Set theory for the working mathematician. (English) Zbl 0938.03067

Ciesielski, Krzysztof

London Mathematical Society Student Texts. 39. Cambridge: Cambridge University Press. xi, 236 p. (1997).

Although the author starts from the beginning and develops the usual topics of elementary axiomatic set theory ZFC (Zermelo-Fraenkel set theory with the axiom of choice), the reader should be familiar with the intuitive set theory that is found in the standard university mathematics program. The most important part of this book consists of the more advanced topics that are usual to graduate students and research mathematicians. Here, the exposition is clear and accurate, and the author was wise to limit himself to the central results. The scope is best described by listing the contents of the last two-thirds of the book. Part III: The power of recursive definitions. Chapter 6. Subsets of \(\mathbb{R}^n\). 6.1 Strange subsets of \(\mathbb{R}^n\) and the diagonalization argument. 6.2 Closed sets and Borel sets. 6.3 Lebesgue-measurable sets and sets with the Baire property. Chapter 7. Strange real functions. 7.1 Measurable and nonmeasurable functions. 7.2 Darboux functions. 7.3 Additive functions and Hamel bases. 7.4 Symmetrically discontinuous functions. Part IV: When induction is too short. Chapter 8. Martin’s axiom. 8.1 Rasiowa-Sikorski lemma. 8.2 Martin’s axiom. 8.3 Suslin hypothesis and diamond principle. Chapter 9. Forcing. 9.1 Elements of logic and other forcing preliminaries. 9.2 Forcing method and a model for \(\neg\text{CH}\). 9.3 Model for CH and \(\diamondsuit\). 9.4 Product lemma and Cohen model. 9.5 Model for \(\text{MA}+\neg\text{CH}\).

Reviewer: E.Mendelson (Flushing)

MSC:

03Exx	Set theory
03-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03E35	Consistency and independence results
03E50	Continuum hypothesis and Martin’s axiom
03-02	Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Keywords:

subsets of \(\mathbb{R}^n\); strange real functions; forcing; elementary axiomatic set theory ZFC; recursive definitions; induction; Martin’s axiom