Problems and theorems in classical set theory.

*(English)*Zbl 1103.03041
Problem Books in Mathematics. New York, NY: Springer (ISBN 0-387-30293-X/hbk). xii, 514 p. (2006).

The authors have amassed a collection of 1001 problems – not counting parts – in what they call classical set theory. By “classical” they mean sans forcing and independence results. The gathered problems provide evidence that set theory is a living mathematical subject with applicability to other areas of mathematics. A number of problems touch upon other fields, such as algebra, combinatorics, topology, and real analysis. The problems are gathered by subtopic into thirty-one chapters. Most chapters have only brief introductions. There is a fourteen-page glossary of concepts and a two-page glossary of symbols, both at the end of the book. There is also an index.

The bulk of the book consists of the solutions to problems. Very few of these are just immediate from the definitions of the concepts involved in the problem statements. From the preface: “Most problems require work, wit, and inspiration \(\dots\) several of them are published results.” Some solutions also include a reference to a source. But most do not have such information.

Axiomatics have little to do with most solutions, but a reader should be familiar with ZFC. This is not a book from which to learn set theory, but rather it is a book that allows one to savor set theory.

The bulk of the book consists of the solutions to problems. Very few of these are just immediate from the definitions of the concepts involved in the problem statements. From the preface: “Most problems require work, wit, and inspiration \(\dots\) several of them are published results.” Some solutions also include a reference to a source. But most do not have such information.

Axiomatics have little to do with most solutions, but a reader should be familiar with ZFC. This is not a book from which to learn set theory, but rather it is a book that allows one to savor set theory.

Reviewer: J. M. Plotkin (East Lansing)