Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects

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- Jonathan Kirby

*Table of contents : ContentsPrefacePART I LANGUAGES AND STRUCTURES 1 Structures 2 Terms 3 Formulas 4 Definable Sets 5 Substructures and QuantifiersPART II THEORIES AND COMPACTNESS 6 Theories and Axioms 7 The Complex and Real Fields 8 Compactness and New Constants 9 Axiomatisable Classes 10 Cardinality Considerations 11 Constructing Models from SyntaxPART III CHANGING MODELS 12 Elementary Substructures 13 Elementary Extensions 14 Vector Spaces and Categoricity 15 Linear Orders 16 The Successor StructurePART IV CHARACTERISING DEFINABLE SETS 17 Quantifier Elimination for DLO 18 Substructure Completeness 19 Power Sets and Boolean Algebras 20 The Algebras of Definable Sets 21 Real Vector Spaces and Parameters 22 Semi-algebraic SetsPART V TYPES 23 Realising Types 24 Omitting Types 25 Countable Categoricity 26 Large and Small Countable Models 27 Saturated ModelsPART VI ALGEBRAICALLY CLOSED FIELDS 28 Fields and Their Extensions 29 Algebraic Closures of Fields 30 Categoricity and Completeness 31 Definable Sets and Varieties 32 Hilbert’s NullstellensatzBibliographyIndex*

An Invitation to Model Theory Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a branch of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from speciﬁc structures familiar from undergraduate mathematics, paying particular attention to deﬁnable sets throughout. With numerous exercises of varying difﬁculty, this is an accessible introduction to model theory and its place in mathematics. is a Senior Lecturer in Mathematics at the University of East Anglia. His main research is in model theory and its interactions with algebra, number theory, and analysis, with particular interest in exponential functions. He has taught model theory at the University of Oxford, the University of Illinois at Chicago, and the University of East Anglia. j o nat h a n

kirby

An Invitation to Model Theory J O NAT H A N K I R B Y University of East Anglia

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107163881 DOI: 10.1017/9781316683002 © Jonathan Kirby 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A.

A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Kirby, Jonathan, 1979– author. Title: An invitation to model theory / Jonathan Kirby, University of East Anglia. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references and index. Identiﬁers: LCCN 2018052996 | ISBN 9781107163881 (hardback ; alk. paper) | ISBN 1107163889 (hardback ; alk. paper) | ISBN 9781316615553 (pbk. ; alk. paper) | ISBN 1316615553 (pbk. ; alk. paper) Subjects: LCSH: Model theory. Classiﬁcation: LCC QA9.7 .K57 2019 | DDC 511.3/4–dc23 LC record available at https://lccn.loc.gov/2018052996 ISBN 978-1-107-16388-1 Hardback ISBN 978-1-316-61555-3 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Pirita, Lumia, Tapio, and Sakari

