Set Theory (eBook)

Front Cover 1
Set Theory 4
Copyright Page 5
Contents 10
Preface 12
Part I: Sets 14
Chapter 1. Axiomatic Set Theory 14
1. Axioms of Set Theory 14
2. Ordinal Numbers 25
3. Cardinal Numbers 35
4. Real Numbers 42
5. The Axiom of Choice 51
6. Cardinal Arithmetic 55
7. Filters and Ideals. Closed Unbounded Sets 65
8. Singular Cardinals 74
9. The Axiom of Regularity 83
Appendix: Bernays–Gödel Axiomatic Set Theory 89
Chapter 2. Transitive Models of Set Theory 91
10. Models of Set Theory 91
11. Transitive Models of ZF 100
12. Constructible Sets 112
13. Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis 121
14. The Sn Hierarchy of Classes, Relations, and Functions 127
15. Relative Constructibility and Ordinal Definability 139
Part II: More Sets 150
Chapter 3. Forcing and Generic Models 150
16. Generic Models 150
17. Complete Boolean Algebras 157
18. Forcing and Boolean-Valued Models 172
19. Independence of the Continuum Hypothesis and the Axiom of Choice 189
20. More Generic Models 200
21. Symmetric Submodels of Generic Models 210
Chapter 4. Some Applications of Forcing 229
22. Suslin’s Problem 229
23. Martin’s Axiom and Iterated Forcing 242
24. Some Combinatorial Problems 257
25. Forcing and Complete Boolean Algebras 274
26. More Applications of Forcing 296
Part III: Large Sets 308
Chapter 5. Measurable Cardinals 308
27. The Measure Problem 308
28. Ultrapowers and Elementary Embeddings 318
29. Infinitary Combinatorics 334
30. Silver lndiscernibles 350
31. The Model L[U] 372
32. Large Cardinals below a Measurable Cardinal 394
Chapter 6. Other Large Cardinals 411
33. Compact Cardinals 411
34. Real-Valued Measurable Cardinals 429
35. Saturation of Ideals and Generic Ultrapowers 440
36. Measurable Cardinals and the Generalized Continuum Hypothesis 463
37. Some Applications of Forcing in the Theory of Large Cardinals 478
38. More on Ultrafilters 491
Part IV: Sets of Reals 506
Chapter 7. Descriptive Set Theory 506
39. Borel and Analytic Sets 506
40. S1/n and .1/n: Sets and Relations in the Baire Space 522
41. Projective Sets in the Constructible Universe 540
42. A Model Where All Sets Are Lebesgue Measurable 550
43. The Axiom of Determinacy 563
44. Some Applications of Forcing in Descriptive Set Theory 576
Historical Notes and Guide to The Bibliography 592
Bibliography 609
Notation 624
Index 628
Pure and Applied Mathematics 635