Synopses & Reviews
This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.
Review
"The book contains a large amount of interesting and well-presented material on transfinite recursion and transfinite induction and a very good introduction to Martin's axiom. The short discussion of forcing can give the reader a first glimplse at the method. The treatment of mathematical logic is too sketchy to be of any realuse and may be a source of confusion." Journal of Symbolic Logic
Review
"...a very valuable resource for the working mathematician. Postgraduates and established researchers in many (perhaps all) areas of mathematics will benefit from reading it." Proceedings of the Edinburgh Mathematical Society
Synopsis
Presents those methods of modern set theory most applicable to other areas of pure mathematics.
Synopsis
Presents those methods of modern set theory most applicable to other areas of pure mathematics.
Description
Includes bibliographical references (p. 225-227) and index.
Table of Contents
Part I. Basics of Set Theory: 1. Axiomatic set theory; 2. Relations, functions and Cartesian product; 3. Natural, integer and real numbers; Part II. Fundamental Tools of Set Theory: 4. Well orderings and transfinite induction; 5. Cardinal numbers; Part III. The Power of Recursive Definitions: 6. Subsets of Rn; 7. Strange real functions; Part IV. When Induction is Too Short: 8. Martin's axiom; 9. Forcing; Part V. Appendices: A. Axioms of set theory; B. Comments on forcing method; C. Notation.