Contents

page ix

Preface

PART I

LANGUAGES AND STRUCTURES

1

1

Structures

3

2

Terms

9

3

Formulas

13

4

Deﬁnable Sets

19

5

Substructures and Quantiﬁers

24

PART II

29

THEORIES AND COMPACTNESS

6

Theories and Axioms

31

7

The Complex and Real Fields

37

8

Compactness and New Constants

42

9

Axiomatisable Classes

47

10

Cardinality Considerations

53

11

Constructing Models from Syntax

57

PART III

63

CHANGING MODELS

12

Elementary Substructures

65

13

Elementary Extensions

70

vii

viii

Contents

14

Vector Spaces and Categoricity

77

15

Linear Orders

83

16

The Successor Structure

88

PART IV

93

CHARACTERISING DEFINABLE SETS

17

Quantiﬁer Elimination for DLO

95

18

Substructure Completeness

99

19

Power Sets and Boolean Algebras

104

20

The Algebras of Deﬁnable Sets

109

21

Real Vector Spaces and Parameters

115

22

Semi-algebraic Sets

119

PART V

127

TYPES

23

Realising Types

129

24

Omitting Types

133

25

Countable Categoricity

138

26

Large and Small Countable Models

142

27

Saturated Models

147

PART VI

153

ALGEBRAICALLY CLOSED FIELDS

28

Fields and Their Extensions

155

29

Algebraic Closures of Fields

159

30

Categoricity and Completeness

163

31

Deﬁnable Sets and Varieties

167

32

Hilbert’s Nullstellensatz

173

Bibliography

177

Index

179

Preface

This book is designed as an undergraduate or master’s-level course in model theory. It has grown out of courses taught for many years at the University of Oxford and courses taught by me at UEA. The choice of material and presentation are based on pedagogical considerations, and I have tried to resist the temptation to be encyclopedic. In this book, the main programme of model theory is to take a familiar mathematical structure and get an understanding of it in the following way. First, ﬁnd an axiomatisation of its complete theory. Second, if possible, classify all the other models of the theory. Third, describe all the deﬁnable sets. As a result, model theory is presented as a set of tools for understanding structures, and the way the tools are applied to speciﬁc structures is as important as the tools themselves. This gives motivation to the subject and connects it to familiar material. Some readers may be more interested in the theory than in the applications, but my view is that even those who eventually wish to work in abstract model theory will get a better understanding of the basics by seeing them applied to examples. Historically, model theory grew as a branch of mathematical logic, and the focus was mostly on logical issues, such as decidability. As model theory has found more applications and connections to other branches of mathematics, the study of deﬁnable sets has become more central. I have tried to keep the book as self-contained as possible. Model theory requires a level of mathematical sophistication in terms of abstract, rigorous thinking, proofs, and algebraic thinking which students will normally have developed through previous courses in algebra, logic, or geometry, but there are few speciﬁc prerequisites. No topology is used, and almost no set theory is used. A brief chapter explains the basic cardinal arithmetic methods needed for some of the proofs, but with one or two exceptions, everything can be assumed

ix

x

Preface

to be countable. A familiarity with the use of basic algebraic ideas, such as bijections and homomorphisms or embeddings of groups, or rings, or vector spaces, is essential, but when algebraic examples such as rings and vector spaces are introduced, all the necessary deﬁnitions and facts are explained. For the most part I have not given historical references for the material. There are good historical remarks at the end of each chapter in the books of Hodges [Hod93, Hod97] and Marker [Mar02], and I refer the reader to those.

Overview The book is organised into six parts. Part I covers structures, languages, and automorphisms, introduces deﬁnable sets, and proves the essential preservation theorems. Part II introduces theories and the programme of ﬁnding axiomatisations for structures. Context is given with axiomatisations of the complex and real ﬁelds and the natural numbers with addition and multiplication (Peano arithmetic). The compactness theorem is then introduced, and examples of its use are given. Part II concludes with the Henkin proof of the compactness theorem. In Part III we show that even a complete theory can have many different models with the L¨owenheim–Skolem theorems. The notion of categoricity is introduced via the example of vector spaces and is used to prove the completeness of an axiomatisation. Further applications are given to dense linear orders, where the back-and-forth method is introduced, and to the natural numbers with the successor function. The idea of quantiﬁer elimination is introduced in Part IV, and used to characterise the deﬁnable sets in dense linear orders and vector spaces. Boolean algebras are introduced via an investigation into the theory of power sets, partially ordered by the subset relation, and the deﬁnable subsets of a structure in any number n of variables is shown to be a Boolean algebra, called the Lindenbaum algebra of the theory. These Lindenbaum algebras are then worked out in the examples of vector spaces and for the real ordered ﬁeld. Parts V and VI go in different directions and do not depend on each other. Part V develops the notion of types, which is at the heart of modern model theory. The ﬁrst goal is the Ryll–Nardzewski theorem, which characterises countably categorical theories as those for which there are only ﬁnitely many deﬁnable sets in any given number of variables. There is then a brief discussion of saturated models, leading to areas for further reading. Part VI takes the model-theoretic techniques developed in the ﬁrst four parts and applies it to the theory of algebraically closed ﬁelds. Two chapters explain

Preface

xi

the necessary algebraic background, and a third proves categoricity, completeness of the theory, and quantiﬁer elimination. The next chapter explains how the deﬁnable sets correspond to algebraic varieties and constructible sets, and a ﬁnal chapter gives a model theoretic proof of Hilbert’s Nullstellensatz, which gives more information about the deﬁnable sets.

