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**Cours de théorie des modèles. Une introduction à la logique mathématique contemporaine.**
*(French)*
Zbl 0583.03001

Nur Al-Mantiq Wal-Ma’rifah, 1. Villeurbanne, France: Bruno Poizat. IV, 584 p. (1985).

Model theory is a good introduction to mathematical logic for the mathematician: it proves theorems about structures from mathematical practice. It comes later that the reader is able to appreciate some of the finesses of metamathematics, finite coding of formulas, recursiveness and decidability.

The book under review deals for the most part with model theory and the above concepts are used only to complete the picture when appropriate. It starts with the local isomorphisms between structures consisting of one relation on a set, describes languages associated to such structures and then it gives the general concept of a first-order language. Elementary extensions, Löwenheim-Skolem-Tarski theorems, ultraproducts, compactness are explained and the power of the back-and-forth method is shown on \(\omega\)-saturated structures. Utility of these tools is illustrated on various types of structures, e.g. algebraically and differentially closed fields, Boolean algebras, ultrametric spaces and existentially closed modules. Some basics on ordinals and cardinal numbers are recalled with a portion of first-order arithmetic. Saturated and prime models complete the first half of the book.

The second half deals with stability theory - a synonym for contemporary model theory in which the author himself has played a significant role. This part starts with the concept of the fundamental order of types which is the key to the Paris school approach to stability. It covers the stability of saturated models, deviation (forking), strong types and ranks. The book culminates in the study of stability of prime models and classification of models of dimensional (i.e. non-multidimensional) theories.

The book is well organized, the first part can serve as an introduction to model theory and mathematical logic in general, the second part is a good exposition of stability theory which is now the most active part of contemporary research in model theory. The book is a result of courageous undertaking of the author, who has published several hundred copies himself after certain experience with major publishing houses in France and elsewhere. This beautifully written book will be a joy to both novices and experts.

The book under review deals for the most part with model theory and the above concepts are used only to complete the picture when appropriate. It starts with the local isomorphisms between structures consisting of one relation on a set, describes languages associated to such structures and then it gives the general concept of a first-order language. Elementary extensions, Löwenheim-Skolem-Tarski theorems, ultraproducts, compactness are explained and the power of the back-and-forth method is shown on \(\omega\)-saturated structures. Utility of these tools is illustrated on various types of structures, e.g. algebraically and differentially closed fields, Boolean algebras, ultrametric spaces and existentially closed modules. Some basics on ordinals and cardinal numbers are recalled with a portion of first-order arithmetic. Saturated and prime models complete the first half of the book.

The second half deals with stability theory - a synonym for contemporary model theory in which the author himself has played a significant role. This part starts with the concept of the fundamental order of types which is the key to the Paris school approach to stability. It covers the stability of saturated models, deviation (forking), strong types and ranks. The book culminates in the study of stability of prime models and classification of models of dimensional (i.e. non-multidimensional) theories.

The book is well organized, the first part can serve as an introduction to model theory and mathematical logic in general, the second part is a good exposition of stability theory which is now the most active part of contemporary research in model theory. The book is a result of courageous undertaking of the author, who has published several hundred copies himself after certain experience with major publishing houses in France and elsewhere. This beautifully written book will be a joy to both novices and experts.

Reviewer: O.Štěpánková

### MSC:

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03C45 | Classification theory, stability, and related concepts in model theory |

03C07 | Basic properties of first-order languages and structures |