematical disciplines. However, in most mathematical disciplines meta-language
and object language are mixed. Even if this discipline claims to be axiomatic like
geometry, it is neither strictly formalized nor are the logical means specified, in
general. An exception is axiomatic set theory which needs a strict formalization to
explain how the axioms of separation and of replacement look like.
In the period of development of modern mathematical logic in the last century,
some highlights may be distinguished of which we mention just a few. The first
was perhaps the axiomatization of set theory in different ways. The most important
ones are the approach of Zermelo (see [Hej]) and the one by Whitehead and Russell
([WR]). The type theory of Russel was the extract and is all what remained from
the original program to reduce mathematics to logic. Instead it became clear that
mathematics can entirely be based on set theory as a first-order theory. In fact, this
became more salient only after the remnants of hidden assumptions by Zermelo,
Whitehead and Russell 2) were removed around 1915. Right after these axiomatiza-
tions were completed Skolem discovered that there are countable models of the set
theoretic axioms, a drawback for the hope for an axiomatic definition of a set.
2)for instance, that the notion of an ordered pair is a set-theoretical and not a logical one.
Introduction XIII
Just then, two distinguished mathematicians, Hilbert and Brower entered the
scene and started their famous quarrel on the foundations of mathematics. It is
exposed in an excellent manner in [Kl2, Chapter IV] and need not be repeated here.
The next highlight was Godel’s proof of the completeness of the rules for predicate
logic presented for the first time in the first textbook on mathematical logic in
1928 in [HA]. Meanwhile Hilbert had developed his program of a foundation of
mathematics. It aimed at proving the consistency of arithmetics and perhaps of
the whole of mathematics including its infinitistic set theoretic methods by finitary
means of proof theory. But Godel showed by his Incompleteness Theorems ([Go2])
that Hilbert’s original program fails or at least needs thorough revision.
Many logicians consider this theorems to be the top highlight of mathematical logic
of the 20th century. A consequence of these theorems is the existence of consistent
extensions of Peano-Arithmetic (the common basis of number theory and discrete
mathematics) in which true and false sentences live in peaceful coexistence with
each other – called “dream theories” in Section 7.2. It is an intellectual adventure
of holistic beauty to see those wisdoms from number theory, known for ages, like
the Chinese Remainder theorem or simple properties of prime numbers and Euclid’s
characterization of coprimeness (page 193) unexpectedly assume pivotal positions
within the architecture of Godels proofs.
The methods Godel developed in his paper were also basic for the creation of
recursion theory around 1936. Church’s proof of the undecidability of the tauto-
logy problem in [Ch] marks another distinctive achievement. After having collected
enough evidence by his own investigations and by those of Turing, Kleene and some
others, Church formulated his famous thesis, although in 1936 no computers in the
modern sense existed nor was it foreseeable that computability will ever play the
basic role it does today.
As was mentioned, Hilbert’s program had to be revised. A decisive step was under-
taken in [Ge], a paper to be considered to be another groundbreaking achievement
of mathematical logic and is the starting point of contemporary proof theory. The
logical calculi in 1.2 and 3.1 are akin to Gentzen’s calculi of natural deduction.
We further mention the discovery (also by Godel) that it is not the axiom of choice
(AC) which creates the consistency problem in set theory. Set theory with AC is
consistent provided set theory without AC is. This is a top result of mathematical
logic insofar as without strictly formal methods Godel would not have succeeded.
The same remark applies to the independence proof of AC by P. Cohen in 1963.
The above indicates that mathematical logic is closely connected with the aim of
giving mathematics a solid foundation. Nonetheless, we confine ourself to the former.
History shows it is impossible to establish a programmatic view on the foundations
of mathematics which pleases everybody from the mathematical community.
XIV
NotationAlmost all notation in this book is standard. N, Z, Q, R denote the sets of natural
numbers including 0, of integers, rational and real numbers, resp. n,m, i, j, k denote
always natural numbers unless stated otherwise. Hence, extended notation like
n ∈ N is as a rule omitted. N+ denotes always the set of positive natural numbers.
M ∪N , M ∩N and M \N denote as usual union, intersection, set difference of sets
M, N , resp. ⊆ denotes inclusion. M ⊂ N abbreviates M ⊆ N and M 6= N , but
will only be used if the circumstance M 6= N has to be emphasized. If M is fixed
in a consideration, and N varies over subsets of M then the set M \N may also be
denoted by \N or ¬N . ∅ denotes the empty set and PM the power set of M , the
set of all its subsets. A set containing a single element only is called a singleton.
