Problems and Theorems in Classical Set Theory

This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

From the reviews: "The volume contains 1007 problems in (mostly combinatorial) set theory. As indicated by the authors, "most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come from the period, say, 1920--1970. Many problems are also related to other fields of mathematics such as algebra, combinatorics, topology and real analysis." And indeed the topics covered include applications of Zorn's lemma, Euclidean spaces, Hamel bases, the Banach-Tarski paradox and the measure problem. The statement of the problems, which are distributed among 31 chapters, takes 132 pages, and the (fairly detailed) solutions (together with some references) another 357 pages. Some problems are elementary but most of them are challenging. For example, in Chapter 29 the reader is asked in Problem 1 to show that $[\lambda]^{^Baumgartner's result that every closed unbounded subset of $[\omega_2]^{^leph_1}$ is of maximal cardinality $\aleph_2^{\aleph_0}$. This is a welcome addition to the literature, which should be useful to students and researchers alike." (Pierre Matet, Mathematical Reviews) "The book is well written and self contained, a choice collection of hundreds of tastefully selected problems related to classical set theory, a wealth of naturally arising, simply formulated problems … . It is certainly available to students of mathematics major even in their undergraduate years. The solutions contain the right amount of details for the targeted readership. … This is a unique book, an excellent source to review the fundamentals of classical set theory, learn new tricks, discover more and more on the field." (Tamás Erdélyi, Journal of Approximation Theory, 2008)