Axioms of Set Theory

Axioms of set theoryWikipedia

Contents

1 Ackermann set theory 11.1 The language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Relation to ZermeloFraenkel set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Ackermann set theory and Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Aczels anti-foundation axiom 32.1 Accessible pointed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 AD+ 43.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Alternative set theory 54.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Axiom of adjunction 65.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Axiom of constructibility 76.1 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

7 Axiom of determinacy 87.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 87.3 Innite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

i

ii CONTENTS

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

8 Axiom of empty set 118.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9 Axiom of extensionality 139.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.3 In predicate logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.4 In set theory with ur-elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

10 Axiom of global choice 1510.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11 Axiom of innity 1611.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.2 Interpretation and consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.3 Extracting the natural numbers from the innite set . . . . . . . . . . . . . . . . . . . . . . . . . 17

11.3.1 Alternative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

12 Axiom of limitation of size 1912.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1912.2 Zermelos models and the axiom of limitation of size . . . . . . . . . . . . . . . . . . . . . . . . . 20

12.2.1 The model V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012.2.2 The models V where is a strongly inaccessible cardinal . . . . . . . . . . . . . . . . . . 21

12.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

13 Axiom of pairing 2513.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2513.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2513.3 Non-independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

13.5 Another alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14 Axiom of power set 2814.1 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

15 Axiom of real determinacy 30

16 Axiom of regularity 3116.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

16.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.1.2 No innite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 3116.1.3 Simpler set-theoretic denition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 3216.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 32

16.2 The axiom of dependent choice and no innite descending sequence of sets implies regularity . . . . 3216.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.4 Regularity and Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

17 Axiom of union 3617.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3617.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3617.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3717.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

18 Axiom schema of predicative separation 38

19 Axiom schema of replacement 3919.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.2 Axiom schema of collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4019.3 Example applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4119.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4119.5 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.6 Relation to the axiom schema of separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

20 Axiom schema of specication 44

iv CONTENTS

20.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4420.2 Relation to the axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.3 Unrestricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.4 In NBG class theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.5 In higher-order settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.6 In Quines New Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

21 Baumgartners axiom 4721.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

22 Constructive set theory 4822.1 Intuitionistic ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

22.1.1 Predicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4822.2 Myhills constructive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.3 Aczels constructive ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.4 Interpretability in type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5022.5 Interpretability in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5022.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5022.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5022.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

23 Freilings axiom of symmetry 5123.1 Freilings argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.2 Relation to the (Generalised) Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 5123.3 Objections to Freilings argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5223.4 Connection to graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5223.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

24 Fuzzy set 5424.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5424.2 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5424.3 Fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.4 Fuzzy interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.5 Fuzzy relation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.6 Axiomatic denition of credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.7 Credibility inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5624.9 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5624.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5624.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

CONTENTS v

24.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25 General set theory 6125.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6225.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6225.5 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6225.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6325.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

26 Ground axiom 6426.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

27 Internal set theory 6527.1 Intuitive justication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

27.1.1 Principles of the standard predicate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.2 Formal axioms for IST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

27.2.1 I: Idealisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.2.2 S: Standardisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.2.3 T: Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

27.3 Formal justication for the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.4 Related theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

28 KripkePlatek set theory 6928.1 The axioms of KP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6928.2 Proof that Cartesian products exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.3 Admissible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

29 KripkePlatek set theory with urelements 7129.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7129.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

29.2.1 Additional assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

30 Martins axiom 7330.1 Statement of Martins axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vi CONTENTS

30.2 Equivalent forms of MA(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7330.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

31 MorseKelley set theory 7531.1 MK axioms and ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7531.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

31.2.1 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.2.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

31.3 The axioms in Kelleys General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7831.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7931.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7931.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

32 Naive set theory 8032.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

32.1.1 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.1.2 Cantors theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.1.3 Axiomatic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.1.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.1.5 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

32.2 Sets, membership and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.2.1 Note on consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.2.2 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.2.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.2.4 Empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

32.3 Specifying sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.4 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.5 Universal sets and absolute complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.6 Unions, intersections, and relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.7 Ordered pairs and Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532.8 Some important sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532.9 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8632.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8732.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8732.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

33 Near sets 8933.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

CONTENTS vii

33.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.4 Efremovi proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9433.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9433.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.8 Descriptive -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9833.10Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

33.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9933.11Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10033.12Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 10133.13Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10133.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.17Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

34 New Foundations 10834.1 The type theory TST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10834.2 Quinean set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

34.2.1 Axioms and stratication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.2.2 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.2.3 Admissibility of useful large sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

34.3 Finite axiomatizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.4 Cartesian closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.5 The consistency problem and related partial results . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.6 How NF(U) avoids the set-theoretic paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.7 The system ML (Mathematical Logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11134.8 Models of NFU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

34.8.1 Self-suciency of mathematical foundations in NFU . . . . . . . . . . . . . . . . . . . . 11234.8.2 Facts about the automorphism j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

34.9 Strong axioms of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11334.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11534.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11534.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11534.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

35 Non-well-founded set theory 11735.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11835.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

viii CONTENTS

35.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11835.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11935.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11935.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

36 On Numbers and Games 12036.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12036.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12036.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

37 Pocket set theory 12237.1 Arguments supporting PST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.2 The theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.3 Remarks on the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12337.4 Some PST theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12337.5 Possible extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

38 Positive set theory 12538.1 Interesting properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12538.2 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12638.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12638.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

39 Proper forcing axiom 12739.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12739.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12739.3 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12739.4 Other forcing axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12739.5 The Fundamental Theorem of Proper Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12839.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12839.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

40 Rough set 12940.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

40.1.1 Information system framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12940.1.2 Example: equivalence-class structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12940.1.3 Denition of a rough set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13040.1.4 Denability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13140.1.5 Reduct and core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13240.1.6 Attribute dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

40.2 Rule extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

CONTENTS ix

40.2.1 Decision matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13340.2.2 LERS rule induction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

40.3 Incomplete data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13640.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13640.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13640.6 Extensions and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

40.6.1 Rough membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13740.6.2 Other generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

40.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13840.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13840.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14040.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

41 S (set theory) 14141.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14141.2 Primitive notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14141.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14241.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14241.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14341.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

42 ScottPotter set theory 14442.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

42.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14442.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14542.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

42.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14642.2.1 Scotts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14642.2.2 Potters theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

42.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14742.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14742.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

43 Semiset 14943.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

44 TarskiGrothendieck set theory 15044.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15044.2 Implementation in the Mizar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15144.3 Implementation in Metamath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15144.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15144.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15144.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

x CONTENTS

44.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

45 Universal set 15345.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

45.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15345.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

45.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15345.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15445.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

45.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15445.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15445.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

46 Vague set 15646.1 Mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15646.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15646.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

47 Von NeumannBernaysGdel set theory 15747.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15747.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15747.3 Axiomatizating NBG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

47.3.1 With Class Comprehension schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15947.3.2 Replacing Class Comprehension with nite instances thereof . . . . . . . . . . . . . . . . 160

47.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16147.4.1 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16247.4.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

47.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16247.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16247.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16347.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48 Zermelo set theory 16548.1 The axioms of Zermelo set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16548.2 Connection with standard set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16548.3 The aim of Zermelos paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16648.4 The axiom of separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16648.5 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16748.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16748.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

49 ZermeloFraenkel set theory 16849.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

CONTENTS xi

49.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16949.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16949.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 16949.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16949.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17049.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17049.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17149.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17249.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17249.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

49.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17349.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

49.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17449.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17449.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17549.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17549.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17649.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 177

49.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17749.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18049.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Chapter 1

Ackermann set theory

Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.

1.1 The languageAckermann set theory is formulated in rst-order logic. The language LA consists of one binary relation 2 and oneconstant V (Ackermann used a predicate M instead). We will write x 2 y for 2 (x; y) . The intended interpretationof x 2 y is that the object x is in the class y . The intended interpretation of V is the class of all sets.

1.2 The axiomsThe axioms of Ackermann set theory, collectively referred to as A, consists of the universal closure of the followingformulas in the language LA1) Axiom of extensionality:

8x8y(8z(z 2 x$ z 2 y)! x = y):

2) Class construction axiom schema: Let F (y; z1; : : : ; zn) be any formula which does not contain the variable x free.

8y(F (y; z1; : : : ; zn)! y 2 V )! 9x8y(y 2 x$ F (y; z1; : : : ; zn))

3) Reection axiom schema: Let F (y; z1; : : : ; zn) be any formula which does not contain the constant symbol V orthe variable x free. If z1; : : : ; zn 2 V then

8y(F (y; z1; : : : ; zn)! y 2 V )! 9x(x 2 V ^ 8y(y 2 x$ F (y; z1; : : : ; zn))):

4) Completeness axioms for V

x 2 y ^ y 2 V ! x 2 V

x y ^ y 2 V ! x 2 V:5) Axiom of regularity for sets:

x 2 V ^ 9y(y 2 x)! 9y(y 2 x ^ :9z(z 2 y ^ z 2 x)):

1

2 CHAPTER 1. ACKERMANN SET THEORY

1.3 Relation to ZermeloFraenkel set theoryLet F be a rst-order formula in the language L2 = f2g (so F does not contain the constant V ). Dene therestriction of F to the universe of sets (denoted FV ) to be the formula which is obtained by recursively replacingall sub-formulas of F of the form 8xG(x; y1 : : : ; yn) with 8x(x 2 V ! G(x; y1 : : : ; yn)) and all sub-formulas ofthe form 9xG(x; y1 : : : ; yn) with 9x(x 2 V ^G(x; y1 : : : ; yn)) .In 1959 Azriel Levy proved that if F is a formula of L2 and A proves FV , then ZF proves FIn 1970 William Reinhardt proved that if F is a formula of L2 and ZF proves F , then A proves FV .

1.4 Ackermann set theory and Category theoryThe most remarkable feature of Ackermann set theory is that, unlike Von NeumannBernaysGdel set theory, aproper class can be an element of another proper class (see Fraenkel, Bar-Hillel, Levy(1973), p. 153).An extension (named ARC) of Ackermann set theory was developed by F.A. Muller(2001), who stated that ARCfounds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole ofmathematics.

1.5 See also Zermelo set theory

1.6 References Ackermann, Wilhelm Zur Axiomatik der Mengenlehre in Mathematische Annalen, 1956, Vol. 131, pp.

336-345. Levy, Azriel, On Ackermanns set theory Journal of Symbolic Logic Vol. 24, 1959 154-166 Reinhardt, William, Ackermanns set theory equals ZF Annals of Mathematical Logic Vol. 2, 1970 no. 2,

189-249 A.A.Fraenkel, Y. Bar-Hillel, A.Levy, 1973. Foundations of Set Theory, second edition, North-Holand, 1973. F.A. Muller, Sets, Classes, and Categories British Journal for the Philosophy of Science 52 (2001) 539-573.

Chapter 2

Aczels anti-foundation axiom

In the foundations of mathematics, Aczels anti-foundation axiom is an axiom set forth by Peter Aczel (1988), as analternative to the axiom of foundation in ZermeloFraenkel set theory. It states that every accessible pointed directedgraph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex witha loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom isnecessarily a non-well-founded set theory.

2.1 Accessible pointed graphsAn accessible pointed graph is a directed graph with a distinguished vertex (the root) such that for any node in thegraph there is at least one path in the directed graph from the root to that node.The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of aunique set. For example, the directed graph with only one node and an edge from that node to itself corresponds toa set of the form x = {x}.

2.2 See also von Neumann universe

2.3 References Aczel, Peter (1988). Non-well-founded sets. CSLI Lecture Notes 14. Stanford, CA: Stanford University,

Center for the Study of Language and Information. ISBN 0-937073-22-9. MR 0940014.

Goertzel, Ben (1994). Self-Generating Systems. Chaotic Logic: Language, Thought and Reality From thePerspective of Complex Systems Science. Plenum Press. ISBN 978-0-306-44690-0. Retrieved 2007-01-15.

Akman, Varol; Pakkan, Mujdat (1996). Nonstandard set theories and information management. Journal ofIntelligent Information Systems 6 (1): 531. doi:10.1007/BF00712384.

3

Chapter 3

AD+

In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which isto be understood in the context of ZF plus DCR (the axiom of dependent choice for real numbers), states two things:

1. Every set of reals is -Borel.2. For any ordinal less than , any subset A of , and any continuous function :, the preimage 1[A]

is determined. (Here is to be given the product topology, starting with the discrete topology on .)

The second clause by itself is referred to as ordinal determinacy.

3.1 See also Suslins problem

3.2 References W.H. Woodin The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (1999 Walter de

Gruyter) p. 618

4

Chapter 4

Alternative set theory

Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposedalternative to the standard set theory.Some of the alternative set theories are:

the theory of semisets the set theory New Foundations Positive set theory Internal set theory

Specically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s byPetr Vopnka and his students. It builds on some ideas of the theory of semisets, but also introduces more radicalchanges: for example, all sets are formally nite, which means that sets in AST satisfy the law of mathematicalinduction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalentto the ZermeloFraenkel (or ZF) set theory, in which the axiom of innity is replaced by its negation). However,some of these sets contain subclasses that are not sets, which makes them dierent from Cantor (ZF) nite sets andthey are called innite in AST.

4.1 See also Non-well-founded set theory

4.2 References Vopnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979. Proceedings of the 1st Symposium Mathematics in the Alternative Set Theory. JSMF, Bratislava, 1989. Holmes, Randall M. Alternative Axiomatic Set Theories in the Stanford Encyclopedia of Philosophy.

5

Chapter 5

Axiom of adjunction

In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x {y} givenby adjoining the set y to the set x.

8x 8y 9w 8z [z 2 w $ (z 2 x _ z = y)]:

Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of settheory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as generalset theory or nitary set theory. The adjunction operation is also used as one of the operations of primitive recursiveset functions.Tarski and Smielew showed that Robinson arithmetic can be interpreted in a weak set theory whose axioms areextensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).

5.1 References Bernays, Paul (1937), A System of Axiomatic Set Theory--Part I, The Journal of Symbolic Logic (Association

for Symbolic Logic) 2 (1): 6577, doi:10.2307/2268862, JSTOR 2268862

Kirby, Laurence (2009), Finitary Set Theory, Notre Dame J. Formal Logic 50 (3): 227244, MR 2572972 Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amster-

dam: North-Holland Publishing Company, MR 0058532 Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMS

Colloquium Publications, v. 41.

6

Chapter 6

Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set isconstructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and theconstructible universe, respectively. The axiom, rst investigated by Kurt Gdel, is inconsistent with the propositionthat zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties). Generalizations ofthis axiom are explored in inner model theory.

6.1 ImplicationsThe axiom of constructibility implies the axiom of choice over ZermeloFraenkel set theory. It also settles manynatural mathematical questions independent of ZermeloFraenkel set theory with the axiom of choice (ZFC). Forexample, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslins hypoth-esis, and the existence of an analytical (in fact, 12 ) non-measurable set of real numbers, all of which are independentof ZFC.The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater orequal to 0#, which includes some relatively small large cardinals. Thus, no cardinal can be 1-Erds in L. WhileL does contain the initial ordinals of those large cardinals (when they exist in a supermodel of L), and they are stillinitial ordinals in L, it excludes the auxiliary structures (e.g. measures) which endow those cardinals with their largecardinal properties.Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as anaxiom for set theory in the same way as the ZFC axioms. Among set theorists of a realist bent, who believe thatthe axiom of constructibility is either true or false, most believe that it is false. This is in part because it seemsunnecessarily restrictive, as it allows only certain subsets of a given set, with no clear reason to believe that theseare all of them. In part it is because the axiom is contradicted by suciently strong large cardinal axioms. This pointof view is especially associated with the Cabal, or the California school as Saharon Shelah would have it.

6.2 See also Statements true in L

6.3 References Devlin, Keith (1984). Constructibility. Springer-Verlag. ISBN 3-540-13258-9.

6.4 External links How many real numbers are there?, Keith Devlin, Mathematical Association of America, June 2001

7

Chapter 7

Axiom of determinacy

The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski andHugo Steinhaus in 1962. It refers to certain two-person games of length with perfect information. AD states thatevery such game in which both players choose natural numbers is determined; that is, one of the two players has awinning strategy.The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies thatall subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property.The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the samecardinality as the full set of reals).Furthermore, AD implies the consistency of ZermeloFraenkel set theory (ZF). Hence, as a consequence of theincompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

7.1 Types of game that are determinedNot all games require the axiom of determinacy to prove them determined. Games whose winning sets are closed aredetermined. These correspond to many naturally dened innite games. It was shown in 1975 by Donald A. Martinthat games whose winning set is a Borel set are determined. It follows from the existence of sucient large cardinalsthat all games with winning set a projective set are determined (see Projective determinacy), and that AD holds inL(R).

