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

Literature

[Ba] J. Barwise (Editor), Handbook of Mathematical Logic, North-Holland 1977.

[BD] A. Berarducci, P. D’Aquino, ∆0-complexity of y =∏

i6n F (i), Ann. PureAppl. Logic 75 (1995), 49-56.

[Be1] L. Beklemishev, On the classification of propositional provability logics, Izvestiya35 (1990), 247-275.

[Be2] Iterated local reflection versus iterated consistency, Ann. Pure Appl. Logic75 (1995), 25-48.

[Be3] Bimodal Logics for Extensions of Arithmetical Theories, Journ. Symb. Logic61 (1996), 91-124.

[Be4] Parameter free Induction and Reflection, in Computational Logic and ProofTheory, LN Comp. Science 1289, Springer 1997.

[Ben] M. Ben-Ari, Mathematical logic for computer science, 2nd edition, Springer 2001.

[Bi] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Phil. Soc.50 (1935), 433 - 455.

[BJ] G. Boolos, R. Jeffrey, Computability and Logic, 3rd ed. Cambridge Univ. Press1992.

[BGG] E. Borger, E. Gradel, J. Gurevich, The classical decision problem, Springer1997.

[BM] J. Bell, M. Machover, A Course in Mathematical Logic, North-Holland 1977.

[Boo] G. Boolos, The Logic of Provability, Cambridge Univ. Press 1993.

[BP] P. Benacerraf, H. Putnam (Editor), Philosophy of Mathematics, SelectedReadings, 2nd ed. Cambridge Univ. Press 1993.

[Bu] S.R. Buss (editor), Handbook of Proof Theory, Elsevier 1998.

[Bue] S. Buechler, Essential Stability Theory, Springer 1996.

242 Literature

[Ca] G. Cantor, Gesammelte Abhandlungen, Springer 1980.

[CZ] A Chagrov, M. Zakharyashev, Modal Logic, Claredon Press 1997.

[CK] C.C. Chang, H.J. Keisler, Model Theory, 3rd ed. North-Holland 1990.

[Ch] A. Church, A Note on the Entscheidungsproblem, Jour. Symb. Logic 1 (1936),40-41.

[Da] D. van Dalen, Logic and Structure, 2nd ed. Springer 1983.

[Dav] M. Davis (editor), The Undecidable, Raven New York 1965.

[De] R. Dedekind, Was sind und was sollen die Zahlen?, (Braunschweig 1888), Vieweg1969.

[Do] K. Doets From Logic to Logical Programming, MIT-Press 1994.

[EFT] H. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic, Springer 1996.

[En] H. Enderton, A Mathematical Introduction to Logic, Acad. Press 1972. 2nd edi-tion 2001.

[Fe] W. Felscher, Lectures on mathematical logic, Gordon and Breach 2000.

[Fr] G. Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprachedes reinen Denkens (Halle 1879), Olms 1971.

[FS] H. Friedman, M. Sheard, Elementary decent Recursion and Proof Theory, Arch.Pure & Appl. Logic 71 (1995), 1-47.

[Ga] D. Gabbay, Decidability results in non-classical logic III, Israel Jour. Math 10(1971)), 135-146.

[Ge] G. Gentzen, Untersuchungen uber das logische Schlie”sen, Math. Z. 39 (1934/35),176-210 and 405-431.

[GJ] M. Garey, D. Johnson, Computers and Intractability, A Guide to the Theoryof NP-Completeness, Freeman 1979.

[Go1] K. Godel, Die Vollstandigkeit der Axiome des logischen Funktionenkalkuls,Monatshefte Math. u. Physik 37 (1930), 349-360, or Collected Works I.

[Go2] , Uber formal unentscheidbare Satze der Principia Mathematica und ver-wandter Systeme I, Monatshefte fur Mathematik und Physik 38 (1931).

[Go3] , Collected Works, Oxford Univ. Press, Vol. I 1986, Vol. II 1990, Vol. III1995.

Literature 243

[Gor] S.N. Goryachev, On the Interpretability of some Extensions of Arithmetic, Math.Notes 40 (1986), 561-572.

[Gr] G. Gratzer, Universal Algebra, 2nd ed. Springer 1979.

[HA] D. Hilbert, W. Ackermann, Grundzuge der theoretischen Logik, Springer 1928.

