# zbMATH — the first resource for mathematics

Model theory. (English) Zbl 0789.03031
Encyclopedia of Mathematics and Its Applications. 42. Cambridge: Cambridge University Press. xiii, 772 p. (1993).
The author writes in the introduction that the idea to write a book on model theory arose twelve years ago. Working on this project, so much material was accumulated that he wrote three other books on special parts of model theory.
But it is more than enough what was left for the present book. Scattered through the 790 pages are nine hundred exercises. The references contain one thousand items. This gives a hint on the immense material collected by the author.
What is the main difference to other books on model theory, for instance to the famous book of Chang and Keisler? It is the concentration on constructions rather than classification. Many results and methods appear for the first time in a book.
Now let us describe the contents. The book has twelve chapters and an appendix with special examples. Each chapter ends with a history and bibliography. Here the reader finds information for his further reading. This is important if the reader wants to go into details. Not all results presented in the book are proved.
The first chapter gives a short introduction. The author defines structures, term algebras, homomorphisms and so on. Diagrams are used to build models.
The second chapter covers essential material on languages. The author gives examples of axiomatizable classes of structures and introduces some of the fundamental notions like type and categoricity and deals with chains of models and quantifier elimination.
In chapter 3 the author defines Skolem functions and deals with Skolem hulls. He describes games for elementary equivalence. He also describes infinitary languages and gives a proof of Scott’s isomorphism theorem.
In chapter 4 the author demonstrates the influence of other branches of mathematics to model theory. So methods of group theory are explained to study automorphism groups of structures. Then he goes on with imaginary elements, minimal sets and strongly minimal sets. He shows the connection to geometry. This chapter includes the result of Zil’ber and Hrushovski on groups interpretable in modular geometries.
In chapter 5 the author discusses interpretations. They are used to show undecidability of theories. The author introduces the Morley rank and shows Macintyre’s result that every totally transcendental field is algebraically closed.
Chapter 6 is wholly devoted to the model theory of first-order languages. It starts with the compactness theorem for first-order logic, Boolean algebras, Stone spaces and types. It goes on with the amalgamation property and preservation theorems and ends with some facts from stability.
In chapter 7 the author discusses omitting types and the Fraïssé construction and countable categoricity.
Chapter 8 is mainly connected with the work of Robinson. Here the author deals with existentially closed structures, model theoretic forcing, model completeness and model companions.
Chapter 9 deals with products. The author describes direct products, reduced products, ultraproducts and Boolean powers. He introduces Horn formulas and discusses the Feferman-Vaught theorem. This chapter also includes a description of Word constructions.
In chapter 10 the author shows how to build saturated structures by means of chains. He discusses recursively saturated structures and atomic compact models.
Chapter 11 deals with Ehrenfeucht-Mostowski models. The author shows the influence of infinite combinatorics to model theory. He discusses non- standard methods and the problem of defining well-orderings. Using Baumgartner’s orderings the author shows non-structure theorems.
Chapter 12 discusses uncountably categorical theories. So the cohomology of expansions is examined.
The appendix assembles results of various interesting theories. So the Baur-Monk invariants for modules are introduced and used for the stability classification of modules. The Szmierlew invariants for Abelian groups are described and used to show the decidability of the theory of Abelian groups. Further examples deal with several classes of groups, fields and linear orderings.
Reviewer: M.Weese (Berlin)

##### MSC:
 03Cxx Model theory 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations