large cardinals

Large cardinalsFrom Wikipedia, the free encyclopedia

Contents

1 Ackermann ordinal 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Additively indecomposable ordinal 22.1 Multiplicatively indecomposable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Admissible ordinal 33.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Aleph number 44.1 Aleph-naught . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Aleph-one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.4 Aleph- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.5 Aleph- for general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.6 Fixed points of omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.7 Role of axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Axiom of determinacy 95.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 95.3 Innite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 BachmannHoward ordinal 12

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6.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Beth number 137.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Relation to the aleph numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Specic cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7.3.1 Beth null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3.2 Beth one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3.3 Beth two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3.4 Beth omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Burali-Forti paradox 178.1 Stated in terms of von Neumann ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Stated more generally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Resolution of the paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Cantors diagonal argument 199.1 Uncountable set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.1.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.1.2 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9.2 General sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.2.1 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.2.2 Version for Quines New Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Cantors theorem 2510.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.2 A detailed explanation of the proof when X is countably innite . . . . . . . . . . . . . . . . . . . 2610.3 Related paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11 Cardinal assignment 30

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11.1 Cardinal assignment without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12 Cardinal characteristic of the continuum 3112.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

12.2.1 non(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2.2 Bounding number b and dominating number d . . . . . . . . . . . . . . . . . . . . . . . . 3112.2.3 Splitting number s and reaping number r . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2.4 Ultralter number u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2.5 Almost disjointness number a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12.3 Cicho's diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Cardinal function 3413.1 Cardinal functions in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3413.2 Cardinal functions in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.2.1 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.3 Cardinal functions in Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.4 Cardinal functions in algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Cardinal number 3814.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

14.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

15 Cardinality 4615.1 Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.1.1 Denition 1: | A | = | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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15.1.2 Denition 2: | A | | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4615.1.3 Denition 3: | A | < | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.3 Finite, countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.4 Innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

15.4.1 Cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.5 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.6 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16 Cardinality of the continuum 5116.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16.1.1 Uncountability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.1.2 Cardinal equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.1.3 Alternative explanation for c = 2@0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16.2 Beth numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.4 Sets with cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.5 Sets with greater cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17 ChurchKleene ordinal 5617.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 Cicho's diagram 5718.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2 Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

19 Club set 5919.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.2 The closed unbounded lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

20 Conality 6120.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.3 Conality of ordinals and other well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.4 Regular and singular ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.5 Conality of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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20.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

21 Continuous function (set theory) 6421.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

22 Continuum function 6522.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

23 Continuum hypothesis 6623.1 Cardinality of innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.2 Independence from ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.3 Arguments for and against CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.4 The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23.4.1 Implications of GCH for cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . 6923.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

24 Core model 7124.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.2 Construction of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.3 Properties of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.4 Construction of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25 Countable set 7325.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7425.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

26 Critical point (set theory) 8126.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

27 Dedekind-innite set 8227.1 Comparison with the usual denition of innite set . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.2 Dedekind-innite sets in ZF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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27.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.4 Relation to the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.5 Proof of equivalence to innity, assuming axiom of countable choice . . . . . . . . . . . . . . . . . 8427.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

28 Diagonal intersection 8528.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

29 Eastons theorem 8629.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8629.2 No extension to singular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8729.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8729.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

30 0 8830.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8830.2 Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9030.3 Surreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9030.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9030.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

31 Equiconsistency 9231.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9331.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

32 Equinumerosity 9432.1 Reexivity, symmetry, and transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9432.2 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9532.3 Compatibility with set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9532.4 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9532.5 Dedekind-innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9632.6 Categorial denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9632.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9632.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

33 Erds cardinal 9733.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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34 Even and odd ordinals 9834.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

35 Extender (set theory) 9935.1 Formal denition of an extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9935.2 Dening an extender from an elementary embedding . . . . . . . . . . . . . . . . . . . . . . . . . 9935.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

36 Extendible cardinal 10136.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10136.2 Variants and relation to other cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10136.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10136.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

37 FefermanSchtte ordinal 10337.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10337.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

