The book under review is a well-organized textbook, short and readable, that gives a modern and almost self-contained introduction to model theory, including some important recent developments in categoricity, stability and simplicity. The book starts with a quick review of the basic facts of first-order logic and its structures from the definition of languages, models and theories to the compactness and Löwenheim-Skolem theorems. It must be mentioned that, in contrast to older treatments and following recent trends, important topics such as categoricity and Vaught’s test are introduced early, on page 25 of Chapter 2.
Chapter 3 studies the important class of theories where every formula is equivalent to a quantifier-free formula. The examples included in this chapter range from the theory of algebraically closed fields to separably closed fields and differentially closed fields. As the authors remark, perhaps a notable omission is the theory of valued fields, for which a classic reference is Chang and Keisler’s Model Theory, recently reissued by Dover.
Chapter four starts by proving the omitting type theorem for countable theories and then introduces a topology on the set on n-types: for a formula f the closed set [f] is the set of all n-types containing f (the analogy with the Zariski topology on the prime spectrum of a commutative ring is clear). After introducing the amalgamation method to construct models, the chapter ends with a section on theories that admit a smallest model realizing only the necessary types.
Chapters five and six are devoted to Morley’s theorem and the notion of Morley rank. Morley’s theorem (a theory with only one model in some uncountable cardinality has a unique model in every uncountable cardinality) is proved in Chapter five, and Chapter six is devoted to the Morley rank, a sort of dimension for definable sets, viewing them as solution sets of formulas. Again, since the main example is the model theory of fields, it is important to observe that for the theory of algebraically closed fields, definable sets are just the constructible sets and the Morley rank of a constructible set is just its Krull dimension; this is proved in the last section of this chapter. Thus, we are led to the study of the so-called Zariski pre-geometries.
Next to consider a larger class of stable theories, in Chapter seven. After introducing the class of simple theories, Kim and Pillay characterize them in terms of a suitable notion of independence. The most important example in this context is the theory of pseudo-finite fields (perfect fields K such that every affine variety defined over it and irreducible over its algebraic closure has a K-rational point).
Chapter eight focuses on stable theories; one the main results proven here is that they are simple. To do this, the authors develop some classical results of stability theory describing forking in these theories.
The last two chapters return to more classical results of stability theory. Some of the highlights of Chapter nine are the uniqueness of prime extensions for totally transcendental theories and that for countable stable theories if prime extensions exist they are unique. The book ends with a variant of Hrushovski’s construction of a strongly minimal set, a counterexample to Zilber’s conjecture.
Three appendices collect some results from set theory, fields and pre-geometries that are used throughout the book. At the end of each one of its sections the book has set of exercises, over 200 of them, and Appendix D contains the solutions to some of those exercises, in particular the ones that are used in the book. With its focus on some of the most active areas in the intersection of model theory and Diophantine geometry, this book could be of interest to anyone looking for a fast-paced introduction to this area of mathematical logic. The book could be used as a textbook for an introductory course on model theory or for self-study.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.
1. The basics
2. Elementary extensions and compactness
3. Quantifier elimination
4. Countable models
5. Aleph-1-categorical theories
6. Morley rank
7. Simple theories
8. Stable theories
9. Prime extensions
10. The fine structure of 1-categorical theories
A. Set theory
D. Solutions of exercises