Suggestions for Using This Book as a Textbook This book can be used for a course in many different ways. Figure 0.1, illustrating the dependencies between chapters, can be used as a guide to planning a course according to the lecturer’s preferences. Each chapter is intended to be of such a length that it can be taught in one hour (perhaps covering only the essentials) or in two hours (sometimes using material from the exercises). Everything depends on the ﬁrst three chapters, so they should be covered ﬁrst, unless the students have the prerequisite knowledge from predicate logic. The emphasis is on semantic ideas, including automorphisms, which is somewhat different from the usual emphasis in a ﬁrst logic course, and I have successfully taught this material to a class of students of whom some had seen logic before and some not. For those who have not seen this material before, Chapter 3 in particular goes rather quickly. Chapters 4 and 5 carry on the study of formulas to give students more opportunity to consolidate their understanding. Each chapter has several exercises at the end, which range from very easy consolidations of deﬁnitions to more substantial projects. Some exercises need more background than the other material in the chapter, and some exercises develop extension material. A short model theory section of a mathematical logic course could have the compactness theorem and some applications as its goal and consist of Chapters 1, 2, 3, and 6, Section 8.1, and Chapter 9. This could be ﬁlled out with any of Chapters 4, 5, and 7, the rest of Chapter 8 or Chapter 11, or with applications from Chapters 14, 15, 16, or 19. The back-and-forth method is another highlight that could be reached quickly, by covering only the essential sections of Chapters 1, 3, 6, and 17. The heart of this book is the study of deﬁnable sets. A course centred on those would consist of Parts I to IV, possibly omitting Chapters 5, 9, 11, and 16, and possibly adding Part VI. A course aiming to cover the basics of types and the Ryll–Nardzewski theorem, without the emphasis on deﬁnable sets, could consist of Chapters 1, 2, 3, 6, 8, 10–15, and 23–25.

xii

Preface

31

32 30

5 27

29 7

13

28

26 25 20

24 23

22 21

7

20 18

19

17

16

15 14 13 12 10 7

19

9 6

4

8

11 10

5 3 2 1

Figure 0.1 Chapter dependencies: In order to make the diagram planar, Chapters 5, 7, 10, 13, 19, and 20 have been drawn in two (or even three) places. As the diagram shows, Parts IV, V, and VI are largely independent of each other, with much of Part V also independent of Part III. While Chapter 25 uses the concept of Lindenbaum algebras from Chapter 20, that material does not rely on the rest of Parts III or IV.

Preface

xiii

For research students, Part VI, on algebraically closed ﬁelds, gives a pattern for carrying out the programme of understanding deﬁnable sets in a theory which is similar to the pattern which works in many other theories of ﬁelds, such as real-closed ﬁelds, differentially closed ﬁelds, separably closed ﬁelds, algebraically closed valued ﬁelds, exponentially closed ﬁelds, algebraically closed ﬁelds with an automorphism, and so on.

Further Reading There are several more advanced textbooks in model theory, including those of Marker [Mar02], Hodges [Hod93, Hod97], Poizat [Poi00], Tent and Ziegler [TZ12], Sacks [Sac10], and Chang and Keisler [CK90]. Rothmaler [Rot00] is more at the level of this book, but there is more emphasis on algebra, especially groups. V¨aa¨ n¨anen [V¨aa¨ 11] introduces model theory via back-and-forth games, which complements the approach in this book. Bridge [Bri77] was based on the incarnation of the Oxford model theory course from the early 1970s, and the emphasis is much more on logic. Other suggestions for further reading are given through the book, particularly at the end of Parts IV, V, and VI.