If one wants to emphasize that all elements of a set F are sets, F is also called a
family or system of sets.⋃
F denotes the union of a set family F , that is, the set of
elements belonging to at least one M ∈ F , and⋂
F stands (whenever F 6= ∅) for the
intersection of F , i.e. the set of elements belonging to all M ∈ F . If F = Mi | i ∈ Ithen
⋃F and
⋂F are mostly denoted by
⋃i∈I Mi and
⋂i∈I Mi, resp.
The cross product M ×N is the set of all ordered pairs (a, b) with a ∈ M and
b ∈ N . A relation between M and N is a subset of M ×N . Is f ⊆ M ×N and is
there to each a ∈ M precisely one b ∈ N with (a, b) ∈ f , then f is called a function
or a mapping from M to N . Then b is also denoted by f(a) or fa or by af and
called the value of f at a. In addition, ran f = fx | x ∈ M is called the range
(or image) of f , while dom f = M is the domain of f . In case that dom f ⊆ M is
assumed only, f is called a partial function from M to N .
A function f with dom f = M and ran f ⊆ N is often denoted by f : M → N ,
also by f : x 7→ t(x), provided f(x) = t(x) for some term t and for all x ∈ M .
Furthermore, the phrase ‘let f be a function from M to N ’ is often shortened to ‘let
f : M → N ’. A function f is injective (or reversible), if fx = fy ⇒ x = y, for all
x, y ∈ M , surjective (or onto), if ran f is the whole of N , and bijective, if f is both,
injective and surjective. Whenever N = M then the identical mapping idM : x 7→ x
is an example. The set of all f : M → N is denoted by NM .
If f, g are mappings with ran g ⊆ dom f then h : x 7→ f(g(x)) is named their
composition or their product, denoted by h = f g. It is easily seen that f ∈ NM is
bijective if and only if there is some g ∈ MN with f g = idN and g f = idM . By
the way, the notation xf for fx suggests to define f g in such a way that first f
and then g is applied.
Let I and M be sets where I, fairly arbitrary, is called the index set. Then a
function f ∈ M I with i 7→ ai will often be denoted by (ai)i∈I and is called, depending
on the context, a family, an I-tuple or a sequence. f is called finite oder infinite,
Notation XV
according to whether I is finite or infinite. If 0 is identified with ∅ and n > 0 with
0, 1, . . . , n−1 as is common in set theory, then Mn can be understood as the set of
finite sequences or n-tuples (ai)i<n from elements of M of length n. The only element
of M0 (= M∅) is the empty sequence ∅ which has length 0. In concatenating finite
sequences (which has an obvious meaning) the empty sequence plays the role of a
neutral element. An equivalent notation for (ai)i<n is (a0, . . . , an−1). For k < n is
(a0, . . . , ak) called a beginning of (a0, . . . , an−1). It is always non-empty. Is k < n−1,
one speaks of a proper beginning. A sequence of the form (a1, . . . , an) will mostly be
denoted by ~a. Here for n = 0 the empty sequence is meant, similar as a1, . . . , anfor n = 0 always denotes the empty set.
If A is an alphabet, i.e., if the elements of A are symbols or at least called symbols,
then the sequence (a1, . . . , an) is written as a1 · · · an and called a string or a word on
the alphabet A. The empty sequence is then consequently called the empty string.
ξη denotes the concatenation of the strings ξ,η. If a string ξ has a representation
ξ = ξ1ηξ2 for some strings ξ1, ξ2 and η 6= ∅ then η is called a substring of ξ. One or
both of ξ1, ξ2 may be empty so that ξ is a substring of itself.
Subsets P, Q, R, . . . ⊆ Mn are called n-ary predicates of M or n-ary relations.
Unary predicates will be identified with the corresponding subsets of M . Instead
of ~a ∈ P we may write P~a, instead of ~a /∈ P also ¬P~a. Is P ⊆ M2 a symbol like
C, <,∈ one normally writes aPb instead of Pab. Predicates cast in words will often
be distinguished from the surrounding text by ‘. . .’, for instance, if we speak on the
syntactic predicate ‘The variable x occurs in the formula α′.
Let P ⊆ Mn. The function χP , defined by
χP~a =
1 in case P~a,
0 in case ¬P~a
is called the characteristic function of P . It is unessential whether the values 0, 1
are understood as truth values or as natural numbers, or if 0, 1 are permuted in the
definition. What matters is that P is uniquely determined by χP .
