Editorial BoardS. Axler

K.A. Ribet

For other titles in this series, go to http://www.springer.com/series/136

Graduate Texts in Mathematics 53

for Mathematicians A Course in Mathematical Logic

Yu. I. Manin

Second Edition

by Neal Koblitz Chapters I-VIII translated from the Russian

With new chapters by Boris Zilber and Yuri I. Manin

Max-Planck Institut für Mathematik

Germany

Oxford OX1 3LB manin@mpim-bonn.mpg.de United Kingdom

Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98195 USA

Editorial Board: S. Axler K. A. RibetMathematics Department Mathematics Department San Francisco State University San Francisco, CA 94132 USA USAaxler@sfsu.edu

ISSN 0072-5285 ISBN 978-1-4419-0614-4 e-ISBN 978-1-4419-0615-1

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© First edition 1977 by Springer Verlag, New York, Inc.

Yu. I. Manin

© Second edition 2010 by Yu. I. Manin

To Nikita, Fedor and Mitya, with love

Preface to the Second Edition

1. The first edition of this book was published in 1977. The text has been wellreceived and is still used, although it has been out of print for some time.

In the intervening three decades, a lot of interesting things have happenedto mathematical logic:

(i) Model theory has shown that insights acquired in the study of formallanguages could be used fruitfully in solving old problems of conventionalmathematics.

(ii) Mathematics has been and is moving with growing acceleration fromthe set-theoretic language of structures to the language and intuition of(higher) categories, leaving behind old concerns about infinities: a newview of foundations is now emerging.

(iii) Computer science, a no-nonsense child of the abstract computabilitytheory, has been creatively dealing with old challenges and providing newones, such as the P/NP problem.

Planning additional chapters for this second edition, I have decided to focuson model theory, the conspicuous absence of which in the first edition was notedin several reviews, and the theory of computation, including its categorical andquantum aspects.

The whole Part IV: Model Theory, is new. I am very grateful toBoris I. Zilber, who kindly agreed to write it. It may be read directly afterChapter II.

The contents of the first edition are basically reproduced here asChapters I–VIII. Section IV.7, on the cardinality of the continuum, iscompleted by Section IV.7.3, discussing H. Woodin’s discovery.

The new Chapter IX: Constructive Universe and Computation, was writtenespecially for this edition, and I tried to demonstrate in it some basics of cate-gorical thinking in the context of mathematical logic. More detailed commentsfollow.

I am grateful to Ronald Brown and Noson Yanofsky, who read prelimi-nary versions of new material and contributed much appreciated criticism andsuggestions.

2. Model theory grew from the same roots as other branches of logic: prooftheory, set theory, and recursion theory. From the start, it focused on languageand formalism. But the attention to the foundations of mathematics in model

viii Preface to the Second Edition

theory crystallized in an attempt to understand, classify, and study models oftheories of real-life mathematics.

One of the first achievements of model theory was a sequence of localtheorems of algebra proved by A. Maltsev in the late 1930s. They were based onthe compactness theorem established by him for this purpose. The compactnesstheorem in many of its disguises remained a key model-theoretic instrumentuntil the end of the 1950s. We follow these developments in the first two sec-tions of Chapter X, which culminate with a general discussion of nonstandardanalysis discovered by A. Robinson. The third section introduces basic toolsand concepts of the model theory of the 1960s: types, saturated models, andmodern techniques based on these.

We try to illustrate every new model-theoretic result with an application in“real” mathematics. In Section 4 we discuss an algebro-geometric theorem firstproved by J. Ax model-theoretically and re-proved by G. Shimura and A. Borel.Moreover, we explain an application of the Tarski–Seidenberg quantifier elim-ination for R due to L. Hormander. A real gem of model-theoretic techniquesof the 1980s is the calculation by J. Denef of the Poincare series countingp-adic points on a variety based on A. Macintyre’s quantifier eliminationtheorem for Qp.

In the last two sections we present a survey of classification theory, whichstarted with M. Morley’s analysis of theories categorical in uncountable powersin 1964, and was later expanded by S. Shelah and others to a scale that no onecould have envisaged.

