Ascending chain conditionWikipedia

Contents

1 Artinian 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Ascending chain condition 22.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Ascending chain condition on principal ideals 43.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Axiom of limitation of size 64.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Zermelos models and the axiom of limitation of size . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2.1 The model V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.2 The models V where is a strongly inaccessible cardinal . . . . . . . . . . . . . . . . . . 8

4.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Axiom of regularity 125.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.2 No innite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 125.1.3 Simpler set-theoretic denition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 135.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 13

5.2 The axiom of dependent choice and no innite descending sequence of sets implies regularity . . . . 135.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 Regularity and Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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5.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Better-quasi-ordering 176.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Simpsons alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Major theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Binary relation 197.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Class (set theory) 298.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Classes in formal set theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Commutative ring 319.1 Denition and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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9.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.1.2 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9.2 Ideals and the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2.1 Ideals and factor rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2.2 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2.3 Prime ideals and the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.3 Ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.5 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.6 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.7 Constructing commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.7.1 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.8 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.10.1 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 Dicksons lemma 3810.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.3 Generalizations and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11 Element (mathematics) 4111.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12 Empty set 4412.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

12.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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12.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13 Epsilon-induction 5013.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14 Eventually (mathematics) 5114.1 Motivation and Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15 Higmans lemma 5215.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16 Ideal (ring theory) 5316.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.6 Ideal generated by a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.7 Types of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.8 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.9 Ideal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.10Ideals and congruence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

17 Innite descending chain 5917.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18 Integer 6018.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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18.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19 Inverse relation 6619.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

19.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

20 KleeneBrouwer order 6820.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.2 Tree interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.3 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

21 Krull dimension 7021.1 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.2 Krull dimension and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.4 Krull dimension of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.5 Krull dimension for non-commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

22 Kruskals tree theorem 7322.1 Friedmans nite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

23 Knigs lemma 7523.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

23.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.2 Computability aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.3 Relationship to constructive mathematics and compactness . . . . . . . . . . . . . . . . . . . . . . 7623.4 Relationship with the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7723.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7723.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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23.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7723.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

24 Maximal element 7924.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.3 Maximal elements and the greatest element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.4 Directed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

24.5.1 Consumer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8124.6 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

25 Mostowski collapse lemma 8325.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

26 Newmans lemma 8526.1 Diamond lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

26.3.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

27 Noetherian 8727.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

28 Noetherian topological space 8828.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 8828.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8928.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

29 Non-well-founded set theory 9029.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9029.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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30 Order theory 9330.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9330.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

30.2.1 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.2.2 Visualizing a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.2.3 Special elements within an order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.2.5 Constructing new orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

30.3 Functions between orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.4 Special types of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9730.5 Subsets of ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9830.6 Related mathematical areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

30.6.1 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9830.6.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9830.6.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

30.7 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

31 Ordinal number 10131.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10231.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

31.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 10431.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

31.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

31.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

31.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10731.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

31.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

31.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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31.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

32 Partially ordered set 11132.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 11332.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11532.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11532.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11732.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11732.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

33 Prewellordering 11833.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

33.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

33.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11933.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

34 Rewriting 12034.1 Intuitive examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

34.1.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12034.1.2 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

34.2 Abstract rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12134.2.1 Normal forms, joinability and the word problem . . . . . . . . . . . . . . . . . . . . . . . 12134.2.2 The ChurchRosser property and conuence . . . . . . . . . . . . . . . . . . . . . . . . 12134.2.3 Termination and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

34.3 String rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12234.4 Term rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

34.4.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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34.4.2 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12434.4.3 Graph rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

34.5 Trace rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.6 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12634.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12634.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

35 RobertsonSeymour theorem 12835.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12835.2 Forbidden minor characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12935.3 Examples of minor-closed families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12935.4 Obstruction sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12935.5 Polynomial time recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13035.6 Fixed-parameter tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.7 Finite form of the graph minor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

36 ScottPotter set theory 13336.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

36.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13336.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

36.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13536.2.1 Scotts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13536.2.2 Potters theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

36.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

37 Semigroup 13837.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

37.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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37.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.8 Semigroup methods in partial dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.13Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

38 Set theory 14638.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14938.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14938.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15038.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

38.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

38.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 15338.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15438.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

39 Structural induction 15539.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.2 Well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

40 Total order 15840.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

CONTENTS xi

40.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

40.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

40.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 16140.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

41 Transitive set 16341.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16441.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16441.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

42 Universal set 16542.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

42.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

42.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16642.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

42.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16642.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16642.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

43 Well-founded relation 16843.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

44 Well-order 171

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44.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17144.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

44.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

44.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17444.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

45 Well-ordering principle 17545.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

46 Well-quasi-ordering 17646.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17646.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17646.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17646.4 Wqos versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.5 Innite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17846.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17846.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

47 Well-structured transition system 17947.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17947.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

48 ZermeloFraenkel set theory 18048.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18048.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

48.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18148.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 18148.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18148.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18348.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18448.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18448.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

48.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18548.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

CONTENTS xiii

48.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18648.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18648.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18748.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18748.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18848.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 189

48.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19448.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Chapter 1

Artinian

In mathematics, Artinian, named for Emil Artin, is an adjective that describes objects that satisfy particular cases ofthe descending chain condition.

A ring is an Artinian ring if it satises the descending chain condition on (one-sided) ideals A module is an Artinian module if it satises the descending chain condition on submodules.

1.1 See also Araz Artinian, an Armenian-Canadian lm maker and photographer

1

Chapter 2

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.