A function f : Mn → M is called a n-ary operation of M. Almost everywhere f~a
will be written for f(a1, . . . , an). Since M0 = ∅, a 0-ary operation of M is of the
shape (∅, c) with c ∈ M ; it is shorter denoted by c and called a constant.
Each operation f : Mn → M is uniquely described by
graphf := (a1, . . . , an+1) ∈ Mn+1 | f(a1, . . . , an) = an+1.
This is a (n + 1)-ary relation, named the graph of f . Both f and graphf are the
same if Mn+1 is identified with Mn × M . In most situations it is however more
convenient to distinguish between f and graphf .
A generalization of M1×M2 is the direct product N =∏
i∈I Mi of a family of sets
(Mi)i∈I . Each a ∈ N is a function a = (ai)i∈I with ai ∈ Mi defined on I, a so-called
XVI Notation
choice function. The mapping a 7→ ai from N to Mi and sometimes the element
ai itself, is called the i-th projection. One also speaks of the i-th component of a.∏i∈I Mi and M I coincide whenever Mi = M for all i ∈ I. This also holds in case
I = ∅ since then∏
i∈I Mi = ∅ as well as M I = ∅.If A, B are expressions of our meta-language, A ⇔ B stands for A iff B (to mean
A if and only if B). A ⇒ B stands for if A then B, and A & B and A∨∨∨B stand for
A and B, A or B, resp. This notation does not aim at formalizing the meta-language
but serves improved organization of metatheoretic statements. We agree that ⇒ ,
⇔, . . . separate stronger than linguistic binding particles. Therefore, in
T α ⇔ α ∈ T , for all α ∈ L0 (definition page 64)
the comma should not be omitted since ‘α ∈ T for all α ∈ L0 could erroneously be
read as ‘the theory T is inconsistent’.
If s, t are terms with values in an ordered set, then s > t is throughout only
another way of writing t < s. The same remark refers also to some other relation
symbols like 6, ⊆, ⊂. Here the notion term is used to denote certain strings of a
formal language as defined in Section 2.2. s := t means that the term s is defined
by the term t, or, whenever s is a variable, the allocation of the value t to s.
When integrating formulas in the colloquial meta-language one may use certain
abbreviating notation. In the same sense as the conjunction ‘a < b and b ∈ M ’ is
occasionally shortened to a < b ∈ M , also ‘X ` α and α ≡ β’ may be abbreviated
by X ` α ≡ β (‘from the set X of formulas is deducible the formula α, and α is
equivalent to β’). This is allowed as long as the symbol ≡ does not belong to the
formal language from which the formulas α, β are taken. Actually, ≡ will never
belong to an object language in this book but will denote the logical equivalence of
formulas in a formalized language. For instance, the metatheoretical statement ’the
formula α∧α is equivalent to the formula α’ can then be shortened to α∧α ≡ α.
In a textbook on logic one cannot help to clearly distinguish the equality sign used
in the meta-language from an equality symbol in a formal language L. This will be
realized by bold-face printing of the equality symbol in L. For instance, if L is the
language of arithmetic, we shall write x+y ==== y+x instead of x+y = y+x, to express
the equivalence of the terms on the left side and the right side of the equation in
a theory formalized in L, while s = t means throughout syntactic identity of terms
s, t, letter by letter. Thus, x + y ==== y + x is a formula of L while x + y = y + x is a
(false) metatheoretical statement.
A similar distinction is made in denoting the membership relation. In the meta-
language it is denoted by the standard symbol ∈ while in the formal language for
set theory the smaller symbol ∈ is used.
