Elements of Set Theory

Abstract

This book, like almost every other modern mathematics book, develops its subject matter assuming a knowledge of elementary set theory. This assumption is often not justified. In this chapter we offer an informal exposition of set theory. In §1–6, we present the axioms for set theory and develop enough of elemen­tary set theory to formalize basic number theory including, for example, the uniqueness of the natural numbers up to isomorphism as a structure satisfying Peano’s famous axioms for successor as well as the existence and uniqueness of functions defined by induction or recursion on the natural numbers. We also present the basic set-theoretic notions such as functions, sequences and orders needed elsewhere in this book. In §7, we develop the basic notions of cardinality (size) for finite and countable sets. The remaining sections (8–11) establish the principle of transfinite induction and develop the basic theory of infinite ordinals and cardinals, the cumulative hierarchy of sets that forms the natural model of set theory, as well as some of the usual variants of the axiom of choice. Together, this material should supply a sufficient background in set theory for almost any graduate course of studies in computer science or mathematics. The exposition is independent of the rest of the book and repeats a small amount of material fromI.1and the historical appendix.