38 Finite set 10438.1 Denition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10438.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10438.3 Necessary and sucient conditions for niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . 10538.4 Foundational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10638.5 Set-theoretic denitions of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

38.5.1 Other concepts of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10738.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10738.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10738.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10838.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

39 First uncountable ordinal 10939.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10939.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10939.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

40 Fixed-point lemma for normal functions 11040.1 Background and formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11040.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11040.3 Example application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11140.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

41 Gimel function 11241.1 Values of the Gimel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11241.2 Reducing the exponentiation function to the gimel function . . . . . . . . . . . . . . . . . . . . . . 112

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41.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

42 Grothendieck universe 11342.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11342.2 Grothendieck universes and inaccessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . 11442.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11442.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

43 Hartogs number 11643.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11643.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

44 Hereditarily countable set 11744.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11744.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

45 Homogeneous (large cardinal property) 11845.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

46 Huge cardinal 11946.1 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11946.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12046.3 -huge cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12046.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12046.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

47 Inaccessible cardinal 12147.1 Models and consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12147.2 Existence of a proper class of inaccessibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12247.3 -inaccessible cardinals and hyper-inaccessible cardinals . . . . . . . . . . . . . . . . . . . . . . . 12247.4 Two model-theoretic characterisations of inaccessibility . . . . . . . . . . . . . . . . . . . . . . . 12247.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12347.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

48 Indescribable cardinal 12448.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

49 Ineable cardinal 12649.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

50 Innite set 12750.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12750.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12750.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12750.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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51 Iterable cardinal 12951.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12951.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

52 Jnsson cardinal 13052.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

53 Kleenes O 13153.1 Kleenes O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13153.2 Basic properties of

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57.2.4 Unrecursable recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14357.3 Beyond recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

57.3.1 The ChurchKleene ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14457.3.2 Admissible ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14457.3.3 Beyond admissible ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14457.3.4 Unprovable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

57.4 A pseudo-well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

57.5.1 On recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.5.2 Beyond recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.5.3 Both recursive and nonrecursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.5.4 Inline references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

58 Large Veblen ordinal 14658.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

59 Laver function 14759.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14759.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14759.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

60 Limit cardinal 14860.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14860.2 Relationship with ordinal subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14860.3 The notion of inaccessibility and large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 14960.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14960.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14960.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

61 Limit ordinal 15061.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15061.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15061.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15261.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15261.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15261.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

62 List of large cardinal properties 15362.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15462.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

63 Mahlo cardinal 15563.1 Minimal condition sucient for a Mahlo cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . 15563.2 Example: showing that Mahlo cardinals are hyper-inaccessible . . . . . . . . . . . . . . . . . . . 155

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63.3 -Mahlo, hyper-Mahlo and greatly Mahlo cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 15663.4 The Mahlo operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15663.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15763.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

64 Measurable cardinal 15864.1 Measurable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15864.2 Real-valued measurable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15964.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15964.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

65 Natural number 16065.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

65.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

65.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 16365.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 163

65.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16465.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

65.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16465.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

65.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16665.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16665.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16765.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

66 Normal function 17166.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17166.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17166.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17266.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

67 Normal measure 17367.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17367.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

68 Order type 17468.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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68.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17568.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17568.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17568.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17568.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

69 Ordinal analysis 17669.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

69.2.1 Theories with proof theoretic ordinal 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.2.2 Theories with proof theoretic ordinal 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.2.3 Theories with proof theoretic ordinal n . . . . . . . . . . . . . . . . . . . . . . . . . . . 17769.2.4 Theories with proof theoretic ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . 17769.2.5 Theories with proof theoretic ordinal 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17769.2.6 Theories with proof theoretic ordinal the Feferman-Schtte ordinal 0 . . . . . . . . . . . 17769.2.7 Theories with proof theoretic ordinal the Bachmann-Howard ordinal . . . . . . . . . . . . 17769.2.8 Theories with larger proof theoretic ordinals . . . . . . . . . . . . . . . . . . . . . . . . . 177


