WellfoundednessWikipedia

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 Dicksons lemma 165.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Generalizations and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Epsilon-induction 196.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Higmans lemma 207.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Innite descending chain 218.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9 KleeneBrouwer order 229.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.2 Tree interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.3 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10 Kruskals tree theorem 2410.1 Friedmans nite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 Knigs lemma 2611.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2 Computability aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.3 Relationship to constructive mathematics and compactness . . . . . . . . . . . . . . . . . . . . . . 2711.4 Relationship with the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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11.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Mostowski collapse lemma 3012.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

13 Newmans lemma 3213.1 Diamond lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13.3.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14 Noetherian topological space 3414.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 3414.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

15 Non-well-founded set theory 3615.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

16 Ordinal number 3916.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

16.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 4216.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

16.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4316.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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16.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4316.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

16.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

16.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

16.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

17 Prewellordering 4917.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

17.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

17.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

18 RobertsonSeymour theorem 5118.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.2 Forbidden minor characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.3 Examples of minor-closed families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.4 Obstruction sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.5 Polynomial time recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.6 Fixed-parameter tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.7 Finite form of the graph minor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

19 ScottPotter set theory 5619.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

19.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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19.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.2.1 Scotts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.2.2 Potters theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

19.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

20 Structural induction 6120.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.2 Well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

21 Universal set 6421.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

21.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

22 Well-founded relation 6722.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6822.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6822.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

23 Well-order 7023.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

24 Well-ordering principle 74

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24.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

25 Well-quasi-ordering 7525.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.4 Wqos versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.5 Innite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

26 Well-structured transition system 7826.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 79

26.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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

Dicksons lemma

In mathematics, Dicksons lemma states that every set of n -tuples of natural numbers has nitely many minimalelements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, whoused it to prove a result in number theory about perfect numbers.[1] However, the lemma was certainly known earlier,for example to Paul Gordan in his research on invariant theory.[2]

5.1 ExampleLetK be a xed number, and let S = f(x; y) j xy Kg be the set of pairs of numbers whose product is at leastK. When dened over the positive real numbers, S has innitely many minimal elements of the form (x;K/x) , onefor each positive number x ; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbolaare minimal, because it is not possible for a dierent pair that belongs to S to be less than or equal to (x;K/x) inboth of its coordinates. However, Dicksons lemma concerns only tuples of natural numbers, and over the naturalnumbers there are only nitely many minimal pairs. Every minimal pair (x; y) of natural numbers has x K andy K , for if x were greater than K then (x 1,y) would also belong to S, contradicting the minimality of (x,y), andsymmetrically if y were greater than K then (x,y 1) would also belong to S. Therefore, over the natural numbers, Shas at mostK2 minimal elements, a nite number.[3]

5.2 Formal statementLet N be the set of non-negative integers (natural numbers), let n be any xed constant, and let Nn be the set ofn -tuples of natural numbers. These tuples may be given a pointwise partial order, the product order, in which(a1; a2; : : : ; an) (b1; b2; : : : bn) if and only if, for every i , ai bi . The set of tuples that are greater than orequal to some particular tuple (a1; a2; : : : ; an) forms a positive orthant with its apex at the given tuple.With this notation, Dicksons lemma may be stated in several equivalent forms:

In every subset S of Nn , there are nitely many elements that are minimal elements of S for the pointwisepartial order

In every innite set of n -tuples of natural numbers, there exist two tuples (a1; a2; : : : ; an) and (b1; b2; : : : bn)such that, for every i , ai bi .[4]

The partially ordered set (Nn;) is a well partial order.[5]

Every subset S of Nn may be covered by a nite set of positive orthants, whose apexes all belong to S

5.3 Generalizations and applicationsDickson used his lemma to prove that, for any given number n , there can exist only a nite number of odd perfectnumbers that have at most n prime factors.[1] However, it remains open whether there exist any odd perfect numbers

16

5.4. SEE ALSO 17

Innitely many minimal pairs of real numbers x,y (the black hyperbola) but only ve minimal pairs of positive integers (red) havexy 9.

