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1.Introduction 1

Problems for Chapter 1 5

2.Equinumerosity 7

Countable unions of countable sets 9

The reals are uncountable 11

A ＜c P（A） 15

Schroder-Bernstein Theorem 16

Problems for Chapter 2 18

3.Paradoxes and axioms 19

The Russell paradox 21

Axioms （Ⅰ） - （Ⅱ） 24

Axioms for definite conditions and operations 27

Classes 28

Problems for Chapter 3 31

4.Are sets all there is？ 33

Ordered pairs 35

Disjoint union 36

Relations 37

Equivalence relations 38

Functions 39

Cardinal numbers 43

Structured sets 45

Problems for Chapter 4 46

5.The natural numbers 53

Existence of the Natural Numbers 54

Uniqueness of the Natural Numbers 54

Recursion Theorem 55

Addition and multiplication 59

Pigeonhole Principle 64

Strings 67

The continuum 69

Problems for Chapter 5 69

6.Fixed points 73

Posets 73

Partial functions 76

Inductive posets 77

Continuous Least Fixed Point Theorem 79

About topology 81

Graphs 85

Problems for Chapter 6 86

Streams 87

Scott topology 91

Directed-complete posets 91

7.Well ordered sets 93

Transfinite induction 98

Transfinite recursion 100

Iteration Lemma 100

Comparability of well ordered sets 104

Wellfoundedness of ≤0 105

Hartogs’ Theorem 106

Fixed Point Theorem 108

Least Fixed Point Theorem 108

Problems for Chapter 7 110

8.Choices 117

Axiom of Choice 117

Equivalents of AC 120

Countable Principle of Choice， ACN 122

Axiom （Ⅵ） of Dependent Choices， DC 122

The axiomatic theories ZDC， ZAC 125

Consistency and independence results 126

Problems for Chapter 8 127

9.Choice’s consequences 131

Trees 132

Konig’s Lemma 133

Fan Theorem 134

Wellfoundedness of ＜c 134

Best wellorderings 135

Absorption laws 138

Konig’s Theorem 140

Coninality，regular cardinals 141

Problems for Chapter 9 142

10.Baire space 147

Cardinality of perfect pointsets 150

Cantor-Bendixson Theorem 151

Property P 152

Analytic pointsets 153

Perfect Set Theorem 157

Borel sets 160

Counterexample to the general property P 162

Consistency and independence results 164

Problems for Chapter 10 165

Borelisomorphisms 166

11.Replacement and other axioms 169

Replacement Axiom （Ⅷ） 170

The axiomatic theories ZFDC， ZFAC 170

Grounded Recursion Theorem 172

Transitive classes 174

Basic Closure Lemma 175

Hereditarily 176

nite sets 176

Zermelo universes 177

The least Zermelo universe 179

Grounded sets 180

Principle of Foundation 180

The axiomatic theory Zermelo-F raenkel， ZFC 181

Z-F universes 183

von Neumann’s class V 183

Mostowski Collapsing Lemma 183

Consistency and independence results 184

Problems for Chapter 11 185

12.Ordinal numbers 189

Characterization of the ordinal assignment 193

Characterization of the ordinals 194

Ordinal recursion 197

Ordinal addition， multiplication 197

von Neumann cardinals 198

The operation ？ 200

The cumulative rank hierarchy 201

Problems for Chapter 12 203

The operation ？α 205

Strongly inaccessible cardinals 206

Frege cardinals 206

Quotients of equivalence conditions 207

A.The real numbers 209

Congruences 209

Fields 211

Ordered Fields 212

Uniqueness of the rationals 214

Existence of the rationale 215

Countable， dense， linear orderings 219

The archimedean property 221

Nested interval property 226

Dedekind cuts 229

Existence of the real numbers 231

Uniqueness of the real numbers 234

Problems for Appendix A 236

B.Axioms and universes 239

Set universes 242

Propositions and relativizations 243

Rieger universes 248

Rieger’s Theorem 248

Antifoundation Principle， AFA 254

Bisimulations 255

The antifounded universe 259

Aczel’s Theorem 259

Problems for Appendix B 262

Index 267