Set Theory (eBook)

Front Cover 1
Set Theory: An Introduction to Large Cardinals 4
Copyright Page 5
Contents 11
Preface 8
Chapter 1. Introduction: Sets and Languages 14
§1. What are sets?—The cumulative type structure 14
§2. The first-order language of set theory 16
§3. The Zermelo–Fraenkel axioms 21
§4. A note on paradoxes 26
§5. More general languages 27
§6. The hereditarily finite sets–an example 31
Notes to Chapter 1 32
Chapter 2. Thedevelopment of ZFC 34
§1. Elementary definitions 34
§2. Ordinals 37
§3. Transfinite induction 43
§4. Cardinals: introduction 58
§5. Cardinal arithmetic 63
§6. The axiom of choice 67
§7. The generalized continuum hypothesis inaccessible cardinals
§8. Ramsey’s theorem 83
Notes to Chapter 2 86
Chapter 3. The Lévy Hierarchy And The Reflection Principle 88
§1. Transitive €-structures 88
§2. Lévy’s hierarchy 89
§3. Delta and transfinite induction 95
§4. Absoluteness 97
§5. Delta-definability of the satisfaction relation 102
§6. The reflection principle of ZF 111
§7. Cardinality and Sigma-formulas 116
Notes to Chapter 3 119
Chapter 4. Inaccessible and Mahlocardinals 120
§1. Properties of Va 120
§2. Normal functions 126
§3. Mahlo cardinals 129
§4. Reflection principles for Mahlo cardinals 134
Notes to Chapter 4 137
Chapter 5. The Constructible Universe 140
§1. Constructible sets 140
§2. Gödel’s theorems on L: AC and GCH 147
§3. Constructible orders 152
§4. On reducing proofs to ZFC 155
§5. The minimal model of ZF 158
§6. Relative constructibility 162
§7. The analytical hierarchy and constructible sets 165
§8. Ordinal definable sets 179
Notes to Chapter 5 184
Chapter 6. Measurable Cardinals 186
§1. Measures: classical properties 186
§2. The ultrapower construction for measurable cardinals 192
§3. Normal measures 200
§4. Measurable cardinals and constructible sets 205
§5. Measurable cardinals and the GCH 209
Notes to Chapter 6 212
Chapter 7. Trees and Partition Properties 214
§1. Trees 214
§2. Generalizations of Ramsey’s theorem 216
§3. Partition cardinals: K .( K )squre 222
§4. Partition cardinals: K .(a)< w
§5. Souslin and Kurepa trees 235
Notes to Chapter 7 246
Chapter 8. Partition Cardinals and Model Theory: Silver‘s Results 248
§1. Indiscernibles in a structure 248
§2. K (a)< w and indiscernibles
§3. Constructing models using indiscernibles 256
§4. Implications for the constructible universe 263
§5. A delta non-constructible set 269
§6. Further properties of Lµ 273
Notes to Chapter 8 277
Chapter 9.Indescribable Cardinals 280
§1. pai nm and Sigma nm-indescribables 280
§2. Enforceable classes 290
§3. Indescribability of measurable cardinals 294
§4. v-indescribable cardinals 297
Notes to Chapter 9 301
Chapter 10. Infinitarylanguages and Large Cardinals 302
§1. The languages Laß 302
§2. Weakly compact cardinals 305
§3. Strongly compact cardinals 315
§4. Summary of large cardinals 327
Notes to Chapter 10 331
Bibliography 332
Index 352
List of Symbols and Abbreviations Used and Page Where Introduced 361