John L. BELL. Set theory: Boolean-valued models and independence proofs. Oxford: Clarendon press, 2005. Oxford logic guides, no. 47. pp. XXII + 191. ISBN 0-19-856852-5, 987-0-19-856852-0 (pbk) [Book Review]

This is the third edition of a book originally published in the 1970s; it provides a systematic and nicely organized presentation of the elegant method of using Boolean-valued models to prove independence results. Four things are new in the third edition: background material on Heyting algebras, a chapter on ‘Boolean-valued analysis’, one on using Heyting algebras to understand intuitionistic set theory, and an appendix explaining how Boolean and Heyting algebras look from the perspective of category theory. The book presents results from a number of set theorists and includes an insightful and informative foreword by Dana Scott. Bell's presentation is lively and pleasant to read, and the material is given in a nicely cohesive way.One obvious reason to be interested in independence proofs is that they concern the important question, what is the set-theoretic hierarchy like? The proofs in Bell's book cover some of the most basic and fundamental independence results, such as those concerning the size of the continuum, the independence of the Axiom of Choice from ZF, cardinal collapsing, Souslin's hypothesis, and Martin's Axiom.Since Gödel's incompleteness theorem, it has been known that for any serious candidate list of set-theoretic axioms, there will be statements neither provable nor disprovable from those axioms. What is really interesting, however, and what the independence proofs discussed here show, is how many of the most natural questions about sets are not decided by the standard axioms of ZFC, and how …