Well-founded relationWikipedia

Contents

1 Ascending chain condition 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Axiom of limitation of size 32.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Zermelos models and the axiom of limitation of size . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 The model V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 The models V where is a strongly inaccessible cardinal . . . . . . . . . . . . . . . . . . 5

2.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Axiom of regularity 93.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 No innite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3 Simpler set-theoretic denition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 103.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 10

3.2 The axiom of dependent choice and no innite descending sequence of sets implies regularity . . . . 103.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Regularity and Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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4 Better-quasi-ordering 144.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Simpsons alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Major theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Binary relation 165.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Class (set theory) 266.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Classes in formal set theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Dicksons lemma 287.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.3 Generalizations and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Element (mathematics) 318.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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8.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Empty set 349.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10 Epsilon-induction 4010.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11 Higmans lemma 4111.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12 Innite descending chain 4212.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

13 Inverse relation 4313.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14 KleeneBrouwer order 4514.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.2 Tree interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.3 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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14.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15 Kruskals tree theorem 4715.1 Friedmans nite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

16 Knigs lemma 4916.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.2 Computability aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.3 Relationship to constructive mathematics and compactness . . . . . . . . . . . . . . . . . . . . . . 5016.4 Relationship with the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

17 Maximal element 5317.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.3 Maximal elements and the greatest element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.4 Directed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17.5.1 Consumer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.6 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 Mostowski collapse lemma 5718.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

19 Newmans lemma 5919.1 Diamond lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

19.3.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

20 Noetherian topological space 61

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20.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 6120.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

21 Non-well-founded set theory 6321.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6321.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

22 Order theory 6622.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

22.2.1 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.2.2 Visualizing a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.2.3 Special elements within an order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.2.5 Constructing new orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

22.3 Functions between orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.4 Special types of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.5 Subsets of ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.6 Related mathematical areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

22.6.1 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.6.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.6.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

22.7 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

23 Ordinal number 7423.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

23.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7723.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 7723.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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23.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

23.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

23.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

23.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

23.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24 Partially ordered set 8424.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 8624.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

25 Prewellordering 9125.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

25.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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25.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9225.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

26 Rewriting 9326.1 Intuitive examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

26.1.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9326.1.2 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

26.2 Abstract rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9426.2.1 Normal forms, joinability and the word problem . . . . . . . . . . . . . . . . . . . . . . . 9426.2.2 The ChurchRosser property and conuence . . . . . . . . . . . . . . . . . . . . . . . . 9426.2.3 Termination and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

26.3 String rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526.4 Term rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

26.4.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9626.4.2 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.4.3 Graph rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

26.5 Trace rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.6 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

27 RobertsonSeymour theorem 10127.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10127.2 Forbidden minor characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.3 Examples of minor-closed families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.4 Obstruction sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.5 Polynomial time recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327.6 Fixed-parameter tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.7 Finite form of the graph minor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

28 ScottPotter set theory 10628.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

28.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10628.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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28.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10728.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

28.2.1 Scotts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10828.2.2 Potters theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

28.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10928.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10928.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

29 Set theory 11129.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

29.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

29.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 11829.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11929.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

30 Structural induction 12030.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12030.2 Well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

31 Total order 12331.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12331.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12431.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

31.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12431.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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31.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

31.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 12631.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12631.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12631.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12631.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

32 Transitive set 12832.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

33 Universal set 13033.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

33.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

33.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13133.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

33.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13133.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13133.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

34 Well-founded relation 13334.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13334.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13434.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13434.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13534.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

35 Well-order 13635.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13635.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

35.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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35.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13935.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

36 Well-ordering principle 14036.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

37 Well-quasi-ordering 14137.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.4 Wqos versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.5 Innite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38 Well-structured transition system 14438.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

39 ZermeloFraenkel set theory 14539.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14539.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

39.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14639.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 14639.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14639.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14739.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14739.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14839.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14939.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14939.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

39.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15039.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

39.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

CONTENTS xi

39.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 154

39.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 1

Ascending chain condition

In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are nitenessproperties satised by some algebraic structures, most importantly, ideals in certain commutative rings.[1][2][3] Theseconditions played an important role in the development of the structure theory of commutative rings in the works ofDavid Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so thatthey make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory dueto Gabriel and Rentschler.

