## The Joy of Sets: Fundamentals of Contemporary Set TheoryThis book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a 'naive' manner. Here the notions of 'set', 'union', 'intersection', 'power set', 'rela tion', 'function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast. |

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### Contents

Naive Set Theory | 1 |

12 Operations on Sets | 4 |

13 Notation for Sets | 6 |

14 Sets of Sets | 7 |

15 Relations | 10 |

16 Functions | 12 |

17 WellOrderings and Ordinals | 16 |

18 Problems | 25 |

42 Closed Unbounded Sets | 103 |

43 Stationary Sets and Regressive Functions | 106 |

44 Trees | 109 |

45 Extensions of Lebesgue Measure | 113 |

46 A Result About the GCH | 116 |

The Axiom of Constructibility | 120 |

52 The Constructible Hierarchy | 123 |

53 The Axiom of Constructibility | 124 |

The ZermeloPraenkel Axioms | 29 |

21 The Language of Set Theory | 30 |

22 The Cumulative Hierarchy of Sets | 35 |

23 The ZermeloFraenkel Axioms | 40 |

24 Classes | 46 |

25 Set Theory as an Axiomatic Theory | 50 |

26 The Recursion Principle | 51 |

27 The Axiom of Choice | 56 |

28 Problems | 63 |

Ordinal and Cardinal Numbers | 66 |

32 Addition of Ordinals | 68 |

33 Multiplication of Ordinals | 69 |

34 Sequences of Ordinals | 71 |

35 Ordinal Exponentiation | 74 |

36 Cardinality Cardinal Numbers | 75 |

37 Arithmetic of Cardinal Numbers | 82 |

38 Regular and Singular Cardinals | 88 |

39 Cardinal Exponentiation | 91 |

310 Inaccessible Cardinals | 95 |

311 Problems | 98 |

Topics in Pure Set Theory | 101 |

54 The Consistency of V L | 127 |

55 Use of the Axiom of Constructibility | 128 |

Independence Proofs in Set Theory | 130 |

63 The BooleanValued Universe | 133 |

64 VB and V | 136 |

65 BooleanValued Sets and Independence Proofs | 137 |

66 The Nonprovability of the CH | 139 |

NonWeilFounded Set Theory | 143 |

71 SetMembership Diagrams | 145 |

72 The AntiFoundation Axiom | 151 |

73 The Solution Lemma | 156 |

74 Inductive Definitions Under AFA | 159 |

75 Graphs and Systems | 163 |

76 Proof of the Solution Lemma | 168 |

77 CoInductive Definitions | 169 |

78 A Model of ZF +AFA | 173 |

185 | |

Glossary of Symbols | 186 |

189 | |

### Other editions - View all

The Joy of Sets: Fundamentals of Contemporary Set Theory Keith Devlin,Professor Keith Devlin No preview available - 1993 |

### Common terms and phrases

abbreviates assume Axiom of Choice Axiom of Constructibility Axiom of Foundation Axiom of Infinity Axiom of Replacement Axiom of Subset axioms of set basic bijection binary relation bisimulation boolean algebra boolean-valued chM(a Clearly cofinal concept consistent constructible set theory Corollary countable define denote easily seen elements equivalence existence fact finite sets fixed-point formula of LAST Fraenkel function f graph Hence inaccessible cardinal induction infinite cardinal isomorphic Let f limit ordinal logic mathematical maximal means non-well-founded sets nonempty sets notation notion operation ordered pair ordinal number system poset power set proper class prove recursion principle result Second edition Set Axiom set-theoretic Solution Lemma Subset Selection successor ordinal Suppose system map system of equations tagged Theorem theory of sets top node uncountable unique decoration well-founded well-ordering woset Zermelo hierarchy Zermelo-Fraenkel set theory ZFC axioms ZFCA

### Popular passages

Page 185 - London, 1987. [3] JL Bell, Boolean- Valued Models and Independence Proofs in Set Theory, Oxford University Press, London, 1977.