[HB] D. Hilbert, P. Bernays, Grundlagen der Mathematik, Band I Springer 1934,Band II Springer 1939.

[He] L. Henkin, The Completeness of the First-order Functional Calculus, Journ.Symb. Logic 14 (1949), 159-166.

[Hej] J. Heijenoort, From Frege to Godel, Harvard Univ. Press 1967.

[HeR] B. Herrmann, W. Rautenberg, Finite Replacement and Hilbert-style Axiom-atizability, Zeitsch. Math. Log. Grundl. Math. 38 (1982), 327-344.

[Her] J. Herbrand, Recherches sur la theorie de la demonstration, in [Hej].

[Ho] W. Hodges, Model Theory, Cambridge Univ. Press 1993.

[HP] P. Hajek, P. Pudlak, Metamathematics of First-order Arithmetic, Springer1993.

[HR] H. Herre, W. Rautenberg, Das Basistheorem und einige Anwendungen in derModelltheorie, Wiss. Zeitsch. Humboldt-Univ. 19 (1970), 575-577.

[Id] P. Idziak, A Characterization of Finitely Decidable Congruence Modular Vari-eties, Trans. Am. Math. Soc. 349 (1997), 903-934.

[Ig] K. Ignatiev, On Strong Provability Predicates, Journ. Symb. Logic 58 (1993),249-290.

[JK] R. Jensen, C. Karp, Primitive recursive Set Theory, in Axiomatic Set Theory,AMS 1971, 143-167.

[Ka] R. Kaye, Models of Peano Arithmetic, Clarendon Press 1991.

[Ke] H.J. Keisler, Logic with the Quantifier “there exist uncountably many”, Ann.Pure Appl. Logic 1 (1970), 1-93.

[Kl1] S. Kleene, Introduction to Metamathematics, North-Holland 1952, 2nd edition,Wolters-Noordhoff 1988.

[Kl2] , Mathematical Logic, Wiley & Sons 1968.

[KR] I. Korec, W. Rautenberg, Model Interpretability into Trees and Applications,Arch. Math. Logik 17 (1976), 97-104.

244 Literature

[KK] G. Kreisel, J. Krivine, Elements of Mathematical Logic, North-Holland 1971.

[Kr] J. Krajicek, Bounded Arithmetic, Propositional Logic, and Complexity Theory,Cambridge Univ. Press 1995.

[Ku] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland1980.

[Li] P. Lindstrom, On Extensions of Elementary Logic, Theoria 35 (1969), 1-11.

[Ll] J.W. Lloyd, Foundations of Logic Programming, Springer 1987.

[Lo] M. Lob, Solution of a Problem of Leon Henkin, Journ. Symb. Logic 20 (1955),115-118.

[Ma] A.I. Malcev, The Metamathematics of Algebraic Systems, North-Holland 1971.

[Mal] J. Malitz, Introduction to Mathematical Logic, Springer 1979.

[Mat] Y. Matiyasevich, Hilbert’s Tenth Problem, MIT Press 1993.

[Me] E. Mendelson, Introduction to Mathematical Logic, Van Nostrand 1979.

[Mo] D. Monk, Mathematical Logic, Springer 1976.

[ML] G. Muller, W. Lenski (editors) The Ω-Bibliography of Mathematical Logic,Springer 1987.

[MS] A. Macintyre, H. Simmons, Godel’s diagonalization technique and related prop-erties of theories, Colloq. Math. 28, 1973.

[MW] A. Macintyre, A. Wilkie, On the Decidability of the Real Exponential Field,Preprint 1995.

[MV] R. McKenzie, M. Valeriote, The Structure of Decidable Locally Finite Vari-eties, Progress in Math. 79, Birkhauser 1989.

[Po] W. Pohlers, Proof Theory – An Introduction, Springer Lecture Notes 1407 (1989)

[Pr] M. Presburger, Uber die Vollstandigkeit eines gewisses Systems der Arithmetikganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Congr.Math. Pays Slaves (1) (1929), 92-101.

[Pz] B. Poizat, A course in model theory. An introduction to contemporary mathemat-ical logic, Springer 2000.

[RS] H. Rasiowa, R. Sikorski, The Mathematics of Metamathematics, Polish Scien-tific Publ. 1968.

