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Full Description
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.
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.