at all.The divisibility relation among the P-smooth numbers, natural numbers whose prime factors all belong to the niteset P, gives these numbers the structure of a partially ordered set isomorphic to (NjP j;) . Thus, for any set S ofP-smooth numbers, there is a nite subset of S such that every element of S is divisible by one of the numbers inthis subset. This fact has been used, for instance, to show that there exists an algorithm for classifying the winningand losing moves from the initial position in the game of Sylver coinage, even though the algorithm itself remainsunknown.[6]

The tuples (a1; a2; : : : ; an) inNn correspond one-for-onewith themonomialsxa11 xa22 : : : xann over a set ofn variablesx1; x2; : : : xn . Under this correspondence, Dicksons lemma may be seen as a special case of Hilberts basis theoremstating that every polynomial ideal has a nite basis, for the ideals generated by monomials. Indeed, Paul Gordanused this restatement of Dicksons lemma in 1899 as part of a proof of Hilberts basis theorem.[2]

5.4 See also

Gordans lemma

18 CHAPTER 5. DICKSONS LEMMA

5.5 References[1] Dickson, L. E. (1913), Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors,

American Journal of Mathematics 35 (4): 413422, doi:10.2307/2370405, JSTOR 2370405.

[2] Buchberger, Bruno; Winkler, Franz (1998), Grbner Bases and Applications, London Mathematical Society Lecture NoteSeries 251, Cambridge University Press, p. 83, ISBN 9780521632980.

[3] With more care, it is possible to show that one of x and y is at mostpK , and that there is at most one minimal pair for

each choice of one of the coordinates, from which it follows that there are at most 2pK minimal elements.

[4] Figueira, Diego; Figueira, Santiago; Schmitz, Sylvain; Schnoebelen, Philippe (2011), Ackermannian and primitive-recursive bounds with Dicksons lemma, 26th Annual IEEE Symposium on Logic in Computer Science (LICS 2011), IEEEComputer Soc., Los Alamitos, CA, pp. 269278, arXiv:1007.2989, doi:10.1109/LICS.2011.39, MR 2858898.

[5] Onn, Shmuel (2008), Convex Discrete Optimization, in Floudas, Christodoulos A.; Pardalos, Panos M., Encyclopedia ofOptimization, Vol. 1 (2nd ed.), Springer, pp. 513550, arXiv:math/0703575, ISBN 9780387747583.

[6] Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2003), 18 The Emperor and his Money, Winning Ways foryour Mathematical Plays, Vol. 3, Academic Press, pp. 609640. See especially Are outcomes computable, p. 630.

Chapter 6

Epsilon-induction

In mathematics, 2 -induction (epsilon-induction) is a variant of transnite induction that can be used in set theoryto prove that all sets satisfy a given property P[x]. If the truth of the property for x follows from its truth for allelements of x, for every set x, then the property is true of all sets. In symbols:

8x8y(y 2 x! P [y])! P [x]

! 8xP [x]

This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity giventhe other ZF axioms. 2 -induction is a special case of well-founded induction.The name is most often pronounced epsilon-induction, because the set membership symbol2 historically developedfrom the Greek letter .

6.1 See also Mathematical induction Transnite induction Well-founded induction

19

Chapter 7

Higmans lemma

In mathematics, Higmans lemma states that the set of nite sequences over a nite alphabet, as partially orderedby the subsequence relation, is well-quasi-ordered. That is, if w1; w2; : : : is an innite sequence of words over somexed nite alphabet, then there exist indices i < j such that wi can be obtained from wj by deleting some (possiblynone) symbols. More generally this remains true when the alphabet is not necessarily nite, but is itself well-quasi-ordered, and the subsequence relation allows the replacement of symbols by earlier symbols in the well-quasi-orderingof labels. This is a special case of the later Kruskals tree theorem. It is named after Graham Higman, who publishedit in 1952.

