Stabilité en théorie des modèles. (Stability in model theory).

*(French)*Zbl 0655.03021
Monographies de Mathématique, 2. Louvain-la-Neuve: Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée; Cabay Libraire-Éditeur S. A. II, 231 p. (1986).

The book is an introduction to stability theory with special emphasis on the classification of bounded (nonmultidimensional) theories and those with NDOP. It is accessible to those with a basic knowledge of model theory.

After the introduction with historical notes in Chapter I, Chapter II provides a review of basic model-theoretical notions, starting with types and the Stone space. It contains a nice proof of the omitting types theorem, in a topological setting, just one of the many examples throughout the book of the elegance and simplicity of this approach.

Stability theory starts in Chapter III with the fundamental order, heirs and definable types, and continues in Chapter IV with forking and the open mapping and finite equivalence relation theorems. Chapter V introduces U-rank, Morley rank and the continuous rank R(p,L,\(\infty)\). It ends with an example in algebraically closed fields due to C. Berline, demonstrating the connection between the Krull dimension of a prime ideal and the U-rank of the associated type. Chapter VI deals with indiscernibles, prime and saturated models and \(\aleph_{\epsilon}\)- saturated models.

Chapter VII leads into the second half of the book, which obtains a classification of particular classes of theories. Starting with Morley’s theorem and the Baldwin-Lachlan theorem, the reader learns to classify models by their dimensions, in this case the dimension of the strongly minimal sets. Chapter VIII generalizes this approach to superstable theories. The author introduces domination, regular types, orthogonality, dimensions of regular types and weight. Lachlan’s theorem on the number of countable models is obtained as an application. Chapter IX contains the corresponding results for \(\omega\)-stable theories and the Rudin- Keisler (R-K) order. In Chapter X the author introduces bounded types, which are later used to give his new (but equivalent to Shelah’s) definitions of nonmultidimensional (bounded) theories and depth. For an \(\omega\)-stable bounded theory he classifies the extensions of a model by the dimensions of their minimal types in the R-K-ordering. Chapter XI deals with theories which do not have the dimensional order property (NDOP theories), the presentable theories. Upper bounds are given for the number of \((\aleph_{\epsilon}\)-saturated) models in the \(\omega\)-stable [superstable] case. The lower bounds are not dealt with. The exposition concludes with two extensive examples: the first for the DOP when the classification fails; the second for a presentable theory.

The book contains a subject index and a notation index. There are only a few examples, which for the most part are fully explained. Exercises are very rarely suggested. The exposition is carefully streamlined to include only those topics needed for the development of the classification; e.g. Lascar’s rank inequalities are not included. Yet the presentation is not terse but rather creates a very reasonable and readable description of the landscape below the main gap.

After the introduction with historical notes in Chapter I, Chapter II provides a review of basic model-theoretical notions, starting with types and the Stone space. It contains a nice proof of the omitting types theorem, in a topological setting, just one of the many examples throughout the book of the elegance and simplicity of this approach.

Stability theory starts in Chapter III with the fundamental order, heirs and definable types, and continues in Chapter IV with forking and the open mapping and finite equivalence relation theorems. Chapter V introduces U-rank, Morley rank and the continuous rank R(p,L,\(\infty)\). It ends with an example in algebraically closed fields due to C. Berline, demonstrating the connection between the Krull dimension of a prime ideal and the U-rank of the associated type. Chapter VI deals with indiscernibles, prime and saturated models and \(\aleph_{\epsilon}\)- saturated models.

Chapter VII leads into the second half of the book, which obtains a classification of particular classes of theories. Starting with Morley’s theorem and the Baldwin-Lachlan theorem, the reader learns to classify models by their dimensions, in this case the dimension of the strongly minimal sets. Chapter VIII generalizes this approach to superstable theories. The author introduces domination, regular types, orthogonality, dimensions of regular types and weight. Lachlan’s theorem on the number of countable models is obtained as an application. Chapter IX contains the corresponding results for \(\omega\)-stable theories and the Rudin- Keisler (R-K) order. In Chapter X the author introduces bounded types, which are later used to give his new (but equivalent to Shelah’s) definitions of nonmultidimensional (bounded) theories and depth. For an \(\omega\)-stable bounded theory he classifies the extensions of a model by the dimensions of their minimal types in the R-K-ordering. Chapter XI deals with theories which do not have the dimensional order property (NDOP theories), the presentable theories. Upper bounds are given for the number of \((\aleph_{\epsilon}\)-saturated) models in the \(\omega\)-stable [superstable] case. The lower bounds are not dealt with. The exposition concludes with two extensive examples: the first for the DOP when the classification fails; the second for a presentable theory.

The book contains a subject index and a notation index. There are only a few examples, which for the most part are fully explained. Exercises are very rarely suggested. The exposition is carefully streamlined to include only those topics needed for the development of the classification; e.g. Lascar’s rank inequalities are not included. Yet the presentation is not terse but rather creates a very reasonable and readable description of the landscape below the main gap.

##### MSC:

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