## Set Theory and its Philosophy: A Critical IntroductionMichael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. |

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### Contents

3 | |

6 | |

12 The background logic | 11 |

13 Schemes | 13 |

14 The choice of logic | 16 |

15 Definite descriptions | 18 |

Notes | 20 |

Collections | 21 |

94 The axiom of countable choice | 161 |

Notes | 165 |

Basic cardinal arithmetic | 167 |

102 Cardinal arithmetic | 168 |

103 Infinite cardinals | 170 |

104 The power of the continuum | 172 |

Notes | 174 |

Ordinals | 175 |

22 Membership | 23 |

23 Russells paradox | 25 |

24 Is it a paradox? | 26 |

25 Indefinite extensibility | 27 |

26 Collections | 30 |

Notes | 32 |

The hierarchy | 34 |

32 Construction | 36 |

33 Metaphysical dependence | 38 |

34 Levels and histories | 40 |

35 The axiom scheme of separation | 42 |

36 The theory of levels | 43 |

37 Sets | 47 |

38 Purity | 50 |

39 Wellfoundedness | 51 |

Notes | 53 |

The theory of sets | 55 |

42 The initial level | 57 |

43 The empty set | 58 |

44 Cutting things down to size | 60 |

45 The axiom of creation | 61 |

46 Ordered pairs | 63 |

47 Relations | 65 |

48 Functions | 67 |

49 The axiom of infinity | 68 |

410 Structures | 72 |

Notes | 75 |

Conclusion to Part I | 76 |

Numbers | 79 |

Introduction to Part II | 81 |

Arithmetic | 88 |

52 Definition of natural numbers | 89 |

53 Recursion | 92 |

54 Arithmetic | 95 |

55 Peano arithmetic | 98 |

Notes | 101 |

Counting | 103 |

62 The ancestral | 106 |

63 The ordering of the natural numbers | 108 |

64 Counting finite sets | 110 |

65 Counting infinite sets | 113 |

66 Skolems paradox | 114 |

Notes | 116 |

Lines | 117 |

72 Completeness | 119 |

73 The real line | 121 |

74 Souslin lines | 125 |

75 The Baire line | 126 |

Notes | 128 |

Real numbers | 129 |

82 Integral numbers | 130 |

83 Rational numbers | 132 |

84 Real numbers | 135 |

85 The uncountability of the real numbers | 136 |

86 Algebraic real numbers | 138 |

87 Archimedean ordered fields | 140 |

88 Nonstandard ordered fields | 144 |

Notes | 147 |

Conclusion to Part II | 149 |

Cardinals and Ordinals | 151 |

Introduction to Part III | 153 |

Cardinals | 155 |

92 The partial ordering | 157 |

93 Finite and infinite | 159 |

112 Ordinals | 179 |

113 Transfinite induction and recursion | 182 |

114 Cardinality | 184 |

115 Rank | 186 |

Notes | 189 |

Ordinal arithmetic | 191 |

122 Ordinal addition | 192 |

124 Ordinal exponentiation | 199 |

125 Normal form | 202 |

Notes | 204 |

Conclusion to Part III | 205 |

Further Axioms | 207 |

Introduction to Part IV | 209 |

Orders of infinity | 211 |

131 Goodsteins theorem | 212 |

132 The axiom of ordinals | 218 |

133 Reflection | 221 |

134 Replacement | 225 |

135 Limitation of size | 227 |

136 Back to dependency? | 230 |

137 Higher still | 231 |

138 Speedup theorems | 234 |

Notes | 236 |

The axiom of choice | 238 |

142 Skolems paradox again | 240 |

143 The axiom of choice | 242 |

144 The wellordering principle | 243 |

145 Maximal principles | 245 |

146 Regressive arguments | 250 |

147 The axiom of constructibility | 252 |

148 Intuitive arguments | 256 |

Notes | 259 |

Further cardinal arithmetic | 261 |

152 The arithmetic of alephs | 262 |

153 Counting wellorderable sets | 263 |

154 Cardinal arithmetic and the axiom of choice | 266 |

155 The continuum hypothesis | 268 |

156 Is the continuum hypothesis decidable? | 270 |

157 The axiom of determinacy | 275 |

158 The generalized continuum hypothesis | 280 |

Notes | 283 |

Conclusion to Part IV | 284 |

Appendices | 289 |

Traditional axiomatizations | 291 |

A2 Cardinals and ordinals | 292 |

A3 Replacement | 296 |

Notes | 298 |

Classes | 299 |

B1 Virtual classes | 300 |

B2 Classes as new entities | 302 |

B3 Classes and quantification | 303 |

B4 Classes quantified | 306 |

B5 Impredicative classes | 307 |

B6 Impredicativity | 308 |

B7 Using classes to enrich the original theory | 310 |

Sets and classes | 312 |

C2 The difference between sets and classes | 313 |

C3 The metalinguistic perspective | 315 |

Notes | 316 |

317 | |

List of symbols | 336 |

338 | |

343 | |

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### Common terms and phrases

aleph algebraic argument arithmetic assert assume the axiom axiom of choice axiom of constructibility axiom of countable axiom of determinacy axiom of infinity axiom of ordinals axiom scheme axiomatization called Cantor card(A chapter claim classes collection consistent constructivist continuum hypothesis Contradiction corollary countable choice Dedekind defined definition denoted determinacy disjoint domain equinumerous equivalent example exists express finite follows formal formula fusion Godel hence hierarchy impredicative individuals infinite sets initial subset instance intuitive isomorphic large cardinal least element lemma limit ordinal logic mathematicians mathematics maximal element method natural numbers notion objects ordered field ordered pair paradox partially ordered set platonist possible Proof properties proposition provable prove quantifiers real line real numbers reason recursion regressive relation result scheme of separation second-order sentence sequence set theory set-theoretic sort structure Suppose theorem theorists theory of sets totally ordered trivial true uncountable unique well-orderable Zermelo's