The striking feature of these developments is the depth of the very abstract“pure” model theory underlying the classification, in combination with thediversity of mathematical theories affected by it, from algebraic andDiophantine geometry to real analysis and transcendental number theory.

3. The formal languages with which we work in the first, and in most ofthe second, edition of this book are exclusively linear in the following sense.Having chosen an alphabet consisting of letters, we proceed to define classesof well-formed expressions in this alphabet that are some finite sequences ofletters. At the next level, there appear well-formed sequences of words, such asdeductions and descriptions. Church’s λ-calculus furnishes a good example ofstrictures imposed by linearity.

Nonlinear languages have existed for centuries. Geometers andcomposers could not perform without using the languages of drawings, resp.musical scores; when alchemy became chemistry, it also evolved its owntwo-dimensional language. For a logician, the basic problem about nonlinearlanguages is the difficulty of their formalization.

This problem is addressed nowadays by relegating nonlinear languages ofcontemporary mathematics to the realm of more conventional mathematicalobjects, and then formally describing such languages as one would describe anyother structure, that is, linearly.

Preface to the Second Edition ix

Such a strategy probably cannot be avoided. But one must be keenly awarethat some basic mathematical structures are “linguistic” at their core. Recog-nition or otherwise of this fact influences the problems that are chosen, thequestions that are asked, and the answers that are appreciated.

It would be difficult to dispute nowadays that category theory as a languageis replacing set theory in its traditional role as the language of mathematics.Basic expressions of this language, commutative diagrams, are one-dimensional,but nonlinear: they are certain decorated graphs, whose topology is that of1-dimensional triangulated spaces.

When one iterates the philosophy of category theory, replacing sets ofmorphisms by objects of a category of the next level, commutative diagramsbecome two-dimensional simplicial sets (or cell complexes), and so on. Arguably,in this way the whole of homotopy topology now develops into the language ofcontemporary mathematics, transcending its former role as an important andactive, but reasonably narrow research domain. Much remains to be recognizedand said about this emerging trend in foundations of mathematics.

The first part of Chapter IX in this edition is a very brief and tentativeintroduction to this way of thinking, oriented primarily to some reshuffling ofclassical computability theory, as was explained in the Part II of the first edition.

4. The second part of the new Chapter IX is dedicated to some theoreticalproblems of classical and quantum computing. It introduces the P/NP problem,classical and quantum Boolean circuits, and presents several celebrated resultsof this early stage of theoretical quantum computing, such as Shor’s factoringand Grover’s search algorithms.

The main reason to include these topics is my conviction that at least sometheoretical achievements of modern computer science must constitute an organicpart of contemporary mathematical logic.

Already in the first edition, the manuscript for which was completed inSeptember 1974, “quantum logic” was discussed at some length; cf.Section II.12.

A Russian version of the Part II of first edition was published as a sepa-rate book, Computable and Uncomputable, by “Soviet Radio” in 1980. For thisRussian publication, I had written a new introduction, in which, in particular,I suggested that quantum computers could be potentially much more powerfulthan classical ones, if one could use the exponential growth of a quantum phasespace as a function of the number of degrees of freedom of the classical system.

When a mathematical implementation of this idea, massive quantumparallelism, made possible by quantum entanglement, gradually matured, Igave a talk at a Bourbaki seminar in June 1999, explaining the basic ideas andresults.

Chapter IX is a revised and expanded version of this talk.5. Finally, a few words about the last digression in Chapter II, “Truth as

Value and Duty: Lessons of Mathematics.”

x Preface to the Second Edition

“Mathematical truth” was the central concept of the first part of the book,“Provability.” Writing this part, I felt that if I did not compensate somehow thearidity and sheer technicality of the analysis of formal languages, I would not beable to convince people–the readers that I imagined, working mathematicianslike me—that it is worth studying at all. The literary device I used to strugglewith this feeling of helplessness was this: from time to time I allowed myself freeassociations, and wrote the outcome in a series of six digressions, with whichthe first two Chapters were interspersed.