1.1 DenitionA partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every strictly ascendingsequence of elements eventually terminates. Equivalently, given any sequence

a1 a2 a3 ;there exists a positive integer n such that

an = an+1 = an+2 = :Similarly, P is said to satisfy the descending chain condition (DCC) if every strictly descending sequence of elementseventually terminates, that is, there is no innite descending chain. Equivalently every descending sequence

a3 a2 a1of elements of P, eventually stabilizes.

1.1.1 Comments A subtly dierent and stronger condition than containing no innite ascending/descending chains is containsno arbitrarily long ascending/descending chains (optionally, 'based at a given element')". For instance, thedisjoint union of the posets {0}, {0,1}, {0,1,2}, etc., satises both the ACC and the DCC, but has arbitrarilylong chains. If one further identies the 0 in all of these sets, then every chain is nite, but there are arbitrarilylong chains based at 0.

The descending chain condition on P is equivalent to P being well-founded: every nonempty subset of P has aminimal element (also called theminimal condition).

Similarly, the ascending chain condition is equivalent to P being converse well-founded: every nonempty subsetof P has a maximal element (themaximal condition).

1

2 CHAPTER 1. ASCENDING CHAIN CONDITION

Every nite poset satises both ACC and DCC.

A totally ordered set that satises the descending chain condition is called a well-ordered set.

1.2 See also Artinian Noetherian Krull dimension Ascending chain condition for principal ideals Maximal condition on congruences

1.3 Notes[1] Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.

[2] Fraleigh & Katz (1967), p. 366, Lemma 7.1

[3] Jacobson (2009), p. 142 and 147

1.4 References Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9

Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko. Algebras, rings and modules. Kluwer AcademicPublishers, 2004. ISBN 1-4020-2690-0

John B. Fraleigh, Victor J. Katz. A rst course in abstract algebra. Addison-Wesley Publishing Company. 5ed., 1967. ISBN 0-201-53467-3

Nathan Jacobson. Basic Algebra I. Dover, 2009. ISBN 978-0-486-47189-1

Chapter 2

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.

2.1 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

3

4 CHAPTER 2. 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]

2.2 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]

2.2.1 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.

2.2. ZERMELOS MODELS AND THE AXIOM OF LIMITATION OF SIZE 5

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. Wewill 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.

2.2.2 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.

6 CHAPTER 2. 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]

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

2.4 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).

2.5. REFERENCES 7

[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 the well-ordering of Vto choose the least element of each set.

[21] Von Neumann 1925, p. 223. English translation: p. 398. Von Neumanns proof, which only uses axioms, has the advantageof applying to all models rather than just to V.

[22] Kunen 1980, p. 95.

[23] Zermelo introduced strongly inaccessible cardinals so that V would satisfy ZFC. The axioms of power set and re-placement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, p. 3135; English translation: p.12211224.) Independently, Sierpiski and Tarski also introduced these cardinals in 1930.

[24] Zermelo used this sequence of cardinals to obtain a sequence of models that explains the paradoxes of set theory suchas, the Burali-Forti paradox and Russells paradox. He stated that the paradoxes depend solely on confusing set theoryitself with individual models representing it. What appears as an 'ultranite non- or super-set' in one model is, in thesucceeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundationstone for the construction of a new domain [model]. (Zermelo 1930, p. 4647; English translation: p. 1233.)

[25] Zermelo proved that ZFC without the axiom of innity is satised by V for = and strongly inaccessible. To provethe class existence axioms of NBG (Gdel 1940, p. 5), note that V is a set when viewed from the set theory that constructsit. Therefore, the axiom of specication produces subsets of V that satisfy the class existence axioms.

