The Joy of Sets: Fundamentals of Contemporary Set Theory (Undergraduate Texts in Mathematics)

This book is intended to provide an account of those parts of contemporary set theory that are relevant to other areas of pure mathematics. Intended for advanced undergraduates and beginning graduate students, the text is written in an easy-going style, with a minimum of formalism. The book begins with a review of "naive" set theory; it then develops the Zermelo-Fraenkel axioms of the theory, showing how they arise naturally from a rigorous answer to the question, "what is a set?" After discussing the ordinal and cardinal numbers, the book then delves into contemporary set theory, covering such topics as: the Borel hierarchy, stationary sets and regressive functions, and Lebesgue measure. Two chapters present an extension of the Zermelo-Fraenkel theory, discussing the axiom of constructibility and the question of provability in set theory. A final chapter presents an account of an alternative conception of set theory that has proved useful in computer science, the non-well-founded set theory of Peter Aczel. The author is a well-known mathematician and the editor of the "Computers in Mathematics" column in the AMS Notices and of FOCUS, the magazine published by the MAA. "This book provides an account of those parts of contemporary set theory that are relevant to other areas of pure mathematics. Intended for advanced undergraduates and beginning graduate students, the text is written in an easy-going style, with a minimum of formalism. The book begins with a review of "naive" set theory; it then develops the Zermelo-Fraenkel axioms of the theory, showing how they arise naturally from a rigorous answer to the question, "what is a set?" After discussing the ordinal and cardinal numbers, the book then delves into contemporary set theory, covering such topics as: the Borel hierarchy, stationary sets and regressive functions, and Lebesgue measure. Two chapters present an extension of the Zermelo-Fraenkel theory, discussing the axiom of constructibility and the question of provability in set theory. A final chapter presents an account of an alternative conception of set theory that has proved useful in computer science, the non-well-founded set theory of Peter Aczel"--Back cover This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a'naive'manner. Here the notions of'set','union','intersection','power set','rela­ tion','function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con­ sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast. This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "nave" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science. Keith Devlin. Rev. Ed. Of: Fundamentals Of Contemporary Set Theory. C1979. Includes Bibliographical References (p. 185) And Index. Zermelo-Fraenkel set theory, which forms the main topic of the book, is a rigorous theory, based on a precise set of axioms.