Set theory and its philosophy: a critical introduction

Michael Potter0199269734, 9781423788850, 9780199269730, 0199270414

Michael 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. 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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Table of contents :
Contents……Page 10
I: Sets……Page 16
Introduction to Part I……Page 18
1.1 The axiomatic method……Page 21
1.2 The background logic……Page 26
1.3 Schemes……Page 28
1.4 The choice of logic……Page 31
1.5 Definite descriptions……Page 33
Notes……Page 35
2.1 Collections and fusions……Page 36
2.2 Membership……Page 38
2.3 Russell’s paradox……Page 40
2.4 Is it a paradox?……Page 41
2.5 Indefinite extensibility……Page 42
2.6 Collections……Page 45
Notes……Page 47
3.1 Two strategies……Page 49
3.2 Construction……Page 51
3.3 Metaphysical dependence……Page 53
3.4 Levels and histories……Page 55
3.5 The axiom scheme of separation……Page 57
3.6 The theory of levels……Page 58
3.7 Sets……Page 62
3.8 Purity……Page 65
3.9 Well-foundedness……Page 66
Notes……Page 68
4.1 How far can you go?……Page 70
4.2 The initial level……Page 72
4.3 The empty set……Page 73
4.4 Cutting things down to size……Page 75
4.5 The axiom of creation……Page 76
4.6 Ordered pairs……Page 78
4.7 Relations……Page 80
4.8 Functions……Page 82
4.9 The axiom of infinity……Page 83
4.10 Structures……Page 87
Notes……Page 90
Conclusion to Part I……Page 91
II: Numbers……Page 94
Introduction to Part II……Page 96
5.1 Closure……Page 103
5.2 Definition of natural numbers……Page 104
5.3 Recursion……Page 107
5.4 Arithmetic……Page 110
5.5 Peano arithmetic……Page 113
Notes……Page 116
6.1 Order relations……Page 118
6.2 The ancestral……Page 121
6.3 The ordering of the natural numbers……Page 123
6.4 Counting finite sets……Page 125
6.5 Counting infinite sets……Page 128
6.6 Skolem’s paradox……Page 129
Notes……Page 131
7.1 The rational line……Page 132
7.2 Completeness……Page 134
7.3 The real line……Page 136
7.4 Souslin lines……Page 140
7.5 The Baire line……Page 141
Notes……Page 143
8.1 Equivalence relations……Page 144
8.2 Integral numbers……Page 145
8.3 Rational numbers……Page 147
8.4 Real numbers……Page 150
8.5 The uncountability of the real numbers……Page 151
8.6 Algebraic real numbers……Page 153
8.7 Archimedean ordered fields……Page 155
8.8 Non-standard ordered fields……Page 159
Notes……Page 162
Conclusion to Part II……Page 164
III: Cardinals and Ordinals……Page 166
Introduction to Part III……Page 168
9.1 Definition of cardinals……Page 170
9.2 The partial ordering……Page 172
9.3 Finite and infinite……Page 174
9.4 The axiom of countable choice……Page 176
Notes……Page 180
10.1 Finite cardinals……Page 182
10.2 Cardinal arithmetic……Page 183
10.3 Infinite cardinals……Page 185
10.4 The power of the continuum……Page 187
Notes……Page 189
11.1 Well-ordering……Page 190
11.2 Ordinals……Page 194
11.3 Transfinite induction and recursion……Page 197
11.4 Cardinality……Page 199
11.5 Rank……Page 201
Notes……Page 204
12.1 Normal functions……Page 206
12.2 Ordinal addition……Page 207
12.3 Ordinal multiplication……Page 211
12.4 Ordinal exponentiation……Page 214
12.5 Normal form……Page 217
Notes……Page 219
Conclusion to Part III……Page 220
IV: Further Axioms……Page 222
Introduction to Part IV……Page 224
13 Orders of infinity……Page 226
13.1 Goodstein’s theorem……Page 227
13.2 The axiom of ordinals……Page 233
13.3 Reflection……Page 236
13.4 Replacement……Page 240
13.5 Limitation of size……Page 242
13.6 Back to dependency?……Page 245
13.7 Higher still……Page 246
13.8 Speed-up theorems……Page 249
Notes……Page 251
14.1 The axiom of countable dependent choice……Page 253
14.2 Skolem’s paradox again……Page 255
14.3 The axiom of choice……Page 257
14.4 The well-ordering principle……Page 258
14.5 Maximal principles……Page 260
14.6 Regressive arguments……Page 265
14.7 The axiom of constructibility……Page 267
14.8 Intuitive arguments……Page 271
Notes……Page 274
15.1 Alephs……Page 276
15.2 The arithmetic of alephs……Page 277
15.3 Counting well-orderable sets……Page 278
15.4 Cardinal arithmetic and the axiom of choice……Page 281
15.5 The continuum hypothesis……Page 283
15.6 Is the continuum hypothesis decidable?……Page 285
15.7 The axiom of determinacy……Page 290
15.8 The generalized continuum hypothesis……Page 295
Notes……Page 298
Conclusion to Part IV……Page 299
Appendices……Page 304
A.1 Zermelo’s axioms……Page 306
A.2 Cardinals and ordinals……Page 307
A.3 Replacement……Page 311
Notes……Page 313
B: Classes……Page 314
B.1 Virtual classes……Page 315
B.2 Classes as new entities……Page 317
B.3 Classes and quantification……Page 318
B.4 Classes quantified……Page 321
B.5 Impredicative classes……Page 322
B.6 Impredicativity……Page 323
B.7 Using classes to enrich the original theory……Page 325
C.1 Adding classes to set theory……Page 327
C.2 The difference between sets and classes……Page 328
C.3 The metalinguistic perspective……Page 330
Notes……Page 331
References……Page 332
List of symbols……Page 351
C……Page 353
H……Page 354
O……Page 355
S……Page 356
Z……Page 357
G……Page 358
Q……Page 359
Z……Page 360