2.5 References Bernays, Paul (1991), Axiomatic Set Theory, Dover Publications, ISBN 0-486-66637-9.

William B. Easton (1964), Powers of Regular Cardinals, Ph.D. thesis, Princeton University.

Ferreirs, Jos (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought(2nd revised ed.), Basel, Switzerland: Birkhuser, ISBN 3-7643-8349-6.

Fraenkel, Abraham (1922), Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre,Mathematische An-nalen 86: 230237, doi:10.1007/bf01457986.

Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of Set Theory (2nd revised ed.),Basel, Switzerland: Elsevier, ISBN 0-7204-2270-1.

Gdel, Kurt (1940), The Consistency of the Continuum Hypothesis, Princeton University Press.

Kanamori, Akihiro, Stanislaw Ulam, http://math.bu.edu/people/aki/9.pdf Missing or empty |title= (help) in:Solomon Fefermann and John W. Dawson, Jr. (editors-in-chief) (2003), Kurt Gdel Collected Works, VolumeV, Correspondence H-Z, Clarendon Press, pp. 280300.

Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 0-444-85401-0.* Hallett, Michael (1984), Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press,ISBN 0-444-86839-9.

Mirimano, Dmitry (1917), Les antinomies de Russell et de Burali-Forti et le probleme fondamental de latheorie des ensembles, L'Enseignement Mathmatique 19: 3752.

8 CHAPTER 2. AXIOM OF LIMITATION OF SIZE

Moore, Gregory H. (1982), Zermelos Axiom of Choice: Its Origins, Development, and Inuence, Springer,ISBN 0-387-90670-3.

Sierpiski, Wacaw; Tarski, Alfred (1930), Sur une proprit caractristique des nombres inaccessibles,Fundamenta Mathematicae 15: 292300, ISSN 0016-2736.

Skolem, Thoralf (1922), Einige Bemerkungen zur axiomatischen Begrndung der Mengenlehre,Matematik-erkongressen i Helsingfors den 4-7 Juli, 1922, pp. 217232. English translation: van Heijenoort, Jean (1967),Some remarks on axiomatized set theory, From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 290301, ISBN 978-0-674-32449-7.

von Neumann, John (1925), Eine Axiomatisierung der Mengenlehre, Journal fr die Reine und AngewandteMathematik 154: 219240. English translation: van Heijenoort, Jean (1967), An axiomatization of set the-ory, From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp.393413, ISBN 978-0-674-32449-7.

von Neumann, John (1928), Die Axiomatisierung der Mengenlehre,Mathematische Zeitschrift 27: 669752,doi:10.1007/bf01171122.

Zermelo, Ernst (1930), "ber Grenzzahlen und Mengenbereiche: neue Untersuchungen ber die Grundla-gen der Mengenlehre, Fundamenta Mathematicae 16: 2947. English translation: Ewald, William B. (ed.)(1996), On boundary numbers and domains of sets: new investigations in the foundations of set theory, FromImmanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press,pp. 12081233, ISBN 978-0-19-853271-2.

Zermelo, Ernst (1908), Untersuchungen ber die Grundlagen der Mengenlehre I, Mathematische Annalen65 (2): 261281, doi:10.1007/bf01449999. English translation: van Heijenoort, Jean (1967), Investigationsin the foundations of set theory, From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931,Harvard University Press, pp. 199215, ISBN 978-0-674-32449-7.

Chapter 3

Axiom of regularity

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of ZermeloFraenkelset theory that states that every non-empty set A contains an element that is disjoint from A. In rst-order logic theaxiom reads:

8x (x 6= ?! 9y 2 x (y \ x = ?))

