Set Theory for the Working Mathematician
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.
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A/-generic algebra antichain arbitrary axiom of choice belongs bijection ccc forcing cf(a choose cofinal compatible condition contains continuous function continuum hypothesis Corollary countable transitive model Darboux function define definition denoted disjoint easy equivalence relation example exists a set extends fact family f filter F finish the proof finite formula function F Hamel basis Hence implies intersects Lemma Let f let G limit ordinal linear-order relation linearly ordered set M-generic filter Martin's axiom maximal element model of ZFC Moreover natural numbers Notice obtain a contradiction one-to-one open set order isomorphism order type ordinal number P-name pairwise-disjoint partially ordered set particular Proof Let proof of Theorem proper initial segment Proposition real numbers satisfies scheme axiom set theory smallest element strictly increasing strongly Darboux Suslin line transfinite induction uncountable union well-ordered set ZFC axioms