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Class Notes for Mathematics 571

Spring 2010

Model Theory

written by

C. Ward Henson

Mathematics Department University of Illinois

1409 West Green Street Urbana, Illinois 61801 email: henson@math.uiuc.edu www:

c Copyright by C. Ward Henson 2010; all rights reserved.

Introduction

The purpose of Math 571 is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with significant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research.

Beyond the core techniques and results of model theory, Math 571 places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the field of real numbers (real closed fields) and show how this can be used to obtain the characterization of positive semi-definite rational functions that gives a solution to Hilbert's 17th Problem.

A highlight of Math 571 is a proof of Morley's Theorem: if T is a complete theory in a countable language, and T is -categorical for some uncountable , then T is categorical for all uncountable . The machinery needed for this proof includes the concepts of Morley rank and degree for formulas in -stable theories. The methods needed for this proof illustrate ideas that have become central to modern research in model theory.

To succeed in Math 571, it is necessary to have exposure to the syntax and semantics of first order logic, and experience with expressing mathematical properties via first order formulas. A good undergraduate course in logic will usually provide the necessary background. The canonical prerequisite course at UIUC is Math 570, but this covers many things that are not needed as background for Math 571.

In the lecture notes for Math 570 (written by Prof. van den Dries) the material necessary for Math 571 is presented in sections 2.3 through 2.6 (pages 24?37 in the 2009 version). These lecture notes are available at vddries/410notes/main.dvi.

A standard undergraduate text in logic is A Mathematical Introduction to Logic by Herbert B. Enderton (Academic Press; second edition, 2001). Here the material needed for Math 571 is covered in sections 2.0 through 2.2 (pages 67?104).

This material is also discussed in Model Theory by David Marker (see sections 1.1 and 1.2, and the first half of 1.3, as well as many of the exercises at the end of chapter 1) and in many other textbooks in model theory.

For Math 571 it is not necessary to have any exposure to a proof system for first order logic, nor to G?odel's completeness theorem. Math 571 begins with a proof of the compactness theorem for first order languages, and this is all one needs for model theory.

We close this introduction by discussing a number of books of possible interest to anyone studying model theory.

The first two books listed are now the standard graduate texts in model theory; they can be used as background references for most of what is done in Math 571.

David Marker, Model Theory: an Introduction.

Bruno Poizat, A Course in Model Theory.

The next book listed was the standard graduate text in model theory from its first publication in the 1960s until recently. It is somewhat out of date and incomplete from a modern viewpoint, but for much of the content of Math 571 it is a suitable reference.

C. C. Chang and H. J. Keisler, Model Theory.

Another recent monograph on model theory is Model Theory by Wilfrid Hodges. This book contains many results and examples that are otherwise only available in journal articles, and gives a very comprehensive treatment of basic model theory. However it is very long and it is organized in a complicated way that makes things hard to find. The author extracted a shorter and more straightforward text entitled A Shorter Model Theory, which is published in an inexpensive paperback edition.

In the early days of the subject (i.e., 1950s and 1960s), Abraham Robinson was the person who did the most to make model theory a useful tool in the main body of mathematics. Along with Alfred Tarski, he created much of modern model theory and gave it its current style and emphasis. He published three books in model theory, and they are still interesting to read: (a) Intro. to Model Theory and the Metamathematics of Algebra, 1963; (b) Complete Theories, 1956; new edition 1976; (c) On the Metamathematics of Algebra, 1951.

The final reference listed here is Handbook of Mathematical Logic, Jon Barwise, editor; this contains expository articles on most parts of logic. Of particular interest to students in model theory are the following chapters: A.1. An introduction to first-order logic, Jon Barwise. A.2. Fundamentals of model theory, H. Jerome Keisler. A.3. Ultraproducts for algebraists, Paul C. Eklof. A.4. Model completeness, Angus Macintyre.

Contents

Introduction

3

1. Ultraproducts and the Compactness Theorem

1

Appendix 1.A: Ultrafilters

6

Appendix 1.B: From prestructures to structures

8

2. Theories and Types

12

3. Elementary Maps

18

4. Saturated Models

25

5. Quantifier Elimination

30

6. L?owenheim-Skolem Theorems

35

7. Algebraically Closed Fields

39

8. Z-groups

44

9. Model Theoretic Algebraic Closure

49

10. Algebraic Closure in Minimal Structures

52

11. Real Closed Ordered Fields

58

12. Homogeneous Models

62

13. Omitting Types

68

14. -categoricity

76

15. Skolem Hulls

80

16. Indiscernibles

82

17. Morley rank and -stability

86

18. Morley's uncountable categoricity theorem

96

19. Characterizing Definability

102

Appendix: Systems of Definable Sets and Functions

110

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