Model Theory

Dover’s reissue of this classical book is a wonderful excuse to review it and reassess its value, both historical and pedagogical, in the context of the development of model theory in the decades following its first edition (North-Holland,1973; third edition, 1990). The book under review was the first successful textbook on model theory that collected and organized in a systematic way the basic results of model theory, previously dispersed on the literature or preserved only as “folklore facts”, known to the experts but not otherwise widely available to the mathematical community. The systematic organization provided by the authors was so successful that it could almost be seen in the table of contents of most other books published after this one.

First, the basic notions and results are introduced: Propositional logic, languages, models, first-order theories, and elimination of quantifiers. Along the way, the main results proved include the compactness theorem and the Lindenbaum theorem.

Next, the book introduces several methods to construct models and various model-theoretic constructions, from models constructed from constants or chains to models constructed from ultraproducts. The first four chapters can be used in an introductory course in model theory, and they do not feel overly outdated.

The last three chapters take some advanced topics, from saturated and special models in Chapter five, Horn sentences in chapter six, to categoricity in power and Morley’s theorem in chapter seven.

Either by being at the right place at the right time or by some fine-tuned sense of where the subject was moving, the special topics chosen by the authors have shown their importance in recent developments. In the last ten or twenty years, model theory and its methods, especially the ones related to the mentioned topics, have been used to obtain several important results in arithmetic algebraic geometry and group theory. These applications have helped to show the importance of model theory and raised interest among researchers and students, with a growing number of textbooks and monographs available now for the interested reader. Perhaps there is still a special place for the book that, partly, started these developments. I, for one, am thankful to Dover for keeping such a nice book in print.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.