The axiom implies that no set is an element of itself, and that there is no innite sequence (an) such that ai+1 is anelement of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), thisresult can be reversed: if there are no such innite sequences, then the axiom of regularity is true. Hence, the axiomof regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downwardinnite membership chains.The axiom of regularity was introduced by von Neumann (1925); it was adopted in a formulation closer to the onefound in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics basedon set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makessome properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but alsoon proper classes that are well-founded relational structures such as the lexicographical ordering on f(n; )jn 2! ^ ordinal an is g :Given the other axioms of ZermeloFraenkel set theory, the axiom of regularity is equivalent to the axiom of induc-tion. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones thatdo not accept the law of the excluded middle), where the two axioms are not equivalent.In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of setsthat are elements of themselves.

3.1 Elementary implications of regularity

3.1.1 No set is an element of itself

Let A be a set, and apply the axiom of regularity to {A}, which is a set by the axiom of pairing. We see that theremust be an element of {A} which is disjoint from {A}. Since the only element of {A} is A, it must be that A is disjointfrom {A}. So, since A {A}, we cannot have A A (by the denition of disjoint).

3.1.2 No innite descending sequence of sets exists

Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for eachn. Dene S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema ofreplacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the denitionof S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an

9

10 CHAPTER 3. AXIOM OF REGULARITY

element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets. Sinceour supposition led to a contradiction, there must not be any such function, f.The nonexistence of a set containing itself can be seen as a special case where the sequence is innite and constant.Notice that this argument only applies to functions f that can be represented as sets as opposed to undenable classes.The hereditarily nite sets, V, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom ofinnity). So if one forms a non-trivial ultrapower of V, then it will also satisfy the axiom of regularity. The resultingmodel will contain elements, called non-standard natural numbers, that satisfy the denition of natural numbers inthat model but are not really natural numbers. They are fake natural numbers which are larger than any actualnatural number. This model will contain innite descending sequences of elements. For example, suppose n is anon-standard natural number, then (n 1) 2 n and (n 2) 2 (n 1) , and so on. For any actual natural numberk, (n k 1) 2 (n k) . This is an unending descending sequence of elements. But this sequence is not denablein the model and thus not a set. So no contradiction to regularity can be proved.

3.1.3 Simpler set-theoretic denition of the ordered pairThe axiom of regularity enables dening the ordered pair (a,b) as {a,{a,b}}. See ordered pair for specics. Thisdenition eliminates one pair of braces from the canonical Kuratowski denition (a,b) = {{a},{a,b}}.

3.1.4 Every set has an ordinal rankThis was actually the original form of von Neumanns axiomatization.

3.1.5 For every two sets, only one can be an element of the otherLet X and Y be sets. Then apply the axiom of regularity to the set {X,Y}. We see there must be an element of {X,Y}which is also disjoint from it. It must be either X or Y. By the denition of disjoint then, we must have either Y is notan element of X or vice versa.

3.2 The axiom of dependent choice and no innite descending sequence ofsets implies regularity

Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-emptyintersection with S. We dene a binary relation R on S by aRb :, b 2 S \ a , which is entire by assumption. Thus,by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is aninnite descending chain, we arrive at a contradiction and so, no such S exists.

3.3 Regularity and the rest of ZF(C) axiomsRegularity was shown to be relatively consistent with the rest of ZF by von Neumann (1929), meaning that if ZFwithout regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation seeVaught (2001, 10.1) for instance.The axiom of regularity was also shown to be independent from the other axioms of ZF(C), assuming they areconsistent. The result was announced by Paul Bernays in 1941, although he did not publish a proof until 1954. Theproof involves (and led to the study of) Rieger-Bernays permutation models (or method), which were used for otherproofs of independence for non-well-founded systems (Rathjen 2004, p. 193 and Forster 2003, pp. 210212).

