This volume contains the basics of Zermelo-Fraenkel axiomatic set theory. It is situated between two opposite poles: On one hand there are elementary texts that familiarize the reader with the vocabulary of set theory and build set-theoretic tools for use in courses in analysis, topology, or algebra – but do not get into metamathematical issues. On the other hand are those texts that explore issues of current research interest, developing and applying tools (constructibility, absoluteness, forcing, etc.) that are aimed to analyze the inability of the axioms to settle certain set-theoretic questions.
Much of this volume just “does set theory”, thoroughly developing the theory of ordinals and cardinals along with their arithmetic, incorporating a careful discussion of diagonalization and a thorough exposition of induction and inductive (recursive) definitions. Thus it serves well those who simply want tools to apply to other branches of mathematics or mathematical sciences in general (e.g., theoretical computer science), but alsowant to find out about some of the subtler results of modern set theory.
Moreover, a fair amount is included towards preparing the advanced reader to read the research literature. For example, we pay two visits to Gödel's constructible universe, the second of which concludes with a proof of the relative consistency of the axiom of choice and of the generalized continuum hypothesis with ZF. As such a program requires, I also include a thorough discussion of formal interpretations and absoluteness. The lectures conclude with a short but detailed study of Cohen forcing and a proof of the non-provability in ZF of the continuum hypothesis.