7.1 References Higman, Graham (1952), Ordering by divisibility in abstract algebras, Proceedings of the London Mathe-

matical Society, (3) 2 (7): 326336, doi:10.1112/plms/s3-2.1.326

20

Chapter 8

Innite descending chain

Given a set S with a partial order , an innite descending chain is an innite, strictly decreasing sequence ofelements x1 > x2 > ... > xn > ...As an example, in the set of integers, the chain 1, 2, 3, ... is an innite descending chain, but there exists noinnite descending chain on the natural numbers, as every chain of natural numbers has a minimal element.If a partially ordered set does not possess any innite descending chains, it is said then, that it satises the descendingchain condition. Assuming the axiom of choice, the descending chain condition on a partially ordered set is equiv-alent to requiring that the corresponding strict order is well-founded. A stronger condition, that there be no innitedescending chains and no innite antichains, denes the well-quasi-orderings. A totally ordered set without innitedescending chains is called well-ordered.

8.1 See also Artinian

8.2 References Yiannis N. Moschovakis (2006) Notes on set theory, Undergraduate texts in mathematics (Birkhuser) ISBN0-387-28723-X, p.116

21

Chapter 9

KleeneBrouwer order

In descriptive set theory, the KleeneBrouwer order or LusinSierpiski order[1] is a linear order on nite se-quences over some linearly ordered set (X;

9.3. RECURSION THEORY 23

9.3 Recursion theoryIn recursion theory, the KleeneBrouwer order may be applied to the computation trees of implementations of totalrecursive functionals. A computation tree is well-founded if and only if the computation performed by it is totalrecursive. Each state x in a computation tree may be assigned an ordinal number jjxjj , the supremum of the ordi-nal numbers 1 + jjyjj where y ranges over the children of x in the tree. In this way, the total recursive functionalsthemselves can be classied into a hierarchy, according to the minimum value of the ordinal at the root of a com-putation tree, minimized over all computation trees that implement the functional. The KleeneBrouwer order of awell-founded computation tree is itself a recursive well-ordering, and at least as large as the ordinal assigned to thetree, from which it follows that the levels of this hierarchy are indexed by recursive ordinals.[2]

9.4 HistoryThis ordering was used by Lusin & Sierpinski (1923),[3] and then again by Brouwer (1924).[4] Brouwer does notcite any references, but Moschovakis argues that he may either have seen Lusin & Sierpinski (1923), or have beeninuenced by earlier work of the same authors leading to this work. Much later, Kleene (1955) studied the sameordering, and credited it to Brouwer.[5]

9.5 References[1] Moschovakis, Yiannis (2009), Descriptive Set Theory (2nd ed.), Rhode Island: American Mathematical Society, pp. 148

149, 203204, ISBN 978-0-8218-4813-5

[2] Schwichtenberg, Helmut; Wainer, Stanley S. (2012), 2.8 Recursive type-2 functionals and well-foundedness, Proofs andcomputations, Perspectives in Logic, Cambridge: Cambridge University Press, pp. 98101, ISBN 978-0-521-51769-0,MR 2893891.

[3] Lusin, Nicolas; Sierpinski, Waclaw (1923), Sur un ensemble non measurable B, Journal de Mathmatiques Pure et Ap-pliques 9 (2): 5372.

[4] Brouwer, L. E. J. (1924), Beweis, dass jede volle Funktion gleichmssig stetig ist, Koninklijke Nederlandse Akademie vanWetenschappen, Proc. Section of Sciences 27: 189193. As cited by Kleene (1955).

[5] Kleene, S. C. (1955), On the forms of the predicates in the theory of constructive ordinals. II, American Journal of Math-ematics 77: 405428, doi:10.2307/2372632, JSTOR 2372632, MR 0070595. See in particular section 26, A digressionconcerning recursive linear orderings, pp. 419422.

Chapter 10

Kruskals tree theorem

In mathematics, Kruskals tree theorem states that the set of nite trees over a well-quasi-ordered set of labels isitself well-quasi-ordered (under homeomorphic embedding). The theorem was conjectured by Andrew Vzsonyi andproved by Joseph Kruskal (1960); a short proof was given by Nash-Williams (1963).Higmans lemma is a special case of this theorem, of which there are many generalizations involving trees with aplanar embedding, innite trees, and so on. A generalization from trees to arbitrary graphs is given by the RobertsonSeymour theorem.