3.4 Regularity and Russells paradoxNaive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent dueto Russells paradox. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension

3.5. REGULARITY, THE CUMULATIVE HIERARCHY, AND TYPES 11

with the much weaker axiom schema of separation. However, this makes set theory too weak. So some of thepower of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset,replacement, and innity) which may be regarded as special cases of comprehension. So far, these axioms do notseem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added toexclude models with some undesirable properties. These two axioms are known to be relatively consistent.In the presence of the axiom schema of separation, Russells paradox becomes a proof that there is no set of all sets.The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition isredundant when added to the rest of ZF. If the ZF axioms without regularity were already inconsistent, then addingregularity would not make them consistent.The existence of Quine atoms (sets that satisfy the formula equation x = {x}, i.e. have themselves as their only ele-ments) is consistent with the theory obtained by removing the axiom of regularity fromZFC. Various non-wellfoundedset theories allow safe circular sets, such as Quine atoms, without becoming inconsistent by means of Russellsparadox.(Rieger 2011, pp. 175,178)

3.5 Regularity, the cumulative hierarchy, and typesIn ZF it can be proven that the classS V (see cumulative hierarchy) is equal to the class of all sets. This statementis even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which doesnot satisfy axiom of regularity, a model which satises it can be constructed by taking only sets inS V .Herbert Enderton (1977, p. 206) wrote that The idea of rank is a descendant of Russells concept of type". Com-paring ZF with type theory, Alasdair Urquhart wrote that Zermelos system has the notational advantage of notcontaining any explicitly typed variables, although in fact it can be seen as having an implicit type structure built intoit, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [Zermelo 1930],and again in a well-known article of George Boolos [Boolos 1971]. Urquhart (2003, p. 305)Dana Scott (1974) went further and claimed that:

The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use ofsome form of the theory of types. That was at the basis of both Russells and Zermelos intuitions.Indeed the best way to regard Zermelos theory is as a simplication and extension of Russells. (Wemean Russells simple theory of types, of course.) The simplication was to make the types cumulative.Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowedto accumulate the earlier ones, we can then easily imagine extending the types into the transnitejusthow far we want to go must necessarily be left open. Now Russell made his types explicit in his notationand Zermelo left them implicit. (emphasis in original)

In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchyturns out to be equivalent to ZF, including regularity. (Lvy 2002, p. 73)

3.6 HistoryThe concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimano (1917) cf. Lvy(2002, p. 68) and Hallett (1986, 4.4, esp. p. 186, 188). Mirimano called a set x regular (French: ordinaire) ifevery descending chain x x1 x2 ... is nite. Mirimano however did not consider his notion of regularity (andwell-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 6263); in later papers Mirimanoalso explored what are now called non-well-founded sets (extraordinaire in Mirimanos terminology) (Sangiorgi2011, pp. 1719, 26).According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as superuous (on p. 404 invan Heijenoort 's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) whichexcludes some, but not all, non-well-founded sets (Rieger 2011, p. 179). In a subsequent publication, von Neumann(1928) gave the following axiom (rendered in modern notation by A. Rieger):

8x (x 6= ; ! 9y 2 x (y \ x = ;))

12 CHAPTER 3. AXIOM OF REGULARITY

3.7 See also Non-well-founded set theory

3.8 References 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 Boolos, George (1971), The iterative conception of set, Journal of Philosophy 68: 215231, doi:10.2307/2025204reprinted in Boolos, George (1998), Logic, Logic and Logic, Harvard University Press, pp. 1329

Enderton, Herbert B. (1977), Elements of Set Theory, Academic Press Urquhart, Alasdair (2003), The Theory of Types, in Grin, Nicholas, The Cambridge Companion to Bertrand

Russell, Cambridge University Press

Halbeisen, Lorenz J. (2012), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer Sangiorgi, Davide (2011), Origins of bisimulation and coinduction, in Sangiorgi, Davide; Rutten, Jan, Ad-

vanced Topics in Bisimulation and Coinduction, Cambridge University Press

Lvy, Azriel (2002) [rst published in 1979], Basic set theory, Dover Publications, ISBN 0-486-42079-5 Hallett, Michael (1996) [rst published 1984], Cantorian set theory and limitation of size, Oxford UniversityPress, ISBN 0-19-853283-0