7.2 Incompatibility of the axiom of determinacy with the axiom of choiceThe set S1 of all rst player strategies in an -game G has the same cardinality as the continuum. The same is trueof the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in Gis also the continuum. Let A be the subset of SG of all sequences which make the rst player win. With the axiomof choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portiondoes not have the cardinality of the continuum. We create a counterexample by transnite induction on the set ofstrategies under this well ordering:We start with the set A undened. Let T be the time whose axis has length continuum. We need to consider allstrategies {s1(T)} of the rst player and all strategies {s2(T)} of the second player to make sure that for every strategythere is a strategy of the other player that wins against it. For every strategy of the player considered we will generatea sequence which gives the other player a win. Let t be the time whose axis has length 0 and which is used duringeach game sequence.

1. Consider the current strategy {s1(T)} of the rst player.2. Go through the entire game, generating (together with the rst players strategy s1(T)) a sequence {a(1), b(2),

a(3), b(4),...,a(t), b(t+1),...}.3. Decide that this sequence does not belong to A, i.e. s1(T) lost.

8

7.3. INFINITE LOGIC AND THE AXIOM OF DETERMINACY 9

4. Consider the strategy {s2(T)} of the second player.

5. Go through the next entire game, generating (together with the second players strategy s2(T)) a sequence{c(1), d(2), c(3), d(4),...,c(t), d(t+1),...}, making sure that this sequence is dierent from {a(1), b(2), a(3),b(4),...,a(t), b(t+1),...}.

6. Decide that this sequence belongs to A, i.e. s2(T) lost.

7. Keep repeating with further strategies if there are any, making sure that sequences already considered do notbecome generated again. (We start from the set of all sequences and each time we generate a sequence andrefute a strategy we project the generated sequence onto rst player moves and onto second player moves, andwe take away the two resulting sequences from our set of sequences.)

8. For all sequences that did not come up in the above consideration arbitrarily decide whether they belong to A,or to the complement of A.

Once this has been done we have a game G. If you give me a strategy s1 then we considered that strategy at sometime T = T(s1). At time T, we decided an outcome of s1 that would be a loss of s1. Hence this strategy fails. Butthis is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.

7.3 Innite logic and the axiom of determinacyMany dierent versions of innitary logic were proposed in the late 20th century. One reason that has been given forbelieving in the axiom of determinacy is that it can be written as follows (in a version of innite logic):8G Seq(S) :8a 2 S : 9a0 2 S : 8b 2 S : 9b0 2 S : 8c 2 S : 9c0 2 S::: : (a; a0; b; b0; c; c0:::) 2 G OR9a 2 S : 8a0 2 S : 9b 2 S : 8b0 2 S : 9c 2 S : 8c0 2 S::: : (a; a0; b; b0; c; c0:::) /2 GNote: Seq(S) is the set of all ! -sequences of S. The sentences here are innitely long with a countably innite list ofquantiers where the ellipses appear.In an innitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantiersthat are true for nite formulas, such as 8a : 9b : 8c : 9d : R(a; b; c; d) OR 9a : 8b : 9c : 8d : :R(a; b; c; d) .

7.4 Large cardinals and the axiom of determinacyThe consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinalaxioms. By a theorem of Woodin, the consistency of ZermeloFraenkel set theory without choice (ZF) together withthe axiom of determinacy is equivalent to the consistency of ZermeloFraenkel set theory with choice (ZFC) togetherwith the existence of innitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD isconsistent, then so are an innity of inaccessible cardinals.Moreover, if to the hypothesis of an innite set of Woodin cardinals is added the existence of a measurable cardinallarger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable thatthe axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

7.5 See also Axiom of real determinacy (ADR) AD+, a variant of the axiom of determinacy formulated by Woodin Axiom of quasi-determinacy (ADQ) Martin measure

10 CHAPTER 7. AXIOM OF DETERMINACY

7.6 References Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-

44085-2. Kanamori, Akihiro (2000). The Higher Innite (2nd ed.). Springer. ISBN 3-540-00384-3. Martin, Donald A.; Steel, John R. (Jan 1989). A Proof of Projective Determinacy. Journal of the American

Mathematical Society 2 (1): 71125. doi:10.2307/1990913. JSTOR 1990913. Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Mycielski, Jan; Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice. Bulletin

de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques 10: 13.ISSN 0001-4117. MR 0140430.

Woodin, W. Hugh (1988). Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proceedingsof the National Academy of Sciences of theUnited States of America 85 (18): 65876591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

7.7 Further reading Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, University

of Bonn, Germany, 2001

Telgrsky, R.J. Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J.Math. 17 (1987), pp. 227276. (3.19 MB)

Chapter 8

Axiom of empty set

In axiomatic set theory, the axiom of empty set is an axiom of KripkePlatek set theory and the variant of generalset theory that Burgess (2005) calls ST, and a demonstrable truth in Zermelo set theory and ZermeloFraenkel settheory, with or without the axiom of choice.

8.1 Formal statementIn the formal language of the ZermeloFraenkel axioms, the axiom reads:

9x 8y :(y 2 x)or in words:

There is a set such that no set is a member of it.

8.2 InterpretationWe can use the axiom of extensionality to show that there is only one empty set. Since it is unique we can name it.It is called the empty set (denoted by { } or ). The axiom, stated in natural language, is in essence:

An empty set exists.

