## Set Theory for the Working MathematicianThis 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 |

225 | |

229 | |

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### Common terms and phrases

A/-generic algebra antichain arbitrary axiom of choice belongs bijection Borel 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 interval 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 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 recursion satisfies scheme axiom sequence set theory smallest element strictly increasing strongly Darboux Suslin line transfinite induction uncountable union well-ordered set ZFC axioms