By the end of the second chapter, I realized that I was finally on the fertilesoil of “real mathematics,” and the need for digressions faded away.

Nevertheless, the whole of Part I was left without proper summary.Its role is now played by the “Last Digression,” published here for the first

time. It is a slightly revised text of the talk prepared for a Balzan FoundationInternational Symposium on “Truth in the Humanities, Science and Religion”(Lugano, 2008), where I was the only mathematician speaker among philoso-phers, historians, lawyers, theologians, and physicists. I was confronted with thetask to explain to a distinguished “general audience” what is so different aboutmathematical truth, and what light the usage of this word in mathematics canthrow on its meaning in totally foreign environments.

The main challenge was this: avoid sounding ponderous.

Yu. Manin, Bonn December 31, 2008

Preface to the First Edition

1. This book is above all addressed to mathematicians. It is intended to be atextbook of mathematical logic on a sophisticated level, presenting the readerwith several of the most significant discoveries of the last ten or fifteen years.These include the independence of the continuum hypothesis, the Diophantinenature of enumerable sets, and the impossibility of finding an algorithmic solu-tion for one or two old problems.

All the necessary preliminary material, including predicate logic and thefundamentals of recursive function theory, is presented systematically and withcomplete proofs. We assume only that the reader is familiar with “naive” set-theoretic arguments.

In this book mathematical logic is presented both as a part of mathematicsand as the result of its self-perception. Thus, the substance of the book consistsof difficult proofs of subtle theorems, and the spirit of the book consists ofattempts to explain what these theorems say about the mathematical way ofthought.

Foundational problems are for the most part passed over in silence. Mostlikely, logic is capable of justifying mathematics to no greater extent thanbiology is capable of justifying life.

2. The first two chapters are devoted to predicate logic. The presentationhere is fairly standard, except that semantics occupies a very dominant position,truth is introduced before deducibility, and models of speech in formal languagesprecede the systematic study of syntax.

The material in the last four sections of Chapter II is not completelytraditional. In the first place, we use Smullyan’s method to prove Tarski’s the-orem on the undefinability of truth in arithmetic, long before the introductionof recursive functions. Later, in the seventh chapter, one of the proofs of theincompleteness theorem is based on Tarski’s theorem. In the second place, alarge section is devoted to the logic of quantum mechanics and to a proof ofvon Neumann’s theorem on the absence of “hidden variables” in the quantum-mechanical picture of the world.

The first two chapters together may be considered as a short course in logicapart from the rest of the book. Since the predicate logic has received the widestdissemination outside the realm of professional mathematics, the author has notresisted the temptation to pursue certain aspects of its relation to linguistics,psychology, and common sense. This is all discussed in a series of digressions,which, unfortunately, too often end up trying to explain “the exact meaning

xii Preface to the First Edition

of a proverb” (E. Baratynsky).1 This series of digressions ends with the secondchapter.

The third and fourth chapters are optional. They are devoted to completeproofs of the theorems of Godel and Cohen on the independence of the contin-uum hypothesis. Cohen forcing is presented in terms of Boolean-valued models;Godel’s constructible sets are introduced as a subclass of von Neumann’suniverse. The number of omitted formal deductions does not exceed theaccepted norm; due respects are paid to syntactic difficulties. This ends thefirst part of the book: “Provability.”

The reader may skip the third and fourth chapters, and proceed immedi-ately to the fifth. Here we present elements of the theory of recursive functionsand enumerable sets, formulate Church’s thesis, and discuss the notion of algo-rithmic undecidability.

The basic content of the sixth chapter is a recent result on the Diophantinenature of enumerable sets. We then use this result to prove the existenceof versal families, the existence of undecidable enumerable sets, and, in theseventh chapter, Godel’s incompleteness theorem (as based on the definability ofprovability via an arithmetic formula). Although it is possible todisagree with this method of development, it has several advantages over earliertreatments. In this version the main technical effort is concentrated on provingthe basic fact that all enumerable sets are Diophantine, and not on the morespecialized and weaker results concerning the set of recursive descriptions orthe Godel numbers of proofs.