The axiom of empty set is generally considered uncontroversial, and it or an equivalent appears in just about anyalternative axiomatisation of set theory.In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of innity. However, thereare other formulations of that axiom that do not presuppose the existence of an empty set. The ZF axioms can alsobe written using a constant symbol representing the empty set; then the axiom of innity uses this symbol withoutrequiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.Furthermore, one sometimes considers set theories in which there are no innite sets, and then the axiom of emptyset may still be required. However, any axiom of set theory or logic that implies the existence of any set will implythe existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a subsetof any set consisting of those elements that satisfy a contradictory formula.In many formulations of rst-order predicate logic, the existence of at least one object is always guaranteed. If theaxiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms,and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similartheories), then the existence of the empty set is a theorem.If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement(as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement

11

12 CHAPTER 8. AXIOM OF EMPTY SET

schema. The formulation used in the axiom schema of replacement article only allows to construct the image F[a]when a is contained in the domain of the class function F; then the derivation of separation requires the axiom ofempty set. On the other hand, the constraint of totality of F is often dropped from the replacement schema, in whichcase it implies the separation schema without using the axiom of empty set (or any other axiom for that matter).

8.3 References Burgess, John, 2005. Fixing Frege. Princeton Univ. Press. Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag,

New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN

3-540-44085-2. Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

Chapter 9

Axiom of extensionality

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom ofextensionality, or axiom of extension, is one of the axioms of ZermeloFraenkel set theory.


8A 8B (8X (X 2 A () X 2 B)) A = B)

or in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B,then A is equal to B.(It is not really essential that X here be a set but in ZF, everything is. See Ur-elements below forwhen this is violated.)

The converse, 8A8B (A = B ) 8X (X 2 A () X 2 B)) , of this axiom follows from the substitutionproperty of equality.

9.2 InterpretationTo understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that Aand B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and onlyif they have precisely the same members. The essence of this is:

A set is determined uniquely by its members.

The axiom of extensionality can be used with any statement of the form 9A8X (X 2 A () P (X) ) , where P isany unary predicate that does not mention A, to dene a unique set A whose members are precisely the sets satisfyingthe predicate P . We can then introduce a new symbol for A ; its in this way that denitions in ordinary mathematicsultimately work when their statements are reduced to purely set-theoretic terms.The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or anequivalent appears in just about any alternative axiomatisation of set theory. However, it may require modicationsfor some purposes, as below.

13

14 CHAPTER 9. AXIOM OF EXTENSIONALITY

9.3 In predicate logic without equalityThe axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomaticset theory prefer to do without this, and instead treat the above statement not as an axiom but as a denition of equality.Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this dened symbol.Most of the axioms of equality still follow from the denition; the remaining one is

8A 8B (8X (X 2 A () X 2 B)) 8Y (A 2 Y () B 2 Y ) )

and it becomes this axiom that is referred to as the axiom of extensionality in this context.

9.4 In set theory with ur-elementsAn ur-element is a member of a set that is not itself a set. In the ZermeloFraenkel axioms, there are no ur-elements,but they are included in some alternative axiomatisations of set theory. Ur-elements can be treated as a dierentlogical type from sets; in this case, B 2 A makes no sense if A is an ur-element, so the axiom of extensionalitysimply applies only to sets.Alternatively, in untyped logic, we can require B 2 A to be false whenever A is an ur-element. In this case, the usualaxiom of extensionality would then imply that every ur-element is equal to the empty set. To avoid this consequence,we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

8A 8B (9X (X 2 A)) [8Y (Y 2 A () Y 2 B)) A = B] ):

That is:

Given any set A and any set B, if A is a nonempty set (that is, if there exists a member X of A), then ifA and B have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to dene A itself to be the only element of A whenever A is an ur-element.While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustmentinstead.

9.5 See also Extensionality for a general overview.

9.6 References Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag,

New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN

3-540-44085-2. Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

Chapter 10

Axiom of global choice

In mathematics, specically in class theories, the axiom of global choice is a stronger variant of the axiom of choicethat applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose anelement from every non-empty set.

10.1 StatementThe axiom of global choice states that there is a global choice function , meaning a function such that that for everynon-empty set z, (z) is an element of z.The axiom of global choice cannot be stated directly in the language of ZFC (ZermeloFraenkel set theory withthe axiom of choice), as the choice function is a proper class and in ZFC one cannot quantify over classes. It canbe stated by adding a new function symbol to the language of ZFC, with the property that is a global choicefunction. This is a conservative extension of ZFC: every provable statement of this extended theory that can be statedin the language of ZFC is already provable in ZFC (Fraenkel, Bar-Hillel & Levy 1973, p.72). Alternatively, Gdelshowed that given the axiom of constructibility one can write down an explicit (though somewhat complicated) choicefunction in the language of ZFC, so in some sense the axiom of constructibility implies global choice.In the language of von NeumannBernaysGdel set theory (NBG) and Morse-Kelley set theory, the axiom of globalchoice can be stated directly (Fraenkel, Bar-Hillel & Levy 1973, p.133), and is equivalent to various other statements:

Every class of nonempty sets has a choice function. V \ { } has a choice function (where V is the class of all sets). There is a well-ordering of V. There is a bijection between V and the class of all ordinal numbers.

In von NeumannBernaysGdel set theory, global choice does not add any consequence about sets (not properclasses) beyond what could have been deduced from the ordinary axiom of choice.Global choice is a consequence of the axiom of limitation of size.

10.2 References Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of set theory, Studies in Logic

and the Foundations of Mathematics 67 (Second revised ed.), Amsterdam-London: North-Holland PublishingCo., ISBN 978-0720422702, MR 0345816

Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN3-540-44085-2.

John L. Kelley; General Topology; ISBN 0-387-90125-6

15

Chapter 11

Axiom of innity

In axiomatic set theory and the branches of logic, mathematics, philosophy, and computer science that use it, theaxiom of innity is one of the axioms of ZermeloFraenkel set theory. It guarantees the existence of at least oneinnite set, namely a set containing the natural numbers.


9I (; 2 I ^ 8x 2 I ( (x [ fxg) 2 I)):In words, there is a set I (the set which is postulated to be innite), such that the empty set is in I and such thatwhenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member ofI. Such a set is sometimes called an inductive set.