10.1 Friedmans nite formFriedman (2002) observed that Kruskals tree theorem has special cases that can be stated but not proved in rst-order arithmetic (though they can easily be proved in second-order arithmetic). Another similar statement is theParisHarrington theorem.Suppose that P(n) is the statement

There is some m such that if T1,...,Tm is a nite sequence of trees where Tk has k+n vertices, then Ti Tj for some i < j.

This is essentially a special case of Kruskals theorem, where the size of the rst tree is specied, and the trees areconstrained to grow in size at the simplest non-trivial growth rate. For each n, Peano arithmetic can prove that P(n)is true, but Peano arithmetic cannot prove the statement "P(n) is true for all n". Moreover the shortest proof of P(n)in Peano arithmetic grows phenomenally fast as a function of n; far faster than any primitive recursive function or theAckermann function for example.Friedman also proved the following nite form of Kruskals theorem for labelled trees with no order among sib-lings, parameterising on the size of the set of labels rather than on the size of the rst tree in the sequence (and thehomeomorphic embedding, , now being inf- and label-preserving):

For every n, there is an m so large that if T1,...,Tm is a nite sequence of trees with vertices labelledfrom a set of n labels, where each T has at most i vertices, then Ti Tj for some i < j.

The latter theorem ensures the existence of a rapidly growing function that Friedman called TREE, such that TREE(n)is the length of a longest sequence of n-labelled trees T1,...,Tm in which each T has at most i vertices, and no tree isembeddable into a later tree.The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormouslylarge thatmany other large combinatorial constants, such as Friedmans n(4),[*] are extremely small by comparison.[1]A lower bound for n(4), and hence an extremelyweak lower bound for TREE(3), is A(A(...A(1)...)), where the numberof As is A(187196),[2] and A() is a version of Ackermanns function: A(x) = 2 [x + 1] x in hyperoperation. Grahamsnumber, for example, is approximately A64(4) which is much smaller than the lower bound AA(187196)(1). It can beshown that the growth-rate of the function TREE exceeds that of the function f0 in the fast-growing hierarchy,where 0 is the FefermanSchtte ordinal.

24

10.2. SEE ALSO 25

The ordinal measuring the strength of Kruskals theorem is the small Veblen ordinal (sometimes confused with thesmaller Ackermann ordinal).

10.2 See also Goodsteins theorem ParisHarrington theorem KanamoriMcAloon theorem

10.3 Notes^ * n(k) is dened as the length of the longest possible sequence that can be constructed with a k-letter alphabet suchthat no block of letters x,...,x is a subsequence of any later block x,...,x.[3] n(1) = 3, n(2) = 11 and n(3) > 2 [7199]158386.

10.4 References Friedman, Harvey M. (2002), Internal nite tree embeddings. Reections on the foundations of mathematics

(Stanford, CA, 1998), Lect. Notes Log. 15, Urbana, IL: Assoc. Symbol. Logic, pp. 6091, MR 1943303

Gallier, Jean H. (1991), Whats so special about Kruskals theorem and the ordinal 0? A survey of someresults in proof theory (PDF), Ann. Pure Appl. Logic 53 (3): 199260, doi:10.1016/0168-0072(91)90022-E,MR 1129778

Kruskal, J. B. (May 1960), Well-quasi-ordering, the tree theorem, and Vazsonyis conjecture (PDF), Trans-actions of the AmericanMathematical Society (AmericanMathematical Society) 95 (2): 210225, doi:10.2307/1993287,JSTOR 1993287, MR 0111704

Nash-Williams, C. St.J. A. (1963), On well-quasi-ordering nite trees, Proc. Of the Cambridge Phil. Soc. 59(04): 833835, doi:10.1017/S0305004100003844, MR 0153601

Simpson, StephenG. (1985), Nonprovability of certain combinatorial properties of nite trees, in Harrington,L. A.; Morley, M.; Scedrov, A. et al., Harvey Friedmans Research on the Foundations of Mathematics, Studiesin Logic and the Foundations of Mathematics, North-Holland, pp. 87117

[1] http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html

[2] https://u.osu.edu/friedman.8/files/2014/01/EnormousInt.12pt.6_1_00-23kmig3.pdf

[3] https://u.osu.edu/friedman.8/files/2014/01/LongFinSeq98-2f0wmq3.pdf; p.5, 48 (Thm.6.8)

Chapter 11

Knigs lemma

For other uses, see Knigs theorem (disambiguation).

