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Categoricity. (English) Zbl 1183.03002

University Lecture Series 50. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4893-7/pbk). xi, 235 p. (2009).
Morley’s 1965 proof of categoricity in power was the beginning of modern model theory. This result, that a first-order theory categorical in one uncountable cardinality is categorical in all uncountable cardinalities, changed the focus from logics to theories and led to Shelah’s stability classification of first-order theories, which in turn formed the background of much of subsequent model-theoretic research.
In the volume under review, John Baldwin carefully expounds the attempts of Shelah and others to extend the analysis of first-order theories to classes that are not first-order, such as sentences of \(L_{\omega_1,\omega}\) and \(L_{\omega_1,\omega}(Q)\) (where \(Q\) is the quantifier “uncountably many”). Baldwin’s goal is “to provide a systematic and intelligible account of some of the central aspects of Shelah’s work and related developments.” He proceeds on two tracks: a study of specific logics, such as those cited above, and the study of Abstract Elementary Classes – AEC. An interesting side-effect of Baldwin’s efforts is the isolation of the set-theoretic requirements of his exposition. These are either ZFC or ZFC plus weak variants of the Generalized Continuum Hypothesis.
The text itself is divided into four parts; each part consists of short chapters. The operational goal is that a chapter can be covered in one or two lectures. Part 1 introduces Zilber’s quasiminimal excellent classes and presents a concrete notion of excellence for a combinatorial geometry. Baldwin also examines in detail the example of complex exponentiation. Part 2 develops the basic properties of AECs and includes Shelah’s theorem that every AEC is a pseudoelementary class. The final chapter of Part 2 gives Shelah’s ZFC proof that an \(\aleph_1\)-categorical sentence of \(L_{\omega_1,\omega}(Q)\) has a model of power \(\aleph_2\). Part 3 is mostly devoted to “reasonably well-behaved classes”, i.e., those that have amalgamation and arbitrarily large models. The conjecture under study is that for such classes categoricity should be eventually true or eventually false. AECs without amalgamation are considered near the end of Part 3. Under weak diamond an AEC which is \(\kappa\)-categorical and which fails to have the amalgamation property for models of cardinality \(\kappa\) has many models of cardinality \(\kappa^{+}\). Part 4 returns to a more concrete realm: categoricity in \(L_{\omega_1,\omega}\). This involves the study of the atomic models of a complete first-order theory.
The reader is expected to have had a graduate course in model theory. Appendices A, B, and C develop particular results to assist the reader. Appendix D lists some open problems.

MSC:

03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03C35 Categoricity and completeness of theories
03C48 Abstract elementary classes and related topics
03C52 Properties of classes of models
03C95 Abstract model theory
03C98 Applications of model theory
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