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Model theory: An introduction. (English) Zbl 1003.03034
Graduate Texts in Mathematics. 217. New York, NY: Springer. viii, 342 p. EUR 64.95/net; sFr. 108.00; £45.50; \$ 59.95 (2002).

The author’s intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications. His goal in writing this text is to present the basic material and to illustrate how the two traditional themes of model theory interact. These traditional themes are the investigation concrete mathematical structures and the sets definable in them and the investigation of sets of sentences (theories) and the general structure of their models. Ideally, the reader of this text should already have an acquaintance with first-order logic, be comfortable with basic set theory (Zorn’s lemma, cardinals, ordinals), and have had a year long course in algebra at the graduate level.

The text has eight chapters and two appendices, one on set theory and one on real algebra. Each chapter ends with a section of exercises and remarks. There are numerous exercises that vary in difficulty; some ask for proofs of results mentioned in the text; some work out examples that further illustrate material in the text; some introduce topics not appearing in the text (ultraproducts, for example); some require more outside knowledge and are marked with a dagger. The remarks contain some historical information. They also contain references to useful, mostly secondary, sources. The author also uses this opportunity to describe further results and suggest further reading. There is an extensive bibliography and a brief index.

This text is noteworthy for its wealth of examples and its desire to bring the student to the point where the frontiers of resrearch are visible. The author briefly indicates sections to comprise a one semester course, but there is no dependency graph for its sections. The text is packed and negotiating a path through it and its exercises may require careful thought. In any case this book should be on the shelf of anybody with an interest in model theory.

Here are the chapter headings and a partial indication of their contents. 1. Structures and Theories: first-order languages $ℒ$ and $ℒ$-structures, theories and elementary classes, definable sets and interpretability; 2. Basic Techniques: compactness and Henkin constructions, complete theories, Löwenheim-Skolem theorems, back and forth constructions, ${ℒ}_{{\omega }_{1},\omega }$ and Scott’s isomorphism theorem, Ehrenfeucht-Fraïssé games; 3. Algebraic Examples: quantifier elimination (QE), (ordered) divisible abelian groups, Presburger arithmetic, algebraically closed fields and the elimination of imaginaries, real closed fields; 4. Realizing and Omitting Types: types, omitting types and prime models, prime model extensions of $\omega$-stable theories, saturated and homogeneous models, QE for differentially closed fields, Vaught’s two cardinal theorem, number of countable models, Morley’s analysis of countable models; 5. Indiscernibles: order indiscernibles, Ehrenfeucht-Mostowski models, a many-models theorem, an independence result for Peano arithmetic (Paris-Harrington); 6. $\omega$-Stable Theories: uncountably categorical theories, the Baldwin-Lachlin proof of Morley’s categoricity theorem, Morley rank, forking and independence, uniqueness of prime model extensions, prime models of $\omega$-stable theories; 7. $\omega$-Stable Groups: chain conditions, generic types, indecomposibility theorem, definable groups in algebraically closed fields, algebraic and constructible groups, generically presented groups and Hrushovski’s theorem; 8. Geometry of Strongly Minimal sets: pregeometries, geometry of strongly minimal sets, Zariski geometries, applications to Diophantine geometry (a special case of Hrushovski’s proof of the Mordell-Lang Conjecture for function fields).

##### MSC:
 03Cxx Model theory 03-01 Textbooks (mathematical logic) 03-02 Research monographs (mathematical logic)