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Classification theory and the number of non-isomorphic models. 2nd rev. ed. (English) Zbl 0713.03013

Studies in Logic and the Foundations of Mathematics, 92. Amsterdam etc.: North-Holland. xxxiv, 705 p. $ 242.75; Dfl. 425.00 (1990).
Classification theory is today one of the most important parts of model theory. Its progress was stimulated mainly by the work of Shelah. His book on classification theory, which appeared first in 1978 (see Zbl 0388.03009), has become one of the main sources for further research. Now, twelve years later, the revised edition is available. Four new chapters have been added which give successful applications of the techniques developed in stability theory.
Classification theory deals with the structure of models. Its main tasks are to determine the number I(\(\lambda\),T) of non-isomorphic models of T of cardinality \(\lambda\), and to find invariants which characterize models up to isomorphism. Research in this direction was stimulated by the categoricity problem for theories. A theory is categorical in \(\lambda\) if all its models of cardinality \(\lambda\) are isomorphic. It was shown by Morley in 1965 that a countable theory which is categorical in one uncountable cardinality, is categorical in every uncountable cardinality. Morley investigated the rank of types. This concept led to the development of stability theory and classification theory. Especially the contributions of the author led to progress in this direction.
Chapter I gives the necessary preliminaries, such as compactness, Löwenheim-Skolem theorems, indiscernible sequences and the notion of stability.
Chapter II deals with different notions of rank. The finite cover property, the strict order property and the independence property are introduced. In this chapter a complete description of the stability spectrum for countable theories is given.
In Chapter III the author mainly deals with uncountable theories. The notion of forking of a type over a set of individuals is introduced. In § 1-§ 5 the author deals with stable T only, in § 7 unstable theories are investigated.
In Chapter IV prime models are considered. This is done in an axiomatic setting. The author shows that there are five kinds of prime models. He gives a characterization and uniqueness theorem for prime models.
In Chapter V the author goes on with types and saturated models. Here he deals with stable theories. He investigates regular, orthogonal and minimal types. This chapter contains the deepest results concerning types. The author also investigates cardinality quantifiers and two- cardinal theorems.
Chapter VI deals with ultraproducts. Here the author shows that his classification can be used to characterize Keisler’s order. Also categoricity of pseudo-elementary classes and saturation of ultraproducts is considered.
In Chapter VII the author goes on with the construction of models. He generalizes \(\lambda\)-saturated and Ehrenfeucht-Mostowski models and gives models which are generated by trees of indiscernibles. These models are of importance for unsuperstable theories. He also considers Hanf numbers and omitting types.
In Chapter VIII the number of non-isomorphic models in pseudo-elementary classes is investigated. Using methods from Chapter VII the author constructs many non-isomorphic models. It is shown that for T unsuperstable with \(| T| <\lambda\) there are \(2^{\lambda}\) non- isomorphic models of cardinality \(\lambda\).
Chapter IX is a continuation of Chapter V. Here the categoricity theorems are shown. For some cases the number of models is computed. The author tries to compute the number of the dimension (or dimensions).
In Chapter X the author introduces the dop (dimensional order property) and the notions deep and shallow. If dop holds, T has many \(\aleph_{\epsilon}\)-saturated models, if it fails, there is a structure theory.
In Chapter XI the author shows the decomposition theorems which are needed in the following.
In Chapter XII the book’s main theorem is proved: The Main Gap Theorem: Let T be countable.
(1) If T is not superstable or (is superstable and) deep or with the dop or the otop, then for every uncountable \(\lambda\), \(I(\lambda,T)=2^{\lambda}.\)
(2) If T is shallow, superstable without the dop and without the otop, then for every \(\alpha >0\), \(I(\aleph_{\alpha},T)<\beth_{\omega_ 1}(| \alpha |).\)
In Chapter XIII the author justifies that the Main Gap Theorem is really so important. So he shows that if T is superstable, shallow without dop and otop, then any model can be characterized up to isomorphism by generalized cardinal invariants of countable depth. He shows that if T is superstable, deep without dop and otop, then a weaker result is best possible. In § 3 he proves the Morley conjecture: For countable first order T, \(\aleph_ 0<\lambda <\mu\) implies I(\(\lambda\),T)\(\leq I(\mu,T)\) except when T is categorical in \(\aleph_ 1.\)
In an appendix the reader can find the necessary results from set theory. Many exercises are scattered about the text. The book contains historical remarks, notes added in proof and a list of open problems.
Reviewer: M.Weese

MSC:

03Cxx Model theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
03C20 Ultraproducts and related constructions

Citations:

Zbl 0388.03009
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