Acknowledgements Thanks to Boris Zilber for allowing me to take his lecture notes as a starting point both for my course and for this book. I learned model theory initially from the books of Wilfrid Hodges and David Marker, and from Boris. My writing style was much inﬂuenced by the late Harold Simmons. I would like to thank Cambridge University Press for their help in producing the book and, in particular, Silvia Barbina for initially suggesting that I write it. Many people have made helpful comments on drafts or have otherwise helped me to write the book. I would like to thank Lou van den Dries, David Evans, Åsa Hirvonen, Wilfrid Hodges, Ehud Hrushovski, Tapani Hyttinen, Gareth Jones, Asaf Karagila, David Marker, Alice Medvedev, Charles Steinhorn, Alex Wilkie, and Boris Zilber for sharing their expertise. Thanks to Francesco Parente for reading two drafts and providing many considered comments. Thanks also to the students who attended my courses or read drafts of the book and provided essential feedback. They include Abeer Albalahi, Michael Arnold, Emma Barnes, Matt Gladders, Grant Martin, Oliver Matheau-Raven, Audie Warren, and Tim Zander.

Part I Languages and Structures

In this ﬁrst part of the book, we introduce the basic methods from predicate logic which underpin the rest of the book. Mathematical objects are formalised as L -structures, and statements about these structures are formalised in the notions of L -terms and L -formulas. We introduce the central notion of a deﬁnable set, a subset of an L -structure which is deﬁned by an L -formula. A key notion in mathematical logic is that of a recursive deﬁnition, and both the notions of L -terms and L -formulas give examples of that. These recursive deﬁnitions give rise to methods of proof by induction. Throughout this part, these proofs by induction are used prove preservation theorems, most importantly the preservation of deﬁnable sets by automorphisms of a structure. From this result we get a method to show that certain subsets of a structure are not deﬁnable, an essential tool which we will later exploit in the programme of characterising the deﬁnable sets of a structure.

1

1 Structures

In different mathematical contexts, familiar mathematical objects may actually have different meanings. For example a reference to the integers Z in group theory is likely to mean the inﬁnite cyclic group, whereas in number theory it is likely to mean Z as a ring. In model theory, when we specify a structure, we have to be precise about such things. The integers as an additive group will be written as Zadgp = ±Z ; +, −, 0² , and Z as a ring will be written as Z ring = ±Z; +, ·, −, 0, 1². The integers are also used to index the years in a calendar, and an appropriate structure for that purpose is Z = ±Z;

0

∨ − f (x¯)

>

0)]

120

22 Semi-algebraic Sets

and

Ro-ring |= ∀ x¯[¬( f ( x¯)

>

0)

↔

( f ( x¯)

= 0 ∨ − f ( x¯)

>

0)].

So S is deﬁned by a positive Boolean combination of atomic formulas. Using the disjunctive normal form theorem again, S is deﬁned by a formula of the form

⎞ ⎛ t si r ⎜³ i ³ ² ⎟⎟ ⎜⎜ hi j( x¯) = 0 ∧ g ( x ¯ ) > 0⎟ . i j ⎠ ⎝ j= 1

i= 1

Let h i( x¯) =

∑ ti

2 j=1 h i j( x¯) .

j= 1

Then S is deﬁned by

⎛ ⎞ r ⎜ si ² ³ ⎟⎟ ⎜⎜ g i j( x¯) > 0⎟⎟⎠ , ⎜⎝h i (x¯) = 0 ∧ i=1

j=1

so it is a ﬁnite union of sets of the desired form.

±

Now we restrict to the case n = 1. A formula of the form f ( x ) = 0 deﬁnes a ﬁnite set unless f is a constant polynomial √ when √ it deﬁnes either all of R or ∅. 3 For example, x − 2x = 0 deﬁnes {− 2, 0, 2}. We can work out what set is deﬁned by a formula of the form f ( x) > 0 by considering how the sign of √f ( x ) changes √ between its zeros. For example, 3 x − 2x > 0 deﬁnes the set (− 2, 0) ∪ ( 2, ∞). In general, any ﬁnite union of open intervals can be obtained this way. By combining these intervals with points, we can get closed intervals and half-open intervals. Using quantiﬁer elimination and Lemma 22.2, we get the following.

Proposition 22.3 Every subset of respect to the structure

R o-ring

R

which is parametrically deﬁnable with is a ﬁnite union of points and intervals.