Knigs lemma or Knigs innity lemma is a theorem in graph theory due to Dnes Knig (1927).[1][2] It gives asucient condition for an innite graph to have an innitely long path. The computability aspects of this theorem havebeen thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theoremalso has important roles in constructive mathematics and proof theory.

11.1 Statement of the lemmaIf G is a connected graph with innitely many vertices such that every vertex has nite degree (that is, each vertexis adjacent to only nitely many other vertices) then G contains an innitely long simple path, that is, a path with norepeated vertices.A common special case of this is that every tree that contains innitely many vertices, each having nite degree, hasat least one innite simple path.

11.1.1 Proof

For the proof, assume that the graph consists of innitely many vertices v and is connected.Start with any vertex v1. Every one of the innitely many vertices of G can be reached from v1 with a simple path,and each such path must start with one of the nitely many vertices adjacent to v1. There must be one of thoseadjacent vertices through which innitely many vertices can be reached without going through v1. If there were not,then the entire graph would be the union of nitely many nite sets, and thus nite, contradicting the assumption thatthe graph is innite. We may thus pick one of these vertices and call it v2.Now innitely many vertices of G can be reached from v2 with a simple path which doesn't use the vertex v1. Eachsuch path must start with one of the nitely many vertices adjacent to v2. So an argument similar to the one aboveshows that there must be one of those adjacent vertices through which innitely many vertices can be reached; pickone and call it v3.Continuing in this fashion, an innite simple path can be constructed by mathematical induction. At each step, theinduction hypothesis states that there are innitely many nodes reachable by a simple path from a particular node vthat does not go through one of a nite set of vertices. The induction argument is that one of the vertices adjacent tov satises the induction hypothesis, even when v is added to the nite set.The result of this induction argument is that for all n it is possible to choose a vertex vn as the construction describes.The set of vertices chosen in the construction is then a chain in the graph, because each one was chosen to be adjacentto the previous one, and the construction guarantees that the same vertex is never chosen twice.This proof is not generally considered to be constructive, because at each step it uses a proof by contradiction toestablish that there exists an adjacent vertex from which innitely many other vertices can be reached. Facts aboutthe computational aspects of the lemma suggest that no proof can be given that would be considered constructive by

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11.2. COMPUTABILITY ASPECTS 27

the main schools of constructive mathematics.

11.2 Computability aspectsThe computability aspects of Knigs lemma have been thoroughly investigated. The form of Knigs lemma mostconvenient for this purpose is the one which states that any innite nitely branching subtree of ! Any such tree has a path computable from 00 , the canonical Turing complete set that can decide the haltingproblem.

Any such tree has a path that is low. This is known as the low basis theorem. Any such tree has a path that is hyperimmune free. This means that any function computable from the path isdominated by a computable function.

For any noncomputable subset X of ! the tree has a path that does not compute X.

A weak form of Knigs lemma which states that every innite binary tree has an innite branch is used to dene thesubsystem WKL0 of second-order arithmetic. This subsystem has an important role in reverse mathematics. Here abinary tree is one in which every term of every sequence in the tree is 0 or 1, which is to say the tree is computablybounded via the constant function 2. The full form of Knigs lemma is not provable in WKL0, but is equivalent tothe stronger subsystem ACA0.

