Model theory can be thought as a distilled form of mathematical thought, extremely careful in dealing with syntax (an object language) and semantics (a class of structures and definable objects in this class). Syntax lists the symbols and rules for manipulating them to construct well-formed formulas or sentences in the language. Semantics provides the interpretation of the formulas and sentences of the formal language in an appropriate structure. A model is a structure where each sentence of the theory is true. Consistency, an important syntactic requirement for a theory, is equivalent to having a model for the given theory (this is Gödel’s completeness theorem). The proof of this theorem, satisfying Hilbert’s requirement of being finitary, implies the compactness theorem (a theory is consistent if and only if each finite subset is consistent). The study of particular theories is an important topic in model theory. Of particular interest are theories that admit elimination of quantifiers, i.e., theories for which every formula is equivalent to a formula with no quantifiers.
An important example of one such theory is the theory of algebraically closed fields (Tarski-Robinson), where elimination of quantifiers could be formulated as follows. In algebraic geometry, constructible sets are elements of the Boolean algebra generated by the Zariski open or closed sets over the given algebraically closed field. Elimination of quantifiers for the theory of algebraically closed fields is equivalent to Chevalley’s theorem (the image of a constructible set is constructible).
A variant of these types of theories are the model-complete theories, i.e., theories where every formula is equivalent to one with only (initial) existential quantifiers. And an important notion in this context is Huntington’s notion of categoricity. A theory is categorical if, up to isomorphism, it has only one model. But there is a more useful form of this concept, categoricity in power. A theory is κ-categorical if all its models of cardinality κ are isomorphic. The first important result in this context is Morley’s theorem: If a complete theory in a countable first-order language is κ-categorical for some uncountable cardinal κ, then it is κ-categorical for all uncountable cardinals κ (Trans. AMS 114, 515–538, 1965).
Morley’s paper introduced the notion of Morley rank, a sort of dimension for definable sets, i.e., solution sets of formulas. 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. Morley proved that in an uncountable categorical theory all definable sets have ordinal-valued Morley rank and later on Baldwin showed that this Morley rank is finite. In the early 1970s Boris Zilber noticed that the theory of simple algebraic groups over algebraically closed fields is uncountable categorical, and thus simple algebraic groups over those fields have finite Morley rank. Additional evidence, from Zilber’s “ladder theorem” allowed him to conjecture that any simple group of finite Morley rank is indeed an algebraic group over an algebraically closed field. This algebraicity conjecture has seen lots of activity in recent years, examples of which are the recent monograph of Altinel, Borovik and Cherlin Simple Groups of Finite Morley Rank (AMS, 2007) or Borovik and Nesin’s Groups of Finite Morley Rank (Oxford, 1995).
The book under review is a welcome reprint of a book published in 1972 (Benjamin) at a time when Morley’s theorem was still fresh. The book has 42 short sections. The first sections are introductory, starting with first-order theories and the compactness theorem (sections 1 to 7), model completeness and algebraically closed fields (sections 8 to 9), Löwenheim-Skolem and model completions (sections 10 to 12). The middle part, up to section 24, is devoted to making a more detailed study of structures. Thus we find saturated and homogeneous structures, Tarski’s theorem on elimination of quantifiers for real-closed fields, and Vaught, Chang and Kreisler two-cardinal theorems.
The last part of the book, sections 25 to 41, starts with a brief review of some category theory concepts (direct limits were introduced in section 10) that are later used to deal with the various notions of rank. Next, the author proceeds to Morley’s analysis of 1-types and the interrelations between saturatedness, ω-stability and sets of indiscernibles. The highlight is a proof of Morley’s theorem in section 37, and the last sections are devoted to work related to Morley’s, including an application to the theory of differentially closed fields.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.