11.2 Interpretation and consequencesThis axiom is closely related to the standard construction of the naturals in set theory, in which the successor of xis dened as x {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also auniquely dened set. Successors are used to dene the usual set-theoretic encoding of the natural numbers. In thisencoding, zero is the empty set:

0 = {}.

The number 1 is the successor of 0:

1 = 0 {0} = {} {0} = {0}.

Likewise, 2 is the successor of 1:

2 = 1 {1} = {0} {1} = {0,1},

and so on. A consequence of this denition is that every natural number is equal to the set of all preceding naturalnumbers.This construction forms the natural numbers. However, the other axioms are insucient to prove the existence of theset of all natural numbers. Therefore its existence is taken as an axiomthe axiom of innity. This axiom assertsthat there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each elementof I, the successor of that element is also in I.Thus the essence of the axiom is:

16

11.3. EXTRACTING THE NATURAL NUMBERS FROM THE INFINITE SET 17

There is a set, I, that includes all the natural numbers.

The axiom of innity is also one of the von NeumannBernaysGdel axioms.

11.3 Extracting the natural numbers from the innite setThe innite set I is a superset of the natural numbers. To show that the natural numbers themselves constitute aset, the axiom schema of specication can be applied to remove unwanted elements, leaving the set N of all naturalnumbers. This set is unique by the axiom of extensionality.To extract the natural numbers, we need a denition of which sets are natural numbers. The natural numbers can bedened in a way which does not assume any axioms except the axiom of extensionality and the axiom of inductionanatural number is either zero or a successor and each of its elements is either zero or a successor of another of itselements. In formal language, the denition says:

8n(n 2 N () ([n = ; _ 9k(n = k [ fkg)] ^ 8m 2 n[m = ; _ 9k 2 n(m = k [ fkg)])):Or, even more formally:

8n(n 2 N () ([8k(:k 2 n) _ 9k8j(j 2 n () (j 2 k _ j = k))]^8m(m 2 n) [8k(:k 2 m) _ 9k(k 2 n ^ 8j(j 2 m () (j 2 k _ j = k)))]))):

11.3.1 Alternative methodAn alternative method is the following. Let (x) be the formula that says `x is inductive'; i.e. (x) = (? 2x ^ 8y(y 2 x ! (y [ fyg 2 x))) . Informally, what we will do is take the intersection of all inductive sets. Moreformally, we wish to prove the existence of a unique set W such that

8x(x 2W $ 8I((I)! x 2 I)):For existence, we will use the Axiom of Innity combined with the Axiom schema of specication. Let I be aninductive set guaranteed by the Axiom of Innity. Then we use the Axiom Schema of Specication to dene our setW = fx 2 I : 8J((J) ! x 2 J)g - i.e. W is the set of all elements of I which happen also to be elements ofevery other inductive set. This clearly satises the hypothesis of (*), since if x 2W , then x is in every inductive set,and if x is in every inductive set, it is in particular in I , so it must also be in W .For uniqueness, rst note that any set which satises (*) is itself inductive, since 0 is in all inductive sets, and if anelement x is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set W 0which satised (*) we would have that W 0 W since W is inductive, and W W 0 since W 0 is inductive. ThusW = W 0 . Let ! denote this unique element.This denition is convenient because the principle of induction immediately follows: If I ! is inductive, then also! I , so that I = ! .Both these methods produce systems which satisfy the axioms of second-order arithmetic, since the axiom of powerset allows us to quantify over the power set of ! , as in second-order logic. Thus they both completely determineisomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal.

11.4 IndependenceThe axiom of innity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent.Nor can it be refuted, if all of ZFC is consistent.Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of innity is replacedby its negation. It is V! , the class of hereditarily nite sets, with the inherited element relation. If allowed, the empty

18 CHAPTER 11. AXIOM OF INFINITY

domain also satises the axioms of this modied theory, as all of them are universally quantied, and thus triviallysatised if no set exists.The cardinality of the set of natural numbers, aleph null ( @0 ), has many of the properties of a large cardinal. Thusthe axiom of innity is sometimes regarded as the rst large cardinal axiom, and conversely large cardinal axioms aresometimes called stronger axioms of innity.

11.5 See also Peano axioms Finitism

11.6 References Paul Halmos (1960)Naive Set Theory. Princeton, NJ: D. Van Nostrand Company. Reprinted 1974 by Springer-

Verlag. ISBN 0-387-90092-6. Thomas Jech (2003) Set Theory: The ThirdMillennium Edition, Revised and Expanded. Springer-Verlag. ISBN

3-540-44085-2. Kenneth Kunen (1980) Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9. Hrbacek, Karel; Jech, Thomas (1999). Introduction to Set Theory (3 ed.). Marcel Dekker. ISBN 0-8247-7915-

0.

Chapter 12

Axiom of limitation of size

In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class whichis not a set (an element of other classes), if and only if it can be mapped onto the class V of all sets.[1]

8C:9W (C 2W ) () 9F

8x9W (x 2W )) 9s (s 2 C ^ hs; xi 2 F ) ^8x8y8s (hs; xi 2 F ^ hs; yi 2 F )) x = y:

This axiom is due to John von Neumann. It implies the axiom schema of specication, axiom schema of replacement,axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union[2] at one stroke. The axiom oflimitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is a surjectionfrom the ordinals to the universe, thus an injection from the universe to the ordinals, that is, the universe of sets iswell-ordered.Together the axiom of replacement and the axiom of global choice (with the other axioms of von NeumannBernaysGdel set theory) imply this axiom. This axiom is thus equivalent to the combination of replacement, global choice,specication and union in von NeumannBernaysGdel or MorseKelley set theory.However, the axiom of replacement and the usual axiom of choice (with the other axioms of von NeumannBernaysGdel set theory) do not imply von Neumanns axiom. In 1964, Easton used forcing to build a model that satisesthe axioms of von NeumannBernaysGdel set theory with one exception: the axiom of global choice is replacedby the axiom of choice. In Eastons model, the axiom of limitation of size fails dramatically: the universe of setscannot even be linearly ordered.[3]

It can be shown that a class is a proper class if and only if it is equinumerous to V, but von Neumanns axiom doesnot capture all of the "limitation of size doctrine,[4] because the axiom of power set is not a consequence of it. Laterexpositions of class theories (Bernays, Gdel, Kelley, ...) generally use replacement and a form of the axiom of choicerather than the axiom of limitation of size.