241
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247
Index
∀-formula, 54
a.c., 38
∀-theory, 65
∀∃-sentence, ∀∃-theory, 148
abelian group, 38
divisible, 81
torsion-free, 91
absorption laws, 39
algebra, 34
algebraic, 38
algebras of sets, 40
almost all, 48, 163
alphabet, xv
antisymmetric, 36
arithmetical, 184
arithmetical hierarchy, 205
arithmetizable, 177, 194
Artin, 153
associative, 37
automorphism, 40
Axiom
of Extensionality, 88
of Choice, 90
of Continuity, 85
of Infinity, 90
of Power Set, 89
of Regularity, 90
of Replacement, 89
of Union, 89
axiom system
logical, 29, 95
of a theory, 65
of Peano-Dedekind, 91
axiomatizable, 81
finitely, recursively, 81
β-function, 189
basic instance, 107, 123
basic term, 44
basis Horn formula, 108
Basis Theorem, 140, 160
beginning, xv
Behmann, 98
Birkhoff Rules, 99
Boolean algebra, 39
atomless, 156
Boolean basis
for L in T , 160
for L0 in T , 140
Boolean combination, 45
Boolean function, 2
dual, self-dual, 12
linear, 8
monotonic, 13
Boolean matrix, 40
Boolean signature, 4
cardinal number, 134
cardinality, 134
of a structure, 34
chain, 37
of structures, 148
elementary, 148
of theories, 80
characteristic, 39
248 Stichwortverzeichnis
choice function, xvi
Church, 92, 171
clause, 112, 118
definite, positive, negative, 112
closed under MP, 30
closure (of a model in T ), 152
closure axioms, 200
cofinite, 28
collision-free, 55
collisions of variables, 55
commutative, 37
Compactness Theorem, 24, 82
compatible, 65
complete, 82
Completeness Theorem
Birkhoff’s, 100
Godel’s, 80
propositional, 23
Solovay’s, 223
completion, 93
inductive, 150
composition, xiv, 169
computable, 169
concatenation, xv
arithmetical, 174
congruence relation, 41
connective, 3
connex, 36
consequence relation, 16, 17
finitary, 16
global,local, 63
predicate logical, 51
propositional, 15
consistency extension, 220
consistent, 75, 123
constant, xv
constant-expansion, 76
constant-quantification, 76
continuity schema, 86
Continuum Hypothesis, 135
contradiction, 14
contraposition, 17
converse implication, 3
coprime, 185
course-of-values recursion, 174
cross product, xiv
Cut rule, 20
∆-elementary class, 139
∆0-formula, 185
δ-function, 170
Davis, 199
decidable, 81
(recursively) decidable, 169
Deduction Theorem, 16, 31
deductively closed, 64
definable, 53
explicitly, 53, 69
implicitly, 69
in a structure, 53
in theories, 211
with parameters, 85
degree of a polynomial, 82
DeJongh, 225
Derivability Conditions, 210
derivable, 18, 19, 29
diagram, 132
elementary, 133
universal, 149
direct power, 42
disjunction, 2
distributivity laws, 39
domain, xiv, 34
domain of magnitude, 38
Dzhaparidze, 227
∃-formula, 54
∃-closed, 155
∃-formula
Stichwortverzeichnis 249
simple, 158
Ehrenfeucht’s Game, 142
elementary class, 139
elementary equivalent, 55
elementary type, 139
embedding, 40
elementary, 136
enumerable
effectively or recursively, 92, 174
equation, 45
Diophantine, 184, 198
equipotent, 87
equivalence, 3
logical or semantic, 9
equivalence relation, 37
equivalent, 50
in (or modulo) T , 66
in a structure, 59
logically or semantically, 50
Euclid’s lemma, 193
exclusive or, 2
existentially closed, 148, 155
expansion, 36, 62
explicit definition, 68
extension, 36, 64
conservative, 52, 67
definitorial, 68
elementary, 133
finite, 65
immediate, 153
of a language, 62
of a theory, 65