2.1 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.

2.1.1 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).

2

2.2. SEE ALSO 3

Every nite poset satises both ACC and DCC.

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

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

2.3 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

2.4 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 3

Ascending chain condition on principalideals

In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, orprincipal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principalideals (abbreviated toACCP) is satised if there is no innite strictly ascending chain of principal ideals of the giventype (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.The counterpart descending chain condition may also be applied to these posets, however there is currently no needfor the terminology DCCP since such rings are already called left or right perfect rings. (See Noncommutative ringsection below.)Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings alsosatisfy (ACCP), notably unique factorization domains and left or right perfect rings.

3.1 Commutative ringsIt is well known that a nonzero nonunit in a Noetherian integral domain factors into irreducibles. The proof of thisrelies on only (ACCP) not (ACC), so in any integral domain with (ACCP), an irreducible factorization exists. (Inother words, any integral domains with (ACCP) are atomic. But the converse is false, as shown in (Grams 1974).)Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses Euclids lemma,which requires factors to be prime rather than just irreducible. Indeed one has the following characterization: let Abe an integral domain. Then the following are equivalent.

1. A is a UFD.2. A satises (ACCP) and every irreducible of A is prime.3. A is a GCD domain satisfying (ACCP).

The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closedsubset of A generated by prime elements. If the localization S1A is a UFD, so is A. (Nagata 1975, Lemma 2.1)(Note that the converse of this is trivial.)An integral domain A satises (ACCP) if and only if the polynomial ring A[t] does.[1] The analogous fact is false ifA is not an integral domain. (Heinzer, Lantz 1994)An integral domain where every nitely generated ideal is principal (that is, a Bzout domain) satises (ACCP) ifand only if it is a principal ideal domain.[2]

The ringZ+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actuallya GCD domain) that does not satisfy (ACCP), for the chain of principal ideals

(X) (X/2) (X/4) (X/8); :::

4

3.2. NONCOMMUTATIVE RINGS 5

is non-terminating.

3.2 Noncommutative ringsIn the noncommutative case, it becomes necessary to distinguish the right ACCP from left ACCP. The former onlyrequires the poset of ideals of the form xR to satisfy the ascending chain condition, and the latter only examines theposet of ideals of the form Rx.A theorem of Hyman Bass in (Bass 1960) now known as Bass Theorem P showed that the descending chaincondition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. Jonah showed in (Jonah1970) that there is a side-switching connection between the ACCP and perfect rings. It was shown that if R is rightperfect (satises right DCCP), then R satises the left ACCP, and symmetrically, if R is left perfect (satises leftDCCP), then it satises the right ACCP. The converses are not true, and the above switches from left and rightare not typos.Whether the ACCP holds on the right or left side of R, it implies that R has no innite set of nonzero orthogonalidempotents, and that R is a Dedekind nite ring. (Lam 1999, p.230-231)

3.3 References[1] Gilmer, Robert (1986), Property E in commutative monoid rings, Group and semigroup rings (Johannesburg, 1985),

North-Holland Math. Stud. 126, Amsterdam: North-Holland, pp. 1318, MR 860048.

[2] Proof: In a Bzout domain the ACCP is equivalent to the ACC on nitely generated ideals, but this is known to be equivalentto the ACC on all ideals. Thus the domain is Noetherian and Bzout, hence a principal ideal domain.

Bass, Hyman (1960), Finitistic dimension and a homological generalization of semi-primary rings, Trans.Amer. Math. Soc. 95: 466488, doi:10.1090/s0002-9947-1960-0157984-8, ISSN 0002-9947, MR 0157984

Grams, Anne (1974), Atomic rings and the ascending chain condition for principal ideals, Proc. CambridgePhilos. Soc. 75: 321329, doi:10.1017/s0305004100048532, MR 0340249

Heinzer, William J.; Lantz, David C. (1994), ACCP in polynomial rings: a counterexample, Proc. Amer.Math. Soc. 121 (3): 975977, doi:10.2307/2160301, ISSN 0002-9939, JSTOR 2160301, MR 1653294

Jonah, David (1970), Rings with theminimum condition for principal right ideals have themaximum conditionfor principal left ideals, Math. Z. 113: 106112, doi:10.1007/bf01141096, ISSN 0025-5874, MR 0260779

Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

Nagata, Masayoshi (1975), Some types of simple ring extensions, Houston J. Math. 1 (1): 131136, ISSN0362-1588, MR 0382248

Chapter 4

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.

4.1 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

6

4.2. ZERMELOS MODELS AND THE AXIOM OF LIMITATION OF SIZE 7

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]

4.2 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]

4.2.1 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.

8 CHAPTER 4. AXIOM OF LIMITATION OF SIZE

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.

4.2.2 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.

4.3. SEE ALSO 9

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]

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

4.4 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).

10 CHAPTER 4. AXIOM OF LIMITATION OF SIZE

[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.

4.5 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.

4.5. REFERENCES 11

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 5

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.

5.1 Elementary implications of regularity

5.1.1 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).

5.1.2 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

12

5.2. THEAXIOMOFDEPENDENTCHOICEANDNO INFINITEDESCENDING SEQUENCEOF SETS IMPLIES REGULARITY13

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.

5.1.3 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}}.

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

5.1.5 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.

5.2 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.

5.3 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).

5.4 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

14 CHAPTER 5. AXIOM OF REGULARITY

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)

5.5 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)

5.6 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 = ;))

5.7. SEE ALSO 15

5.7 See also Non-well-founded set theory

5.8 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

5.8.1 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/BF0145910