12.1 HistoryVon Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identies sets viaits set building axioms. However, as Abraham Fraenkel pointed out: The rather arbitrary character of the processeswhich are chosen in the axioms of Z [ZFC] as the basis of the theory, is justied by the historical development ofset-theory rather than by logical arguments.[5]

The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to support his proof of thewell-ordering theorem and to avoid contradictory sets.[6] In 1922, Fraenkel and Skolem pointed out that Zermelosaxioms cannot prove the existence of the set {Z0, Z1, Z2, } where Z0 is the set of natural numbers, and Zn isthe power set of Zn.[7] They also introduced the axiom of replacement, which guarantees the existence of this set.[8]However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor claries thedierence between sets that are safe to use and collections that lead to contradictions.In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identies the sets that are toobig (now called proper classes) and that can lead to contradictions.[9] Von Neumann identied these sets using the

19

20 CHAPTER 12. AXIOM OF LIMITATION OF SIZE

criterion: A set is 'too big' if and only if it is equivalent to the set of all things.[10] He then restricted how these setsmay be used: " in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible aselements.[11] By combining this restriction with his criterion, von Neumann obtained the axiom of limitation of size(which in the language of classes states): A class X is not an element of any class if and only if X is equivalent tothe class of all sets.[12] So von Neumann identied sets as classes that are not equivalent to the class of all sets. VonNeumann realized that, even with his new axiom, his set theory does not fully characterize sets.[13]

Gdel found von Neumanns axiom to be of great interest":

In particular I believe that his [von Neumanns] necessary and sucient condition which a propertymust satisfy, in order to dene a set, is of great interest, because it claries the relationship of axiomaticset theory to the paradoxes. That this condition really gets at the essence of things is seen from the factthat it implies the axiom of choice, which formerly stood quite apart from other existential principles.The inferences, bordering on the paradoxes, which are made possible by this way of looking at things,seem to me, not only very elegant, but also very interesting from the logical point of view.[14] MoreoverI believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, willthe basic problems of abstract set theory be solved.[15]

12.2 Zermelos models and the axiom of limitation of sizeIn 1930, Zermelo published an article on models of set theory, in which he proved that some of his models satisfy theaxiom of limitation of size. These models are built in ZFC by using the cumulative hierarchy V, which is denedby transnite recursion:

1. V0 = .[16]

2. V = V P(V). That is, the union of V and its power set.[17]

3. For limit : V = < V. That is, V is the union of the preceding V.

Zermelo worked with models of the form V where is a cardinal. The classes of the model are the subsets of V,and the models -relation is the standard -relation. The sets of the model V are the classes X such that X V.[18]Zermelo identied cardinals such that V satises:[19]

Theorem 1. A class X is a set if and only if | X | < .Theorem 2. | V | = .

Since every class is a subset of V, Theorem 2 implies that every class X has cardinality . Combining thiswith Theorem 1 proves: Every proper class has cardinality . Hence, every proper class can be put into one-to-onecorrespondence with V, so the axiom of limitation of size holds for the model V.The proof of the axiom of global choice in V is more direct than von Neumanns proof. First note that (beinga von Neumann cardinal) is a well-ordered class of cardinality . Since Theorem 2 states that V has cardinality, there is a one-to-one correspondence between and V. This correspondence produces a well-ordering of V,which implies the axiom of global choice.[20] Von Neumann uses the Burali-Forti paradox to prove by contradictionthat the class of all ordinals is a proper class, and then he applies the axiom of limitation of size to well-order theuniversal class.[21]

12.2.1 The model VTo demonstrate that Theorems 1 and 2 hold for some V, we need to prove that if a set belongs to V then it belongsto all subsequent V, or equivalently: V V for . This is proved by transnite induction on :

1. = 0: V0 V0.

2. For +1: By inductive hypothesis, V V. Hence, V V V P(V) = V.

12.2. ZERMELOS MODELS AND THE AXIOM OF LIMITATION OF SIZE 21

3. For limit : If < , then V < V = V. If = , then V V.

Note that sets enter the hierarchy only through the power set P(V) at step +1. We will need the following denitions:

If x is a set, rank(x) is the least ordinal such that x V.[22]The supremum of a set of ordinals A, denoted by sup A, is the least ordinal such that for all A.

Zermelos smallest model is V. Induction proves that Vn is nite for all n < :

1. | V0 | = 0.2. | Vn | = | Vn P(Vn) | | Vn | + 2 | Vn |, which is nite since Vn is nite by inductive hypothesis.

To prove Theorem 1: since a set X enters V only through P(Vn) for some n < , we have X Vn. Since Vn isnite, X is nite. Conversely: if a class X is nite, let N = sup {rank(x): x X}. Since rank(x) N for all x X, wehave X VN, so X VN V. Therefore, X V.To prove Theorem 2, note that V is the union of countably many nite sets. Hence, V is countably innite andhas cardinality @0 (which equals by von Neumann cardinal assignment).It can be shown that the sets and classes of V satisfy all the axioms of NBG (von NeumannBernaysGdel settheory) except the axiom of innity.