transcendental, 138
f -closed, 35
factor structure, 41
falsum, 4
family (of sets), xiv
Fermat’s Conjecture, 199
Fibonacci, 174
fictional argument, 8
field, 38
algebraically closed, 38
of algebraic numbers, 134
of characteristic 0, 39
of characteristic p, 39
ordered, 39
real closed, 153
filter, 27
proper, 28
finitary, 17
finite model property, 97
Finiteness Theorem, 21, 73, 81
Fixed-Point Lemma, 194
formula, 4, 45
Boolean, 4
closed, 47
defining, 67
dual, 12
first-order, 45
prenex, 60
representable, 184
universal, 54
formula algebra, 34
formula induction, 46
Four-Colour Theorem, 25
Frege, 60
function, xiv
characteristic, xv
partial, xiv
primitive recursive, 169
recursive (= µ-recursive), 169
function term, 44
functionally complete, 12
Godel number
of a proof, 177
of a sequence, 173
of a string, 176
Godel term, 191
250 Stichwortverzeichnis
gap, 37
generalization, 51
anterior, posterior, 62
generally valid, 50
(finitely) generated, 36
Gentzen calculus, 18
goal clause, 123
Goodstein, 219
graph, 37
k-colourable, 25
of an operation, xv
planar, 25
simple, 25
group, 38
ordered, 38
groupoid, 38
H-resolution, 116
Harrington, 219
Henkin set, 77
Herbrand model, 108
minimal, 110
Herbrand structure, 108
Hilbert calculus, 29, 95
homomorphism, 40
natural, 41
strict, 40
Horn clause, 116
Horn formula, 108
positive, negative, 109
universal, 109
Horn resolution, 117
Horn sentence, 109
Horn theory, 109
non-trivial, 110
universal, 110
hyper-exponentiation, 186
I-tuple, xiv
idempotent, 37
identity, 99
identity-free, 80
image, xiv
immediate successor, 37
inclusion, xiv
Incompleteness Theorem
First, 194
Second, 217
inconsistent, 75
independent, 65
independent (in T ), 75
individual variables, 43
<-induction, 86
induction
on ϕ, 6, 46
on t, 44
∆0-induction, 206
induction axiom, 84
induction schema, 83
induction step, 83
infimum, 39
infinitesimal, 86
informally, 63
initial segment, 37
instance, 107, 123
integral domain, 38
(relatively) interpretable, 200
interpretation, 49
intersection, xiv
Invariance Theorem, 54
invertible, 37
irreflexive, 36
isomorphism, 40
partial, 138
ι-term, 68
Jeroslow, 225
κ-categorical, 137
Konig’s Lemma, 26
Stichwortverzeichnis 251
kernel, 60
Kleene, 169
Kreisel, 84, 225
Kripke semantics, 221
L-formula, 46
L-model, 49
Lob’s Axiom, 221
Lob’s Theorem, 218
L-structure (= L-structure), 35
language
first-order (= elementary), 43
of equations, 99
second-order, 102
lattice, 39
distributive, 39
of sets, 39
leaf, 113
legitimate, 68
Lindenbaum, 23
literal, 10, 45
logic program, 122
logical matrix, 40
logically valid, 14, 50
µ-operation, 169
bounded, 172
mapping (function), xiv
bijektive, xiv
identical, xiv
injective, surjective, xiv
Matiyasevich, 198
maximal element, 37
maximally consistent, 22
McAloon, 84
metainduction, xi
metatheory, xi
minimal model, 117
model
free, 110
of a theory, 64
predicate logical, 49
propositional, 7
model companion, 157
model compatible, 150
model complete, 151
model completion, 155
model interpretable, 202
modus ponens, 15, 29
monomorphism, 40
monotonicity rule, 18
Mostowski, 168
mutually satisfiable, 69
mutually valid, 61
n-tuple, xv
negation, 2
neighbor, 25
non-standard analysis, 85
non-standard model, 83
non-standard number, 84
normal form
canonical, 12
disjunctive, conjunctive, 10
prenex, 60
Skolem, 70
ω-consistent, 195
ω-rule, 226
