# Set Theory for the Working Mathematician

Cambridge University Press, Aug 28, 1997 - Mathematics - 236 pages
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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### Contents

 Axiomatic set theory 3 12 The language and the basic axioms 6 Relations functions and Cartesian product 12 22 Functions and the replacement scheme axiom 16 23 Generalized union intersection and Cartesian product 19 24 Partial and linearorder relations 21 Natural numbers integers and real numbers 25 32 Integers and rational numbers 30
 63 Lebesguemeasurable sets and sets with the Baire property 98 Strange real functions 104 72 Darboux functions 106 73 Additive functions and Hamel bases 111 74 Symmetrically discontinuous functions 118 When induction is too short 127 Martins axiom 129 82 Martins axiom 139

 33 Real numbers 31 Fundamental tools of set theory 35 Well orderings and transfinite induction 37 42 Ordinal numbers 44 43 Definitions by transfinite induction 49 44 Zorns lemma in algebra analysis and topology 54 Cardinal numbers 61 52 Cardinal arithmetic 68 53 Cofinality 74 The power of recursive definitions 77 Subsets of ℝⁿ 79 62 Closed sets and Borel sets 89
 83 Suslin hypothesis and diamond principle 154 Forcing 164 92 Forcing method and a model for CH 168 93 Model for CH and 182 94 Product lemma and Cohen model 189 95 Model for MA+CH 196 A Axioms of set theory 211 B Comments on the forcing method 215 C Notation 220 References 225 Index 229 Copyright