11.3 Relationship to constructive mathematics and compactnessThe fan theorem of L. E. J. Brouwer (1927) is, from a classical point of view, the contrapositive of a form of Knigslemma. A subset S of f0; 1g

28 CHAPTER 11. KNIGS LEMMA

11.4 Relationship with the axiom of choiceKnigs lemma may be considered to be a choice principle; the rst proof above illustrates the relationship betweenthe lemma and the Axiom of dependent choice. At each step of the induction, a vertex with a particular propertymust be selected. Although it is proved that at least one appropriate vertex exists, if there is more than one suitablevertex there may be no canonical choice.If the graph is countable, the vertices are well-ordered and one can canonically choose the smallest suitable vertex.In this case, Knigs lemma is provable in second-order arithmetic with arithmetical comprehension, and, a fortiori,in ZF set theory (without choice).Knigs lemma is essentially the restriction of the axiom of dependent choice to entire relations R such that for eachx there are only nitely many z such that xRz. Although the axiom of choice is, in general, stronger than the principleof dependent choice, this restriction of dependent choice is equivalent to a restriction of the axiom of choice. Inparticular, when the branching at each node is done on a nite subset of an arbitrary set not assumed to be countable,the form of Knigs lemma that says Every innite nitely branching tree has an innite path is equivalent to theprinciple that every countable set of nite sets has a choice function.[4] This form of the axiom of choice (and henceof Knigs lemma) is not provable in ZF set theory.

11.5 See also Aronszajn tree, for the possible existence of counterexamples when generalizing the lemma to higher cardi-nalities.

11.6 Notes[1] Note that, although Knigs name is properly spelled with a double acute accent, the lemma named after him is customarily

spelled with an umlaut.

[2] Knig (1927) as explained in Franchella (1997)

[3] Rogers (1967), p. 418.

[4] Truss (1976), p. 273; compare Lvy (1979), Exercise IX.2.18.

11.7 References Brouwer, L. E. J. (1927), On the Domains of Denition of Functions. published in van Heijenoort, Jean, ed.(1967), From Frege to Gdel.

Cenzer, Douglas (1999), " 01 classes in computability theory, Handbook of Computability Theory, Elsevier,pp. 3785, doi:10.1016/S0049-237X(99)80018-4, ISBN 0-444-89882-4, MR 1720779.

Knig, D. (1926), Sur les correspondances multivoques des ensembles (PDF), Fundamenta Mathematicae(in French) (8): 114134.

Knig, D. (1927), "ber eine Schlussweise aus dem Endlichen ins Unendliche, Acta Sci. Math. (Szeged) (inGerman) (3(2-3)): 121130.

Franchella, Miriam (1997), On the origins of Dnes Knigs innity lemma, Archive for History of ExactSciences (51(1)3:2-3): 327, doi:10.1007/BF00376449.

Knig, D. (1936), Theorie der Endlichen und Unendlichen Graphen: Kombinatorische Topologie der Streck-enkomplexe (in German), Leipzig: Akad. Verlag.

Lvy, Azriel (1979), Basic Set Theory, Springer, ISBN 3-540-08417-7, MR 0533962. Reprint Dover 2002,ISBN 0-486-42079-5.

11.8. EXTERNAL LINKS 29

Rogers, Hartley, Jr. (1967), Theory of Recursive Functions and Eective Computability, McGraw-Hill, MR0224462.

Simpson, Stephen G. (1999), Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic,Springer, ISBN 3-540-64882-8, MR 1723993.

Soare, Robert I. (1987), Recursively Enumerable Sets and Degrees: A study of computable functions and com-putably generated sets, Perspectives in Mathematical Logic, Springer, ISBN 3-540-15299-7, MR 0882921.

Truss, J. (1976), Some cases of Knigs lemma, in Marek, V. Wiktor; Srebrny, Marian; Zarach, Andrzej,Set theory and hierarchy theory: a memorial tribute to Andrzej Mostowski, Lecture Notes in Mathematics 537,Springer, pp. 273284, doi:10.1007/BFb0096907, MR 0429557.

11.8 External links Stanford Encyclopedia of Philosophy: Constructive Mathematics The Mizar project has completely formalized and automatically checked the proof of a version of Knigslemma in the le TREES_2.

Chapter 12

Mostowski collapse lemma

In mathematical logic, theMostowski collapse lemma is a statement in set theory named for Andrzej Mostowski.

12.1 StatementSuppose that R is a binary relation on a class X such that

R is set-like: R1[x] = {y : y R x} is a set for every x, R is well-founded: every nonempty subset S of X contains an R-minimal element (i.e. an element x S suchthat R1[x] S is empty),

R is extensional: R1[x] R1[y] for every distinct elements x and y of X

TheMostowski collapse lemma states that for any such R there exists a unique transitive class (possibly proper) whosestructure under the membership relation is isomorphic to (X, R), and the isomorphism is unique. The isomorphismmaps each element x of X to the set of images of elements y of X such that y R x (Jech 2003:69).

12.2 GeneralizationsEvery well-founded set-like relation can be embedded into a well-founded set-like extensional relation. This impliesthe following variant of the Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.A mapping F such that F(x) = {F(y) : y R x} for all x in X can be dened for any well-founded set-like relation R onX by well-founded recursion. It provides a homomorphism of R onto a (non-unique, in general) transitive class. Thehomomorphism F is an isomorphism if and only if R is extensional.The well-foundedness assumption of the Mostowski lemma can be alleviated or dropped in non-well-founded settheories. In Boas set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique)transitive class. In set theory with Aczels anti-foundation axiom, every set-like relation is bisimilar to set-membershipon a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitiveclass.

12.3 ApplicationEvery set model of ZF is set-like and extensional. If the model is well-founded, then by the Mostowski collapselemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.Note that saying that the membership relation of some model of ZF is well-founded is stronger than saying that theaxiom of regularity is true in the model. There exists a modelM (assuming the consistency of ZF) whose domain has

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12.4. REFERENCES 31

a subset A with no R-minimal element, but this set A is not a set in the model (A is not in the domain of the model,even though all of its members are). More precisely, for no such set A there exists x in M such that A = R1[x]. SoM satises the axiom of regularity (it is internally well-founded) but it is not well-founded and the collapse lemmadoes not apply to it.

12.4 References Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

Chapter 13

Newmans lemma

In mathematics, in the theory of rewriting systems, Newmans lemma, also commonly called the diamond lemma,states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are noinnite reduction sequences, is conuent if it is locally conuent. In fact a terminating ARS is conuent preciselywhen it is locally conuent.[1]

Equivalently, for every binary relation with no decreasing innite chains and satisfying a weak version of the diamondproperty, there is a unique minimal element in every connected component of the relation considered as a graph.Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Grard Huet in1980.[2] Newmans original proof was considerably more complicated.[3]

13.1 Diamond lemmaIn general, Newmans lemma can be seen as a combinatorial result about binary relations on a set A (writtenbackwards, so that a b means that b is below a) with the following two properties:

is a well-founded relation: every non-empty subset X of A has a minimal element (an element a of X suchthat a b for no b in X). Equivalently, there is no innite chain a0 a1 a2 a3 .... In the terminologyof rewriting systems, is terminating.

Every covering is bounded below. That is, if an element a in A covers elements b and c in A in the sense thata b and a c, then there is an element d in A such that b* d and c * d, where * denotes the reexivetransitive closure of . In the terminology of rewriting systems, is locally conuent.

If the above two conditions hold, then the lemma states that is conuent: whenever a* b and a* c, there is anelement d such that b* d and c* d. In view of the termination of , this implies that every connected componentof as a graph contains a unique minimal element a, moreover b * a for every element b of the component.[4]

13.2 Notes

[1] Franz Baader, Tobias Nipkow, (1998) Term Rewriting and All That, Cambridge University Press ISBN 0-521-77920-0

[2] Grard Huet, Conuent Reductions: Abstract Properties and Applications to Term Rewriting Systems, Journal of theACM (JACM), October 1980, Volume 27, Issue 4, pp. 797 - 821.

[3] Harrison, p. 260, Paterson(1990), p. 354.

[4] Paul M. Cohn, (1980) Universal Algebra, D. Reidel Publishing, ISBN 90-277-1254-9 (See pp. 25-26)

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13.3. REFERENCES 33

13.3 References M. H. A. Newman. On theories with a combinatorial denition of equivalence. Annals of Mathematics, 43,Number 2, pages 223243, 1942.

Paterson, Michael S. (1990). Automata, languages, and programming: 17th international colloquium. Lecturenotes in computer science 443. Warwick University, Engl