Note that these subsets of R are also parametrically deﬁnable in the reduct R , although not necessarily with the same parameters. However, there are deﬁnable sets in two or more free variables, such as that deﬁned by y = x 2 , which are not deﬁnable in R . The conclusion of Proposition 22.3 is a very useful condition that has been shown to hold in many other structures and has been given a name.

0

∧ (∀

)[ δ > 0

δ

→ (∃y)[| x − y|

< δ

∧ | f (x) −

f (y )| > ± ]]],

so by o-minimality, it is a ﬁnite union of points and open intervals. But by Lemma 22.7, a semi-algebraic function cannot be discontinuous on an open interval. So D is a ﬁnite set of points. ±

22.3 Nash Functions We now characterise the semi-algebraic functions more precisely.

Deﬁnition 22.8 A Nash function is an analytic function f : U → R where U ⊆ Rn is a connected open semi-algebraic subset such that there is a non-zero polynomial h ( x¯, y) ∈ R[ x¯, y ] such that for all x¯ ∈ U , we have h ( x¯, f (x¯)) = 0.

22.3 Nash Functions

123

To say that f is analytic means that it is inﬁnitely differentiable and, furthermore, that for every point a ∈ U , the Taylor series of f at a converges to f in a neighbourhood of a . A polynomial function f is analytic and is a Nash function, taking h ( x¯, y ) to be f ( x¯) − y . The following consequence of the implicit function theorem for analytic functions gives an easy way to check analyticity.

Fact 22.9 Suppose that U ⊆ Rn is a connected open semi-algebraic subset, that f : U → R is continuous, and that there is a polynomial h (x¯, y) ∈ R[ x¯, y ] such that for all x¯ ∈ U, we have h( x¯, f ( x¯)) = 0 and yh ( x¯, f ( x¯)) ² 0. Then f is ∂

∂

a Nash function.

Using this fact, the function f from Example 22.5 can be split into the three points (−3, −1), (−2, 0), and (0, 0) and four Nash functions:

⎧ √3 ⎪ x+3−1 ⎪ ⎪ ⎨ µ 2 y=⎪ ⎪−√ 1 − ( x + 1) ⎪ ⎩ x

∈ (−∞ −3) and for x ∈ (−3 −2) for x ∈ (−2 0) for x ∈ (0 ∞)

for x

,

,

,

,

,

,

.

The functions h ( x, y) are then those appearing in the formula in Example 22.5. Note that f is not differentiable at x = −3, so is not analytic, and this corresponds to the function h ( x, y) = (y + 1)3 − x − 3 having hy (−3, −1) = 0. ∂

∂

Proposition 22.10 Let (a, b) ⊆ R be an open interval, possibly with a = −∞ and / or b = +∞, and let f : (a , b ) → R be a semi-algebraic function. Then f is a piecewise Nash function . That is, there is m ∈ N and a = a 0 < a1 < · · · < a m < a m+1 = b such that, for each k = 0, . . . , m, f ´( ak ak +1) is a Nash function. ,

By quantiﬁer elimination ·for Ro-ring (Fact 18.5) and Lemma 22.2, the ¶ graph Γ f = ( x , y ) ∈ R2 | y = f ( x ) of f is deﬁned by a formula of the form

Proof

⎛ ⎞ r ⎜ si ⎟⎟ ² ³ ⎜⎜ ⎟⎟ . g ( x , y ) > 0 h ( x , y ) = 0 ∧ ij ⎜⎝ i ⎠ i=1

j= 1

The functions hi cannot be the zero polynomial, because otherwise Γ f would contain an open subset of R2 , which is impossible for the graph of a function. Let H be the set consisting of the polynomials h i (x , y) and their iterated t h y -derivatives yti ( x , y ) for all t ∈ N. Since the hi are polynomials, H is a ﬁnite set.¹ For each h ∈º H , let Z h = { x ∈ R | h ( x, f ( x )) = 0 } and let D h = ¸ ∂

∂

x

¹

∈ Z h ¹¹

h y (x,

∂ ∂

f ( x )) ² 0 . These sets are deﬁnable, so by o-minimality, they

are ﬁnite unions of points and open intervals. List all of the isolated points and the endpoints of the intervals for all of the Z h for h ∈ H as a = a 0 < a1 < · · ·