The last section of the sixth chapter stands somewhat apart from the rest.It contains an introduction to the Kolmogorov theory of complexity, which isof considerable general mathematical interest.

The fifth and sixth chapters are independent of the earlier chapters, andtogether make up a short course in recursive function theory. They form thesecond part of the book: “Computability.”

The third part of the book, “Provability and Computability,” relies heavilyon the first and second parts. It also consists of two chapters. All of the seventhchapter is devoted to Godel’s incompleteness theorem. The theorem appearslater in the text than is customary because of the belief that this central resultcan only be understood in its true light after a solid grounding both in formalmathematics and in the theory of computability. Hurried expositions, where1 Nineteenth century Russian poet (translator’s note). The full poem is:

We diligently observe the world,We diligently observe people,And we hope to understand their deepest meaning.But what is the fruit of long years of study?What do the sharp eyes finally detect?What does the haughty mind finally learnAt the height of all experience and thought,What?—the exact meaning of an old proverb.

Preface to the First Edition xiii

the proof that provability is definable is entirely omitted and the mathematicalcontent of the theorem is reduced to some version of the “liar paradox,” canonly create a distorted impression of this remarkable discovery. The proof isconsidered from several points of view. We pay special attention to propertieswhich do not depend on the choice of Godel numbering. Separate sections aredevoted to Feferman’s recent theorem on Godel formulas as axioms, and to theold but very beautiful result of Godel on the length of proofs.

The eighth and final chapter is, in a way, removed from the theme of thebook. In it we prove Higman’s theorem on groups defined by enumerable setsof generators and relations. The study of recursive structures, especially ingroup theory, has attracted continual attention in recent years, and it seemsworthwhile to give an example of a result which is remarkable for its beautyand completeness.

3. This book was written for very personal reasons. After several years ordecades of working in mathematics, there almost inevitably arises the need tostand back and look at this research from the side. The study of logic is, to acertain extent, capable of fulfilling this need.

Formal mathematics has more than a slight touch of self-caricature. Itsstructure parodies the most characteristic, if not the most important, features ofour science. The professional topologist or analyst experiences a strange feelingwhen he recognizes the familiar pattern glaring out at him in stark relief.

This book uses material arrived at through the efforts of many mathemati-cians. Several of the results and methods have not appeared in monographform; their sources are given in the text. The author’s point of view has formedunder the influence the ideas of Hilbert, Godel, Cohen, and especially John vonNeumann, with his deep interest in the external world, his open-mindednessand spontaneity of thought.

Various parts of the manuscript have been discussed withYu. V. Matiyasevic, G. V. Cudnovskiı, and S. G. Gindikin. I am deeply gratefulto all of these colleagues for their criticism.

W. D. Goldfarb of Harvard University very kindly agreed to proofread theentire manuscript. For his detailed corrections and laborious rewriting of partof Chapter IV, I owe a special debt of gratitude.

I wish to thank Neal Koblitz for his meticulous translation.

Yu. I. Manin Moscow, September 1974

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Interdependence of Chapters

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I PROVABILITY

I Introduction to Formal Languages . . . . . . . . . . . . . . . . . . . . . . . . . 31 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 First-Order Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Digression: Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Beginners’ Course in Translation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Digression: Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

II Truth and Deducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Unique Reading Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Interpretation: Truth, Definability . . . . . . . . . . . . . . . . . . . . . . . . . 233 Syntactic Properties of Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Digression: Natural Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Deducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Digression: Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Tautologies and Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 49

Digression: Kennings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 Godel’s Completeness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Countable Models and Skolem’s Paradox . . . . . . . . . . . . . . . . . . . 618 Language Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Undefinability of Truth: The Language SELF . . . . . . . . . . . . . . . 6910 Smullyan’s Language of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 7111 Undefinability of Truth: Tarski’s Theorem . . . . . . . . . . . . . . . . . 74