12.2.2 The models V where is a strongly inaccessible cardinalTo nd models satisfying the axiom of innity, observe that two properties of niteness were used to prove Theorems1 and 2 for V:

1. If is a nite cardinal, then 2 is nite.2. If A is a set of ordinals such that | A | is nite, and is nite for all A, then sup A is nite.

Replacing nite by "< " produces the properties that dene strongly inaccessible cardinals. A cardinal is stronglyinaccessible if > and:

1. If is a cardinal such that < , then 2 < .2. If A is a set of ordinals such that | A | < , and < for all A, then sup A < .

These properties assert that cannot be reached from below. The rst property says cannot be reached by powersets; the second says cannot be reached by the axiom of replacement.[23] Just as the axiom of innity is requiredto obtain , an axiom is needed to obtain strongly inaccessible cardinals. Zermelo postulated the existence of anunbounded sequence of strongly inaccessible cardinals.[24]

If is a strongly inaccessible cardinal, then transnite induction proves | V | < for all < :

1. = 0: | V0 | = 0.2. For +1: | V | = | V P(V) | | V | + 2 | V | = 2 | V | < . Last inequality uses inductive hypothesis

and being strongly inaccessible.3. For limit : | V | = | < V | sup {| V | : < } < . Last inequality uses inductive hypothesis and

being strongly inaccessible.

To prove Theorem 1: since a set X enters V only through P(V) for some < , we have X V. Since | V | < ,we have | X | < . Conversely: if a class X has | X | < , let = sup {rank(x): x X}. Since is strongly inaccessible,| X | < , and rank(x) < for all x X, we have < . Also, rank(x) for all x X implies X V, so X V V. Therefore, X V.

22 CHAPTER 12. AXIOM OF LIMITATION OF SIZE

To prove Theorem 2, we compute: | V | = | < V | sup {| V | : < }. Let be this supremum. Sinceeach ordinal in the supremum is less than , we have . Now cannot be less than . If it were, there would bea cardinal such that < < ; for example, take = 2 | |. Since V and | V | is in the supremum, we have | V | . This contradicts < . Therefore, | V | = = .It can be shown that the sets and classes of V satisfy all the axioms of NBG.[25]

12.3 See also Axiom of global choice Limitation of size Von NeumannBernaysGdel set theory MorseKelley set theory

12.4 Notes[1] This is roughly von Neumanns original formulation, see Fraenkel & al, p. 137.

[2] showing directly that a set of ordinals has an upper bound, see A. Levy, " On von Neumanns axiom system for set theory", Amer. Math. Monthly, 75 (1968), p. 762-763.

[3] Easton 1964.

[4] Fraenkel & al, p. 137. A guiding principle for ZF to avoid set theoretical paradoxes is to restrict to instances of full(contradictory) comprehension scheme that do not give sets too much bigger than the ones they use; it is known aslimitation of size, Fraenkel & al call it limitation of size doctrine, see p. 32.

[5] Historical Introduction in Bernays 1991, p. 31.

[6] "... we must, on the one hand, restrict these principles [axioms] suciently to exclude all contradictions and, on the otherhand, take them suciently wide to retain all that is valuable in this theory. (Zermelo 1908, p. 261; English translation, p.200). Gregory Moore analyzed Zermelos reasons behind his axiomatization and concluded that his axiomatization wasprimarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem " and For Zermelo, theparadoxes were an inessential obstacle to be circumvented with as little fuss as possible. (Moore 1982, p. 159160).

[7] Fraenkel 1922, p. 230231; Skolem 1922 (English translation, p. 296297).

[8] Ferreirs 2007, p. 369. In 1917, Mirimano published a form of replacement based on cardinal equivalence (Mirimano1917, p. 49).

[9] He gave a detailed exposition of his set theory in two articles: von Neumann 1925 and von Neumann 1928.

[10] Hallett 1984, p. 288.

[11] Hallett 1984, p. 290.

[12] Hallett 1984, p. 290. Von Neumann later changed equivalent to the class of all sets to can be mapped onto the class ofall sets.

[13] To be precise, von Neumann investigated whether his set theory is categorical; that is, whether it uniquely determines setsin the sense that any two of its models are isomorphic. He showed that it is not categorical because of a weakness in theaxiom of regularity: this axiom only excludes descending -sequences from existing in the model; descending sequencesmay still exist outside the model. A model having external descending sequences is not isomorphic to a model havingno such sequences since this latter model lacks isomorphic images for the sets belonging to external descending sequences.This led von Neumann to conclude that no categorical axiomatization of set theory seems to exist at all (von Neumann1925, p. 239; English translation: p. 412).

[14] For example, von Neumanns proof that his axiom implies the well-ordering theorem uses the Burali-Forte paradox (vonNeumann 1925, p. 223; English translation: p. 398).

[15] From a Nov. 8, 1957 letter Gdel wrote to Stanislaw Ulam (Kanamori 2003, p. 295).

12.5. REFERENCES 23

[16] This is the standard denition of V0. Zermelo let V0 be a set of urelements and proved that if this set contains a singleelement, the resulting model satises the axiom of limitation of size (his proof also works for V0 = ). Zermelo stated thatthe axiom is not true for all models built from a set of urelements. (Zermelo 1930, p. 38; English translation: p. 1227.)

[17] This is Zermelos denition (Zermelo 1930, p. 36; English translation: p. 1225 & p. 1209), which is equivalent to V =P(V) since V P(V) (Kunen 1980, p. 95; Kunen uses the notation R() instead of V).

[18] In NBG, X is a set if there is a class Y such that X Y. Since Y V, we have X V. Conversely, if X V, then Xbelongs to a class, so X is a set.

[19] These theorems are part of Zermelos Second Development Theorem. (Zermelo 1930, p. 37; English translation: p. 1226.)

[20] The domain of the global choice function consists of the non-empty sets of V; this function uses t

Documents

Axioms of Set Theory