Publisher's Synopsis
This text covers the basic facts about abstract sets, including the axiom of choice, transfinite recursion, cardinals, ordinals and the cumulative hierarchy of well founded sets. It also includes a chapter on Baire space, focusing on results of interest to analysts and introducing the reader to the continuum problem; an appendix with a reasonably detailed construction of the real numbers; and a second appendix introducing set universes, which satisfy conditions that include Aczel's Antifoundation.;Most of the results are derived within Zermelo-Fraenkel set theory with depended choices, which allows atoms and non-well founded sets, with the full axiom of choice and the axiom of foundation assumed explicitly where needed. To clarify the role of set theory as a foundation of mathematics - including computation theory - the book uses the notion of the faithful representation of mathematical objects by structured sets.