Digression: Self-Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7712 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix: The Von Neumann Universe . . . . . . . . . . . . . . . . . . . . . . . . . 89

The Last Digression. Truth as Value and Duty: Lessons ofMathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

III The Continuum Problem and Forcing . . . . . . . . . . . . . . . . . . . . . . 1051 The Problem: Results, Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 A Language of Real Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 The Continuum Hypothesis Is Not Deducible in L2 Real . . . . . . 114

viiPreface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Contents

4 Boolean-Valued Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 The Axiom of Extensionality Is “True” . . . . . . . . . . . . . . . . . . . . . 1246 The Axioms of Pairing, Union, Power Set, and

Regularity Are “True” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277 The Axioms of Infinity, Replacement, and

Choice Are “True” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328 The Continuum Hypothesis Is “False” for Suitable B . . . . . . . . . 1409 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

IV The Continuum Problem and Constructible Sets . . . . . . . . . . 1511 Godel’s Constructible Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512 Definability and Absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553 The Constructible Universe as a Model for Set Theory . . . . . . . 1584 The Generalized Continuum Hypothesis Is L-True . . . . . . . . . . . 1615 Constructibility Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646 Remarks on Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717 What Is the Cardinality of the Continuum? . . . . . . . . . . . . . . . . . 172

II COMPUTABILITY

V Recursive Functions and Church’s Thesis . . . . . . . . . . . . . . . . . . 1791 Introduction. Intuitive Computability . . . . . . . . . . . . . . . . . . . . . . 1792 Partial Recursive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833 Basic Examples of Recursiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 1874 Enumerable and Decidable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915 Elements of Recursive Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 201

VI Diophantine Sets and Algorithmic Undecidability . . . . . . . . . . 2071 The Basic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2072 Plan of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2093 Enumerable Sets Are D-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114 The Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145 Construction of a Special Diophantine Set . . . . . . . . . . . . . . . . . . 2176 The Graph of the Exponential Is Diophantine . . . . . . . . . . . . . . . 2217 The Factorial and Binomial Coefficient Graphs

Are Diophantine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218 Versal Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239 Kolmogorov Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

III PROVABILITY AND COMPUTABILITY

VII Godel’s Incompleteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 2351 Arithmetic of Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2352 Incompleteness Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2403 Nonenumerability of True Formulas . . . . . . . . . . . . . . . . . . . . . . . . 241

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4 Syntactic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435 Enumerability of Deducible Formulas . . . . . . . . . . . . . . . . . . . . . . 2496 The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527 Productivity of Arithmetical Truth . . . . . . . . . . . . . . . . . . . . . . . . 2558 On the Length of Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

VIII Recursive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2631 Basic Result and Its Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2632 Free Products and HNN-Extensions . . . . . . . . . . . . . . . . . . . . . . . . 2663 Embeddings in Groups with Two Generators . . . . . . . . . . . . . . . . 2704 Benign Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715 Bounded Systems of Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2756 End of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

IX Constructive Universe and Computation . . . . . . . . . . . . . . . . . . . 2851 Introduction: A Categorical View of Computation . . . . . . . . . . . 2852 Expanding Constructive Universe: Generalities . . . . . . . . . . . . . . 2893 Expanding Constructive Universe: Morphisms . . . . . . . . . . . . . . . 2934 Operads and PROPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2965 The World of Graphs as a Topological Language . . . . . . . . . . . . 2986 Models of Computation and Complexity . . . . . . . . . . . . . . . . . . . . 3077 Basics of Quantum Computation I: Quantum Entanglement . . 3158 Selected Quantum Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3199 Shor’s Factoring Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210 Kolmogorov Complexity and Growth of Recursive Functions . . 325

IV MODEL THEORY

X Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311 Languages and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 The Compactness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3343 Basic Methods and Constructions . . . . . . . . . . . . . . . . . . . . . . . . . 3424 Completeness and Quantifier Elimination in Some Theories . . . 3505 Classification Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3596 Geometric Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3647 Other Languages and Nonelementary Model Theory . . . . . . . . . 374

Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

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