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Descriptive set theory. 2nd ed. (English) Zbl 1172.03026
Mathematical Surveys and Monographs 155. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4813-5/hbk). xiv, 502 p. (2009).
The book under review is an updated edition of the famous first edition [Studies in Logic and the Foundations of Mathematics, Vol. 100. Amsterdam, New York, Oxford: North-Holland Publishing Company (1980; Zbl 0433.03025)], which became the standard reference for nearly thirty years. Many passages have been rewritten, the notation and terminology is slightly refreshed. The structure of the book is the same and only one brief section 3I (Effective theory on arbitrary (perfect) Polish spaces) on the relativization method of proof is added in Chapter 3. Two sections, 6F (The determinacy of Borel sets) and 7F (Results which depend on the Axiom of Choice), are completely rewritten. Now section 6F presents a version of Martin’s second proof of the determinacy of Borel sets [D. A. Martin, “A purely inductive proof of Borel determinacy”, in: Recursion theory, Proc. AMS-ASL Summer Inst., Ithaca/N.Y. 1982, Proc. Symp. Pure Math. 42, 303–308 (1985; Zbl 0614.03048)]. The author added the most important references to the developments in descriptive set theory since 1980 when they touch questions formulated in the book. This concerns mainly the establishment of the connection between determinacy hypotheses, large cardinals and inner model theory starting with D. A. Martin and J. R. Steel [“Projective determinacy”, Proc. Natl. Acad. Sci. USA 85, No. 18, 6582–6586 (1988; Zbl 0656.03036)] and W. H. Woodin [“Supercompact cardinals, sets of reals, and weakly homogeneous trees”, Proc. Natl. Acad. Sci. USA 85, No. 18, 6587–6591 (1988; Zbl 0656.03037)].

##### MSC:
 03E15 Descriptive set theory 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03E60 Determinacy principles 03E65 Other set-theoretic hypotheses and axioms 03E45 Inner models, including constructibility, ordinal definability, and core models 03D20 Recursive functions and relations, subrecursive hierarchies 03D55 Hierarchies of computability and definability 03D75 Abstract and axiomatic computability and recursion theory 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 26A21 Classification of real functions; Baire classification of sets and functions 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
##### Keywords:
Borel set; projective set; measurable set; determinacy