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영문목차
Ⅰ. Sets=1
Introduction to Part Ⅰ=3
1. Logic=6
1.1. The axiomatic method=6
1.2. The background logic=11
1.3. Schemes=13
1.4. The choice of logic=16
1.5. Definite descriptions=18
Notes=20
2. Collections=21
2.1. Collections and fusions=21
2.2. Membership=23
2.3. Russell's paradox=25
2.4. Is it a paradox?=26
2.5. Indefinite extensibility=27
2.6. Collections=30
Notes=32
3. The hierarchy=34
3.1. Two strategies=34
3.2. Construction=36
3.3. Metaphysical dependence=38
3.4. Levels and histories=40
3.5. The axiom scheme of separation=42
3.6. The theory of levels=43
3.7. Sets=47
3.8. Purity=50
3.9. Well-foundedness=51
Notes=53
4. The theory of sets=55
4.1. How far can you go?=55
4.2. The initial level=57
4.3. The empty set=58
4.4. Cutting things down to size=60
4.5. The axiom of creation=61
4.6. Ordered pairs=63
4.7. Relations=65
4.8. Functions=67
4.9. The axiom of infinity=68
4.13. Structures=72
Notes=75
Conclusion to Part Ⅰ=76
Ⅱ. Numbers=79
Introduction to Part Ⅱ=81
5. Arithmetic=88
5.1. Closure=88
5.2. Definition of natural numbers=89
5.3. Recursion=92
5.4. Arithmetic=95
5.5. Peano arithmetic=98
Notes=101
6. Counting=103
6.1. Order relations=103
6.2. The ancestral=106
6.3. The ordering of the natural numbers=108
6.4. Counting finite sets=110
6.5. Counting infinite sets=113
6.6. Skolem's paradox=114
Notes=116
7. Lines=117
7.1. The rational line=117
7.2. Completeness=119
7.3. The real line=121
7.4. Souslin lines=125
7.5. The Baire line=126
Notes=128
8. Real numbers=129
8.1. Equivalence relations=129
8.2. Integral numbers=130
8.3. Rational numbers=132
8.4. Real numbers=135
8.5. The uncountability of the real numbers=136
8.6. Algebraic real numbers=138
8.7. Archimedean ordered fields=140
8.8. Non-standard ordered fields=144
Notes=147
Conclusion to Part Ⅱ=149
Ⅲ. Cardinals and Ordinals=151
Introduction to Part Ⅲ=153
9. Cardinals=155
9.1. Definition of cardinals=155
9.2. The partial ordering=157
9.3. Finite and infinite=159
9.4. The axiom of countable choice=161
Notes=165
10. Basic cardinal arithmetic=167
10.1. Finite cardinals=167
10.2. Cardinal arithmetic=168
10.3. Infinite cardinals=170
10.4. The power of the continuum=172
Notes=174
11. Ordinals=175
11.1. Well-ordering=175
11.2. Ordinals=179
11.3. Transfinite induction and recursion=182
11.4. Cardinality=184
11.5. Rank=136
Notes=189
12. Ordinal arithmetic=191
12.1. Normal functions=191
12.2. Ordinal addition=192
12.3. Ordinal multiplication=196
12.4. Ordinal exponentiation=199
12.5. Normal form=202
Notes=204
Conclusion to Part Ⅲ=205
Ⅳ. Further Axioms=207
Introduction to Part Ⅳ=209
13. Orders of infinity=211
13.1. Goodstein's theorem=212
13.2. The axiom of ordinals=218
13.3. Reflection=221
13.4. Replacement=225
13.5. Limitation of size=227
13.6. Back to dependency?=230
13.7. Higher still=231
13.8. Speed-up theorems=234
Notes=236
14. The axiom of choice=238
14.1. The axiom of countable dependent choice=238
14.2. Skolem's paradox again=240
14.3. The axiom of choice=242
14.4. The well-ordering principle=243
14.5. Maximal principles=245
14.6. Regressive arguments=250
14.7. The axiom of constructibility=252
14.8. Intuitive arguments=256
Notes=259
15. Further cardinal arithmetic=261
15.1. Alephs=261
15.2. The arithmetic of alephs=262
15.3. Counting well-orderable sets=263
15.4. Cardinal arithmetic and the axiom of choice=266
15.5. The continuum hypothesis=268
15.6. Is the continuum hypothesis decidable?=270
15.7. The axiom of determinacy=273
15.8. The generalized continuum hypothesis=280
Notes=283
Conclusion to Part Ⅳ=284
Appendices=289
A. Traditional axiomatizations=291
A.1. Zermelo's axioms=291
A.2. Cardinals and ordinals=292
A.3. Replacement=296
Notes=298
B. Classes=299
B.1. Virtual classes=300
B.2. Classes as new entities=302
B.3. Classes and quantification=303
B.4. Classes quantified=306
B.5. Impredicative classes=307
B.6. Impredicativity=308
B.7. Using classes to enrich the original theory=310
C. Sets and classes=312
C.1. Adding classes to set theory=312
C.2. The difference between sets and classes=313
C.3. The metalinguistic perspective=315
Notes=316
References=317
List of symbols=329
Index of definitions=331
Index of names=336
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