[Ra1] W. Rautenberg, Klassische und Nichtklassische Aussagenlogik, Vieweg 1979.

Literature 245

[Ra2] , Elementare Grundlagen der Analysis, BI Verlag 1993.

[Ra2] , Einfuhrung in die Mathematische Logic, Vieweg 1995, 2nd edition 2002.

[Ro1] A. Robinson, Introduction to Model Theory and to the Metamathematics of Al-gebra, North-Holland 1974.

[Ro2] A. Robinson, Non-Standard Analysis, North-Holland 1970.

[Ro] J. Robinson, A Machine-oriented Logic based on the Resolution Principle, Journ.Ass. Comp. Machinery 12 (1965), 23-41.

[Rog] H. Rogers, Theory of Recursive Functions and Effective Computability, 2nd ed.MIT Press 1988.

[Ros] J.B. Rosser, Extensions of some Theorems of Godel and Church, Journ. Symb.Logic 1 (1936), 87-91.

[Ry] C. Ryll-Nardzewki, The role of induction in elemenary arithmetic, Fund. Math.39 (1952), 87-91.

[RZ] W. Rautenberg, M. Ziegler Recursive Inseparability in Graph Theory, NoticesAm. Math. Soc. 22 (1975).

[Sa] G. Sacks, Saturated Model Theory, Benjamin Reading 1972.

[Sam] G. Sambin, An Effective Fixed Point Theorem in Intuitionistic DiagonalizableAlgebras, Studia Logica 35 (1976), 345-361.

[Sh] S. Shelah, Classification Theory and the Number of Nonisomorphic Models,North-Holland 1978.

[Se] A. Selman, Completeness of Calculi for Axiomatically Defined Classes, AlgebraUniversalis 2 (1972), 20-32.

[Shoe] J. Shoenfield, Mathematical Logic, Addison-Wesley 1967.

[Si] W. Sieg, Herbrand Analyses, Arch. Math. Logic 30 (1991), 409 -441.

[Sm] C. Smorynski, Self-reference and Modal Logic, Springer 1984.

[Sm1] R. Smullyan, Theory of Formal Systems, Princeton Univ. Press 1961.

[So] R. Solovay, Provability Interpretation of Modal Logic, Israel Jour. Math. 25(1976), 287-304.

[Sz] W. Szmielew, Elementary properties of abelian groups, Fund. Math. 41 (1955),287-304.

246 Literature

[Ta1] A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philo-sophica 1 (1936), in [Ta3].

[Ta2] , A Decision Method for Elementary Algebra and Geometry, Berkeley 1948,1951, Paris 1967.

[Ta3] , Logic, Semantics and Metamathematics, Clarendon Press 1956.

[Ta4] , Introduction to Logic and and to the Methodology of the Deductive Sci-ences, Oxford Univerity Press 1994 (first edition in German, Springer 1937).

[Tak] G. Takeuti, Proof Theory, North Holland, 1975.

[Tan] K. Tanaka, Reverse Mathematics and Subsystems of Second Order Arithmetic,AMS Sugaku Expositions 5, 1992.

[TMR] A. Tarski, A. Mostowski, R.M. Robinson, Undecidable Theories, 2nd ed.North-Holland 1971.

[TS] A. Troelstra, H. Schwichtenberg, Basic Proof Theory, Cambridge Univ.Press 1996.

[Tu] A. Turing, On Computable Numbers, with an Application to the Entschei-dungsproblem, Proceedings of the London Mathematical Society 43 (1937), reprintin [Dav].

[Vi] A. Visser, An Overview of Interpretability Logic, Advances in Modal Logic 1996,Lecture Notes CSLI (ed. M. Kracht et al.), Stanford 1998.

[Wae] B. van der Waerden, Algebra I, Springer 1964.

[WR] A. Whitehead, B. Russell, Principia Mathematica, Cambridge Univ. Press1910.

[Wi] A. Wilkie, Model Completeness Results for Expansions of the Real Field II: theExponential Function, Journ. Am. Math. Soc. 9 (1996), 1051-1094.

[WP] A. Wilkie, J. Paris, On the Scheme of Induction for Bounded Arithmetic For-mulas, Ann. of Pure & Appl. Logic 35 (1987), 261-302.

[Zi] M. Ziegler, Model Theory of Moduls, Ann. Pure Appl. Logic 26 (1984), 149-213.

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