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Introduction to set theory. 2nd ed., rev. and expanded. (English) Zbl 0555.03016
[For a review of the first edition (1978) see Zbl 0378.04001.]
One of the better middle-level set theory texts on the market, this second edition is a marked improvement on the first. The reviewer will list some of the features which the potential user may either approve of or disapprove of. (Philosophies of and attitudes towards mathematics seem to play a great role in what is taught in set theory courses and in how it is taught.)
1. As in the first edition, the basic material is covered thoroughly and well. The approach is axiomatic, rather than naive.
2. Other than \(2^{\omega}\), uncountable cardinals are not emphasized until late in the book, in the next to the last chapter, which is essentially combinatorics.
3. Modern set theory (e.g. forcing, L, large cardinals, determinacy) is discussed in the final chapter and, other than a few remarks here and there, is not integrated into the rest of the text.
4. Descriptive set theory (e.g. Borel sets, perfect sets) appears through much of the book on as a way of applying set theory to a familiar mathematical object without getting into more modern and difficult material. As a corollary, much of the student’s time is spent learning descriptive set theory.
5. In keeping with the second author’s research interests, the combinatorics chapter emphasizes ideals, measure, and large cardinals. Partition calculus appears very briefly (Ramsey’s theorem as an application of filters), Aronszajn trees and Suslin trees not at all.
6. The applications of the axiom of choice (there are six of them) are mostly in analysis (e.g. the Hahn-Banach theorem, or the equivalence of different definitions of continuity). Combinatorial applications (e.g. the existence of non-principal ultrafilters) are either missing or occur in the next to the last chapter.
7. The axiom of choice is introduced well after cardinals, and cardinality without choice (e.g. the Hartogs number) is treated more thoroughly than in most texts.
Reviewer: J.Roitman

03Exx Set theory
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