Rathjen, M. (2004), Predicativity, Circularity, and Anti-Foundation, in Link, Godehard, One Hundred Yearsof Russell s Paradox: Mathematics, Logic, Philosophy (PDF), Walter de Gruyter, ISBN 978-3-11-019968-0

Forster, T. (2003), Logic, induction and sets, Cambridge University Press Rieger, Adam (2011), Paradox, ZF, and the Axiom of Foundation, in David DeVidi, Michael Hallett, PeterClark, Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell., pp. 171187,doi:10.1007/978-94-007-0214-1_9, ISBN 978-94-007-0213-4

Vaught, Robert L. (2001), Set Theory: An Introduction (2nd ed.), Springer, ISBN 978-0-8176-4256-3

3.8.1 Primary sources Mirimano, D. (1917), Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theoriedes ensembles, L'Enseignement Mathmatique 19: 3752

von Neumann, J. (1925), Eine axiomatiserung der Mengenlehre, Journal fr die reine und angewandte Math-ematik 154: 219240; translation in van Heijenoort, Jean (1967), From Frege to Gdel: A Source Book inMathematical Logic, 18791931, pp. 393413

von Neumann, J. (1928), "ber die Denition durch transnite Induktion und verwandte Fragen der allge-meinen Mengenlehre, Mathematische Annalen 99: 373391, doi:10.1007/BF01459102

von Neumann, J. (1929), Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journalfur die reine und angewandte Mathematik 160: 227241, doi:10.1515/crll.1929.160.227

Zermelo, Ernst (1930), "ber Grenzzahlen und Mengenbereiche. Neue Untersuchungen ber die Grundlagender Mengenlehre. (PDF), Fundamenta Mathematicae 16: 2947; translation in Ewald, W.B., ed. (1996),From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2, Clarendon Press, pp. 121933

Bernays, P. (1941), A system of axiomatic set theory. Part II, The Journal of Symbolic Logic 6: 117,doi:10.2307/2267281

3.9. EXTERNAL LINKS 13

Bernays, P. (1954), A system of axiomatic set theory. Part VII, The Journal of Symbolic Logic 19: 8196,doi:10.2307/2268864

Riegger, L. (1957), A contribution to Gdels axiomatic set theory (PDF), Czechoslovak Mathematical Jour-nal 7: 323357

Scott, D. (1974), Axiomatizing set theory,Axiomatic set theory. Proceedings of Symposia in PureMathematicsVolume 13, Part II, pp. 207214

3.9 External links http://www.trinity.edu/cbrown/topics_in_logic/sets/sets.html contains an informative description of the axiomof regularity under the section on Zermelo-Fraenkel set theory.

Axiom of Foundation at PlanetMath.org.

Chapter 4

Better-quasi-ordering

In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array.Every bqo is well-quasi-ordered.

4.1 Motivation

Though wqo is an appealing notion, many important innitary operations do not preserve wqoness. An exampledue to Richard Rado illustrates this.[1] In a 1965 paper Crispin Nash-Williams formulated the stronger notion ofbqo in order to prove that the class of trees of height is wqo under the topological minor relation.[2] Since then,many quasi-orders have been proven to be wqo by proving them to be bqo. For instance, Richard Laver establishedFrass's conjecture by proving that the class of scattered linear order types is bqo.[3] More recently, Carlos Martinez-Ranero has proven that, under the Proper Forcing Axiom, the class of Aronszajn lines is bqo under the embeddabilityrelation.[4]

4.2 Denition

It is common in bqo theory to write x for the sequence x with the rst term omitted. Write [!]4.3 Simpsons alternative denition

Simpson introduced an alternative denition of bqo in terms of Borel maps [!]! ! Q , where [!]! , the set of innitesubsets of ! , is given the usual (product) topology.[5]

Let Q be a quasi-order and endow Q with the discrete topology. A Q -array is a Borel function [A]! ! Q forsome innite subset A of ! . A Q -array f is bad if f(X) 6Q f(X) for every X 2 [A]! ; f is good otherwise.The quasi-order Q is bqo if there is no bad Q -array in this sense.

