Intermediate Set Theory

Preface ......Page 7
1.1 Cantor’s contribution ......Page 9
1.2 Cantor’s concept of a set ......Page 11
1.3 Paradoxes ......Page 13
1.4 Type structures ......Page 15
1.5 Culmination into axiomatics ......Page 18
2.1 Introduction ......Page 21
2.2 The syntax of LST ......Page 22
2.3 Proofs and derivations in LST ......Page 26
2.4 Semantics of LST ......Page 27
2.5 Adding new terms ......Page 32
3.1 The Zermelo-Fraenkel axioms ......Page 35
3.2 Arguments for these axioms ......Page 39
4.1 Countable sets ......Page 45
4.2 Uncountable sets ......Page 49
4.3 The arithmetic of cardinal numbers ......Page 54
5.1 Orderings ......Page 59
5.2 Some properties of ordered sets ......Page 61
5.3 Lattices and Boolean algebras ......Page 64
5.4 Well-ordered sets ......Page 74
6.1 The natural numbers ......Page 79
6.2 The Peano axioms for the natural numbers ......Page 84
6.3 The rational numbers ......Page 93
6.4 The real numbers ......Page 97
6.5 Ordinals in ZF: basic properties ......Page 101
6.6 Transfinite induction ......Page 105
6.7 Cardinals as initial ordinals ......Page 113
7.1 Simple forms ......Page 117
7.2 The well-ordering theorem ......Page 120
7.3 Maximal principles and Zorn’s lemma ......Page 122
7.4 Simple consequences of the axiom of choice ......Page 125
8 Constructible sets and forcing ......Page 135
8.1 Godel’s constructible sets ......Page 136
8.2 The definition of L ......Page 141
8.3 Reflection principles ......Page 146
8.4 Properties of L ......Page 148
8.5 The axiom of choice in L ......Page 153
8.6 The generalized continuum hypothesis in L ......Page 156
8.7 Another presentation ......Page 161
8.8 Forcing models ......Page 162
8.9 Forcing in practice: the ZFC axioms hold in M[G] ......Page 167
8.10 Forcing in practice: some models ......Page 172
8.11 Proofs of the definability and truth lemmas ......Page 179
8.12 Models for the independence of the axiom of choice ......Page 187
8.13 A model with a Dedekind-finite set ......Page 192
8.14 Boolean-valued models: another presentation ......Page 197
9.2 System VNB ......Page 201
9.3 System MK ......Page 205
9.4 Axioms of extent ......Page 207
9.5 Other presentations of set theory ......Page 210
9.6 Remarks on the philosophy of mathematics ......Page 215
10.1 Simplest constructions, and variants ......Page 221
10.2 Ordered pairs, relations, functions, families and sequences ......Page 222
References ......Page 227
Index ......Page 233