Set Theory and the Continuum Problem

I AXIOMATIC SET THEORY 1
1 General background 3
§1 What is infinity? 3
§2 Countable or uncountable? 4
§3 A non-denumerable set 6
§4 Larger and smaller 7
§5 The continuum problem 8
§6 Significance of the results 9
§7 Frege set theory 10
§8 Russell's paradox 11
§9 Zermelo set theory 12
§10 Sets and classes 13
2 Some basics of class-set theory 14
§1 Extensionality and separation 14
§2 Transitivity and supercompleteness 16
§3 Axiom of the empty set 17
§4 The pairing axiom 18
§5 The union axiom 20
§6 The power axiom 22
§7 Cartesian products 22
§8 Relations 23
§9 Functions 23
§10 Some useful facts about transitivity 24
§11 Basic universes 25
3 The natural numbers 27
§1 Preliminaries 27
§2 Definition of the natural numbers 29
§3 Derivation of the Peano postulates and other results 31
§4 A double induction principle and its applications 32
§5 Applications to natural numbers 37
§6 Finite sets 39
§7 Denumerable classes 39
§8 Definition by finite recursion 40
§9 Supplement-optional 41
4 Superinduction, well ordering and choice 43
§1 Introduction to well ordering 43
§2 Superinduction and double superinduction 48
§3 The well ordering of g-towers 51
§4 Well ordering and choice 52
§5 Maximal principles 55
§6 Another approach to maximal principles 58
§7 Cowen's theorem 60
§8 Another characterization of g-sets 62
5 Ordinal numbers 64
§1 Ordinal numbers 64
§2 Ordinals and transitivity 67
§3 Some ordinals 69
6 Order isomorphism and transfinite recursion 70
§1 A few preliminaries 70
§2 Isomorphisms of well orderings 71
§3 The axiom of substitution 73
§4 The counting theorem 74
§5 Transfinite recursion theorems 74
§6 Ordinal arithmetic 78
7 Rank 81
§1 The notion of rank 81
§2 Ordinal hierarchies 82
§3 Applications to the R_a's 83
§4 Zermelo universes 85
8 Foundation, ϵ-induction, and rank 88
§1 The notion of well-foundedness 88
§2 Descending ϵ-chains 89
§3 ϵ-Induction and rank 90
§4 Axiom E and Von Neumann's principle 91
§5 Some other characterizations of ordinals 93
§6 More on the axiom of substitution 94
9 Cardinals 96
§1 Some simple facts 96
§2 The Bernstein-Schroder theorem 97
§3 Denumerable sets 99
§4 Infinite sets and choice functions 100
§5 Hartog's theorem 101
§6 A fundamental theorem 102
§7 Preliminaries 104
§8 Cardinal arithmetic 106
§9 Sierpinski's theorem 109

II CONSISTENCY OF THE CONTINUUM HYPOTHESIS 113
10 Mostowski-Shepherdson mappings 115
§1 Relational systems 115
§2 Generalized induction and r-rank 117
§3 Generalized transfinite recursion 120
§4 Mostowski-Shepherdson maps 122
§5 More on Mostowski-Shepherdson mappings 123
§6 Isomorphisms, Mostowski-Shepherdson maps, and well orderings 124
11 Reflection principles 128
§0 Preliminaries 128
§1 The Tarski-Vaught theorem 131
§2 We add extensionality considerations 133
§3 The class version of the Tarski-Vaught theorems 134
§4 Mostowski, Shepherdson, Tarski, and Vaught 137
§5 The Montague-Levy reflection theorem 137
12 Constructible sets 141
§0 More on first-order definability 141
§1 The class L of constructible sets 142
§2 Absoluteness 143
§3 Constructible classes 148
13 L is a well founded first-order universe 153
§1 First-order universes 153
§2 Some preliminary theorems about first-order universes 156
§3 More on first-order universes 157
§4 Another result 160
14 Constructibility is absolute over L 162
§1 Σ-formulas and upward absoluteness 162
§2 More on Σ definability 164
§3 The relation y = F(x) 166
§4 Constructibility is absolute over L 171
§5 Further results 172
§6 A proof that L can be well ordered 173
15 Constructibility and the continuum hypothesis 176
§0 What we will do 176
§1 The key result 177
§2 Godel's isomorphism theorem (optional) 179
§3 Some consequences of Theorem G 181
§4 Metamathematical consequences of Theorem G 181
§5 Relative consistency of the axiom of choice 183
§6 Relative consistency of GCH and AC in class-set theory 183

III FORCING AND INDEPENDENCE RESULTS 187
16 Forcing, the very idea 189
§1 What is forcing? 189
§2 What is modal logic? 191
§3 What is S4 and why do we care? 195
§4 A classical embedding 196
§5 The basic idea 202
17 The construction of S4 models for ZF 203
§1 What are the models? 203
§2 About equality 208
§3 The well founded sets are present 212
§4 Four more axioms 214
§5 The definability of forcing 218
§6 The substitution axiom schema 220
§7 The axiom of choice 222
§8 Where we stand now 225
18 The axiom of constructibility is independent 226
§1 Introduction 226
§2 Ordinals are well behaved 226
§3 Constructible sets are well behaved too 228
§4 A real S4 model, at last 229
§5 Cardinals are sometimes well behaved 230
§6 The status of the generalized continuum hypothesis 233
19 Independence of the continuum hypothesis 235
§1 Power politics 235
§2 The model 235
§3 Cardinals stay cardinals 236
§4 CH is independent 238
§5 Cleaning it up 239
§6 Wrapping it up 241
20 Independence of the axiom of choice 243
§1 A little history 243
§2 Automorphism groups 243
§3 Automorphisms preserve truth 246
§4 Model and submodel 248
§5 Verifying the axioms 249
§6 AC is independent 254
21 Constructing classical models 259
§1 On countable models 259
§2 Cohen's way 260
§3 Dense sets, filters, and generic sets 261
§4 When generic sets exist 263
§5 Generic extensions 264
§6 The truth lemma 267
§7 Conclusion 269
22 Forcing background 271
§1 Introduction 271
§2 Cohen's version(s) 272
§3 Boolean valued models 273
§4 Unramified forcing 273
§5 Extensions 274
References 277
Subject index 281
Notation index 287