ω-incomplete, 196
ω-term, 194
operation, xv
essentially n-ary, 8
order, 37
dense, 137
discrete, 142
linear, partial, 37
ordered pair, 89
Π1-formula, 184
252 Stichwortverzeichnis
pair set, 89
pairing function, 172
parameter definable, 85
Paris, 219
partial order
irreflexive, reflexive, 37
particularization
anterior, posterior, 62
Peano Arithmetic, 83
persistent, 147
Polish (prefix) notation, 6
power set, xiv
predecessor function, 83
predicate, xv
arithmetical, 184
Diophantine, 185
(primitive) recursive, 169
recursively enumerable, 175
preference order, 229
prefix, 45
premises, 18
Presburger, 158
p.r. (= primitive recursive), 169
prime field, 39
prime formula, 4, 45
prime model, 133
prime term, 44
primitive recursive, 169
Principle of Bivalence, 2
Principle of Extentionality, 2
product
direct, xv, 41
of mappings, xiv
reduced, 163
programming language, 103
projection, xvi
projection function, 169
PROLOG, 122
proof (formal), 29, 95
propositional variables, 3
provable, 18, 29
provably recursive, 212
Putnam, 199
quantification
bounded, 171, 185
quantifier, 33
quantifier compression, 188
quantifier elimination, 157
quantifier rank, 46
quantifier-free, 45
quasi-identity, quasi-variety, 100
query, 122
quotient field, 146
r.e. (recursively enumerable), 174
Rabin, 200
range, xiv
rank (of a formula), 6, 46
reduced formula, 67, 68
reduct, 36, 62
reductio ad absurdum, 19
reflection principle, 220
reflexive, 36
refutable, 65
relation, xiv
P− relativised, 200
renaming, 119
bound, free, 60
free, bound, 60
Replacement Theorem, 10, 59
Representability
of a function, 187
of a predicate, 184
Representability Theorem, 190
representantive independent, 41
resolution calculus, 113
resolution rule, 113
resolution shell, 113
Stichwortverzeichnis 253
Resolution Theorem, 115
resolution tree, 113
resolvent, 113
restriction, 35
ring
ordered, 39
Abraham Robinson, 85
Julia Robinson, 199
Rogers, 225
rule, 18, 72
basic, 18, 72
derivable (provable), 18
Gentzen-style, 20
Hilbert-style, 95
of Horn resolution, 116
sound, 21, 72
rule induction, 21, 73
Σ1-completeness, 186
provable, 215
Σ1-formula, 185
special, 207
S-invariant, 145
Sambin, 225
satisfiability relation, 14, 49
satisfiable, 14, 50, 65, 112
scope, 46
semigroup, 38
ordered, 38
regular, 38
semilattice, 39
semiring, 39
ordered, 39
sentence, 47
separator, 121
sequence, xiv
sequent, 18
initial, 18
set, xiv
countable, uncountable, 87
densely ordered, 137
discretely ordered, 142
finite, 87
ordered, 37
well-ordered, 37
Sheffer’s stroke, 2
signature
algebraic, 45
extralogical, 34
logical, 4
signum function, 170
singleton, xiv
Skolem, 69
Skolem’s Paradox, 91
SLD-resolution, 126
Solovay, 209
solution, 123
soundness, 21, 73
Stone’s Representation Theorem, 40
string, xv
structure, 34
algebraic, 34
relational, 34
subformula, 6, 46
substitution, 47
global, 47
identical, 47
propositional, 15
simple, simultaneous, 47
substitution invariance, 99
Substitution Theorem, 56
substring, xv
substructure, 36
(finitely) generated, 36
elementary, 133
substructure complete, 160
subterm, 44
subtheory, 64
successor function, 82
254 Stichwortverzeichnis
supremum, 39
symbol, xv
symmetric, 36
T -model, 64
Tarski, 17, 131, 168
tautology, 14, 50
Tennenbaum, 84
term, 44
term algebra, 44
term equivalent, 12
term function, 53
term induction, 44
term-model, 106
tertium non datur, 14