14

4.4. MAJOR THEOREMS 15

4.4 Major theoremsMany major results in bqo theory are consequences of the Minimal Bad Array Lemma, which appears in Simpsonspaper[5] as follows. See also Lavers paper,[6] where the Minimal Bad Array Lemma was rst stated as a result. Thetechnique was present in Nash-Williams original 1965 paper.Suppose (Q;Q) is a quasi-order. A partial ranking 0 of Q is a well-founded partial ordering of Q such thatq 0 r ! q Q r . For bad Q -arrays (in the sense of Simpson) f : [A]! ! Q and g : [B]! ! Q , dene:

g f if B A and g(X) 0 f(X) every for X 2 [B]!

g

Chapter 5

Binary relation

Relation (mathematics)" redirects here. For a more general notion of relation, see nitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A A. More generally, a binary relation between two sets A and B is a subsetof A B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include 4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", anddivides in arithmetic, "is congruent to" in geometry, is adjacent to in graph theory, is orthogonal to in linearalgebra and many more. The concept of function is dened as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R A1 An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZZZ is lies between ... and ..., containing e.g. the triples (5,2,8), (5,8,2), and (4,9,7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of is an element of or is a subset of in settheory, without running into logical inconsistencies such as Russells paradox.

5.1 Formal denition

A binary relation R is usually dened as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as dened by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X Y, and from X to Y" must always be either specied or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

16

5.2. SPECIAL TYPES OF BINARY RELATIONS 17

5.1.1 Is a relation more than its graph?According to the denition above, two relations with identical graphs but dierent domains or dierent codomainsare considered dierent. For example, ifG = f(1; 2); (1; 3); (2; 7)g , then (Z;Z; G) , (R;N; G) , and (N;R; G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often dened as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then dened as the set of all x such that there exists at least oney such that (x; y) 2 R , the range of R is dened as the set of all y such that there exists at least one x such that(x; y) 2 R , and the eld of R is the union of its domain and its range.[2][3][4]A special case of this dierence in points of view applies to the notion of function. Many authors insist on distin-guishing between a functions codomain and its range. Thus, a single rule, like mapping every real number x tox2, can lead to distinct functions f : R ! R and f : R ! R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique rst components. This dierence in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivityor being ontoas a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the denitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodenitions usually matters only in very formal contexts, like category theory.

5.1.2 ExampleExample: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation is owned by is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the rst element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by RJ means that the ball is owned by John.Two dierent relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is dierent from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identied or even dened as G(R) and an ordered pair (x, y) G(R)" is usually denoted as"(x, y) R".

5.2 Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be dierent sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = 5and z = +5 to y = 25.

functional (also called univalent[8] or right-unique[7] or right-denite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=5 and z=+5.

18 CHAPTER 5. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is dierentfrom the denition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = 14 to any real number y.

surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = 14.

Uniqueness and totality properties:

5.3. RELATIONS OVER A SET 19

A function: a relation that is functional and left-total. Both the green and the red relation are functions. An injective function: a relation that is injective, functional, and left-total. A surjective function or surjection: a relation that is functional, left-total, and right-total. A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

5.2.1 DifunctionalLess commonly encountered is the notion of difunctional (or regular) relation, dened as a relation R such thatR=RR1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element xX to a set xR = { yY| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can dene the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R x2R implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A Cand g: B C and then dene the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) A B | f(a) = g(b) }. Every difunctional relation R A B arises as the joint kernel of two functionsf: A C and g: B C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justied by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term rectangular to denote any heterogeneous relation whatsoever.[6]

5.3 Relations over a setIf X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph