《NOTES ON SET THEORY》

1.Introduction1

Problems for Chapter 15

2.Equinumerosity7

Countable unions of countable sets9

The reals are uncountable11

A ＜c P（A）15

Schroder-Bernstein Theorem16

Problems for Chapter 218

3.Paradoxes and axioms19

The Russell paradox21

Axioms （Ⅰ） - （Ⅱ）24

Axioms for definite conditions and operations27

Classes28

Problems for Chapter 331

4.Are sets all there is？33

Ordered pairs35

Disjoint union36

Relations37

Equivalence relations38

Functions39

Cardinal numbers43

Structured sets45

Problems for Chapter 446

5.The natural numbers53

Existence of the Natural Numbers54

Uniqueness of the Natural Numbers54

Recursion Theorem55

Addition and multiplication59

Pigeonhole Principle64

Strings67

The continuum69

Problems for Chapter 569

6.Fixed points73

Posets73

Partial functions76

Inductive posets77

Continuous Least Fixed Point Theorem79

About topology81

Graphs85

Problems for Chapter 686

Streams87

Scott topology91

Directed-complete posets91

7.Well ordered sets93

Transfinite induction98

Transfinite recursion100

Iteration Lemma100

Comparability of well ordered sets104

Wellfoundedness of ≤0105

Hartogs’ Theorem106

Fixed Point Theorem108

Least Fixed Point Theorem108

Problems for Chapter 7110

8.Choices117

Axiom of Choice117

Equivalents of AC120

Countable Principle of Choice， ACN122

Axiom （Ⅵ） of Dependent Choices， DC122

The axiomatic theories ZDC， ZAC125

Consistency and independence results126

Problems for Chapter 8127

9.Choice’s consequences131

Trees132

Konig’s Lemma133

Fan Theorem134

Wellfoundedness of ＜c134

Best wellorderings135

Absorption laws138

Konig’s Theorem140

Coninality，regular cardinals141

Problems for Chapter 9142

10.Baire space147

Cardinality of perfect pointsets150

Cantor-Bendixson Theorem151

Property P152

Analytic pointsets153

Perfect Set Theorem157

Borel sets160

Counterexample to the general property P162

Consistency and independence results164

Problems for Chapter 10165

Borelisomorphisms166

11.Replacement and other axioms169

Replacement Axiom （Ⅷ）170

The axiomatic theories ZFDC， ZFAC170

Grounded Recursion Theorem172

Transitive classes174

Basic Closure Lemma175

Hereditarily176

nite sets176

Zermelo universes177

The least Zermelo universe179

Grounded sets180

Principle of Foundation180

The axiomatic theory Zermelo-F raenkel， ZFC181

Z-F universes183

von Neumann’s class V183

Mostowski Collapsing Lemma183

Consistency and independence results184

Problems for Chapter 11185

12.Ordinal numbers189

Characterization of the ordinal assignment193

Characterization of the ordinals194

Ordinal recursion197

Ordinal addition， multiplication197

von Neumann cardinals198

The operation ？200

The cumulative rank hierarchy201

Problems for Chapter 12203

The operation ？α205

Strongly inaccessible cardinals206

Frege cardinals206

Quotients of equivalence conditions207

A.The real numbers209

Congruences209

Fields211

Ordered Fields212

Uniqueness of the rationals214

Existence of the rationale215

Countable， dense， linear orderings219

The archimedean property221

Nested interval property226

Dedekind cuts229

Existence of the real numbers231

Uniqueness of the real numbers234

Problems for Appendix A236

B.Axioms and universes239

Set universes242

Propositions and relativizations243

Rieger universes248

Rieger’s Theorem248

Antifoundation Principle， AFA254

Bisimulations255

The antifounded universe259

Aczel’s Theorem259

Problems for Appendix B262

Index267