Theorem, 64
Cantor’s, 87
Cantor-Bernstein, 134
Herbrand’s, 108
Lowenheim-Skolem, 87
Lagrange’s, 198
Lindenbaum’s, 22
Lindstrom’s, 101
Los’s, 164
Morley’s, 138
Rosser’s, 195
Steinitz’s, 153
Trachtenbrot’s, 98
theory
(finitely) axiomatizable, 81
complete, 137
consistent (satisfiable), 65
countable, 87
decidable, 93, 177
elementary (or first-order), 64
inductive, 148
undecidable, 93
universal, 65
transcendental, 38
transitive, 36, 229
truth, true, 196
truth-function, 2
truth-table, 2
truth-values, 2
Turing machine, 171
U -resolution, 126
U -resolvent, 125
UH-resolution, 126
ultrafilter, 28
non-trivial, 28
Ultrafilter Theorem, 28
ultrapower, 164
ultraproduct, 164
undecidable, 81, 93
strongly, hereditarily, 197
unifiable, 119
unification algorithm, 119
unifier, 119
generic, 119
union, xiv
unit element, 38
universal part, 145
universe, 89
urelement, 88
valuation, 7, 49
variable, 43
free, bound, 46
variety, 99
Vaught, 139
verum, 4
Visser, 224
word, xv
word-semigroup, 38
Z-group, 159
Zorn’s Lemma, 37
255
List of symbols
N, Z, Q, R xiv
∪,∩, \ ,⊆,⊂ xiv
∅, PM xiv⋃F,
⋂F xiv
(a, b), M ×N xiv
dom f, ran f xiv
x 7→ t(x) xiv
f : M → N xiv
idM xiv
NM xiv
P~a, ¬P~a xv
χP xv
graphf xv∏i∈I Mi xv
⇔,⇒, &,∨∨∨ xvi
:= xvi
Bn 2
∧ , ∨ ,¬ 3
F, PV 4
→ ,↔, >, ⊥ 4
Sf α 6
wα 7
Fn, α(n) 7
α ≡ β 9
DNF, CNF 10
w α, α 14
X α, X Y 15
C+, C− 22
MP 29
|∼ 29
rA, fA, cA 35
A ⊆ B 36
2 39
A ' B 40
a/≈ 41∏i∈I Ai, AI 42
==== , = 43
∀, ∃ 43
T (= TL) 44
var ξ 44
6=6=6=6= 45
L, L∈, L==== 45
rk α, qr α 46
free ϕ, bnd ϕ 46
L0,L1, . . . 47
ϕ(x1, . . . , xn) 47
ϕ(~x), t(~x) 47
f~t , r~t 47
ϕ tx , ϕ
~t~x, ϕ~x(~t ) 47
ι 47
M = (A, w) 49
rM, fM, cM 49
tA,w, tM, ~tM 49
Max, M~a
~x 49
M ϕ 49
A ϕ [w] 49
ϕ, α ≡ β 50
A ϕ, A X 51
X ϕ, 51
ϕ∀, X∀ 51
TG, T ====G 51
A ϕ [~a] 53
(A,~a) 53
tA(~a), tA , ϕA 53
∃n, ∃=n 54
>, ⊥ 54
A ≡ B 55
Mσ 56
∃! 57
≡A, ≡K 59
PNF 60
(teilt) 63
∀
63
T, Md T 64
Taut 65
T + α, T + S 65
ThA, ThK 65
K α 66
≡T 66
SNF 70
` 72
mon, fin 73
Lc, LC 76
`T , X `T α 80
ACF 82
N , S, Pd 83
PA, IS, IA 83
n (= Sn0) 83
M ∼ N 87
ZFC, ZF 88
256 List of symbols
z ∈ x | ϕ 89
ω 90
|∼ , MP, MQ 95
Λ, Λ1− Λ10 95
Tautfin 97
`B 99
LII , L∼O 102
F , FX 106
Lk, Vark, Tk 107
FkX 107
GI(X) 107
L∞, T∞, F∞ 108
CU , CT 110
112
K H 112
K, λ; λ, K 113
RR, `RR, Rh 113
HR, `HR116
P, N 116
VP, wP, eP
117
:− , ?− 122
GI(K) 123
UR, `UR125
UHR, `UHR125
UωR, UωHR 125
AA, BA 132
DA 132
DelA 133
A 4 B 133
|M| 134
ℵ0, ℵ1, CH 135
DO, DO00, . . . 137
L, R 138
ACFp 138
〈X〉, ≡X 139
SO, SO00, . . . 142
Γk(A,B) 142
A ∼k B 142
A ≡k B 143
T∀ 145
A ⊆ec B 149
D∀A 149
RCF 153
ZG, ZGE 159
≈F 163∏Fi∈I Ai 163
h[g1, . . . , gm] 169
P [g1, . . . , gm] 169
Oc, Op, Oµ 169
Inν 169
·−, δ, sg 170
prim 171
µkP (~a, k) 172
µk6m[· · · ] 172
℘ 172
lcmaν|ν6n 172
〈a1, . . . , an〉 173
`, ∗ 173
(((a)))k, (((a)))last 173
Oq 174
f = Op(g, h) 174
Lar 176
SL, ξ, ϕ 176
¬, ∧ , → 178
bewT , bwbT 178
==== , ∀, S, . . . 179
Lprim 179
[m]ki 180
Q, N 182
∆0 185
Σ1, Π1, ∆1 185
⊥ (coprime) 185
I∆0 186
rem(a : b) 189
β, beta 189
pϕq, ptq, pΦq 191
bewT , bwbT 191
cf, α~x(~a) 192
sbx, sb~x, sb∅ 193
α~x(~a) 193
prov 196
αP, XP 200
T ∆1 , B∆ 200
ZFCfin 202
Σn, Πn, ∆n 205
(x) 210
(x), α, 3α 210
ConT 210
D0−D3 210
d0, . . . 210
∂, d1, . . . 210
D4∗ 211
[ϕ] 214
PA⊥ 218
D4, D4 218
T n, T ω, nα 220
, n, 3 221
G,`G, G,≡G 221
P H 221
Gn, GS 224
1 , 31 , GD 226
Rf T 228
Gi, Gj 229