Set Theory with a Universal Set: Exploring an Untyped Universe

Cover......Page 1
Title......Page 2
Series......Page 3
Copyright......Page 5
Preface to the First Edition......Page 6
Preface to the Second Edition......Page 8
Contents......Page 10
1 – Introduction......Page 12
1.1 Annotated definitions......Page 15
1.1.1 Quantifier hierarchies......Page 16
1.1.2 Mainly concerning type theory......Page 17
1.1.3 Other definitions......Page 20
1.1.4 Theories......Page 21
1.2.1 Sets as predicates-in-extension......Page 22
1.2.2 Sets as natural kinds......Page 32
1.3 A brief survey......Page 33
1.4 How do theories with V ∈ V avoid the paradoxes?......Page 35
1.5 Chronology......Page 36
2.1 NF......Page 37
2.1.1 The axiom of counting......Page 41
2.1.2 Boffa’s lemma on n-formulae, and the automorphism lemma for set abstracts......Page 44
2.1.3 Miscellaneous combinatorics......Page 46
2.1.4 Well-founded sets......Page 51
2.2 Cardinal and ordinal arithmetic......Page 55
2.2.1 Some remarks on inductive definitions......Page 66
2.2.2 Closure properties of small sets......Page 68
2.3 The Kaye-Specker equiconsistency lemma......Page 69
2.3.1 NF₃......Page 76
2.3.2 NFU......Page 78
2.3.4 KF......Page 83
2.4 Subsystems, term models, and prefix classes......Page 94
2.5 The converse consistency problem......Page 100
3 – Permutation Models......Page 103
3.1 Permutations in NF......Page 107
3.1.1 Inner permutations in NF......Page 108
3.1.2 Outer automorphisms in NF......Page 130
3.2 Applications to other theories......Page 132
4.1 Oswald’s model......Page 133
4.2 Low sets......Page 135
4.2.1 Other definitions of low......Page 136
4.3.1 P-extensions......Page 137
4.3.2 Hereditarily low sets and permutation models......Page 138
4.3.3 Permutation models of CO structures......Page 140
4.4.1 An elementary example......Page 141
4.4.2 P-extending models of Zermelo to models of NFO......Page 143
4.5 Church’s model......Page 147
4.6 Mitchell’s set theory......Page 150
4.7 Conclusions......Page 151
5.1 Permutation models and quantifier hierarchies......Page 154
5.3 KF......Page 155
5.5 Well-founded extensional relations......Page 156
5. 7 Miscellaneous......Page 157
Concerning set theory with a universal set......Page 159
Concerning other matters raised in the text......Page 169
Index of Definitions......Page 172
Author Index......Page 174
General Index......Page 175