Notes on Set Theory

Abstract

The book 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 faithful representation of mathematical objects by structured sets.