Problems and Theorems in Classical Set Theory

Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals.- Ordinal arithmetic.- Cardinals.- Partially ordered sets.- Transfinite enumeration.- Euclidean spaces.- Zorn's lemma.- Hamel bases.- The continuum hypothesis.- Ultrafilters on w.- Families of sets.- The Banach-Tarski paradox.- Stationary sets in w1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- Triangle systems.- Set mappings.- Trees.- The measure problem.- Stationary sets.- The axiom of choice.- Well founded sets and the axiom of foundation.- Solutions: Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals

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 $[\lambda]^{<\kappa}$ is the union of $\kappa$ bounded sets, and in Problem 20 to prove Baumgartner's result that every closed unbounded subset of $[\omega_2]^{<\aleph_1}$ 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)