70 Ordinal arithmetic 17970.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17970.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18070.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18170.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18370.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18670.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18670.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

71 Ordinal collapsing function 18771.1 An example leading up to the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . . . . 187

71.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18771.1.2 Computation of values of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18871.1.3 Ordinal notations up to the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . 18971.1.4 Standard sequences for ordinal notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19171.1.5 A terminating process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

71.2 Variations on the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19371.2.1 Making the function less powerful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19371.2.2 Going beyond the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . . . . . . 193

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71.2.3 A normal variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19471.3 Collapsing large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19571.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19571.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

72 Ordinal logic 19772.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

73 Ordinal notation 19873.1 A simplied example using a pairing function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

73.1.1 -notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19873.2 Systems of ordinal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

73.2.1 Cantor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19973.2.2 Veblen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19973.2.3 Ackermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19973.2.4 Bachmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19973.2.5 Takeuti (ordinal diagrams) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20073.2.6 Fefermans functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20073.2.7 Buchholz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20073.2.8 Kleenes O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200


74 Ordinal number 20274.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20374.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

74.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20574.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 20574.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20574.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

74.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20674.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

74.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20674.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20774.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20774.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20774.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

74.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20874.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

74.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

74.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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74.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

75 Primitive recursive set function 21275.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21275.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

76 Ramsey cardinal 21376.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

77 Rank-into-rank 21477.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

78 Recursive ordinal 21678.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21678.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

79 Reecting cardinal 21779.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

80 Regular cardinal 21880.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21880.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21880.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21980.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

81 Reinhardt cardinal 22081.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22181.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22181.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

82 Remarkable cardinal 22282.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22282.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

83 Rowbottom cardinal 22383.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

84 SchrderBernstein theorem 22484.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22484.2 Original proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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84.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22584.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22684.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22684.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

85 Shelah cardinal 22985.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

86 Shrewd cardinal 23086.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

87 Singular cardinals hypothesis 23187.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

88 Small Veblen ordinal 23388.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

89 Solovay model 23489.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23489.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23489.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23489.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

90 Stationary set 23690.1 Classical notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23690.2 Jechs notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23690.3 Generalized notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23690.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23790.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

91 Strong cardinal 23891.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23891.2 Relationship with other large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23891.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

92 Strong partition cardinal 23992.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

93 Strongly compact cardinal 24093.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

94 Subcompact cardinal 24194.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24194.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

95 Subtle cardinal 242

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95.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24295.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

96 Successor cardinal 24396.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24496.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

97 Successor ordinal 24597.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24597.2 In Von Neumanns model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24597.3 Ordinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24597.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24597.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24697.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

98 Supercompact cardinal 24798.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24798.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24798.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24798.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

99 Superstrong cardinal 24999.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

100Suslin cardinal 250100.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250100.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

101Systems of Logic Based on Ordinals 251101.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251101.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

102Tall cardinal 252102.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

103Tarskis theorem about choice 253103.1Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253103.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

104Tav (number) 254104.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254104.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

105Transnite induction 255105.1Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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105.2Relationship to the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257105.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257105.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257105.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257105.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

106Transnite number 259106.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259106.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260106.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

107Uncountable set 261107.1Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261107.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261107.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261107.4Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262107.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262107.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262107.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

108Unfoldable cardinal 263108.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

109Veblen function 264109.1The Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

109.1.1 Fundamental sequences for the Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . 264109.1.2 The function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

109.2Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265109.2.1 Finitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265109.2.2 Transnitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

109.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

110Von Neumann cardinal assignment 267110.1Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267110.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267110.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

111Vopnkas principle 269111.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269111.2Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269111.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269111.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

112Weakly compact cardinal 271

xviii CONTENTS

112.1Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271112.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272112.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

113Well-order 273113.1Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273113.2Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

113.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274113.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274113.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

113.3Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275113.4Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275113.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276113.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

114Woodin cardinal 277114.1Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277114.2Hyper-Woodin cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278114.3Weakly hyper-Woodin cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278114.4Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278114.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

115Zero dagger 279115.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279115.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279115.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

116Zero sharp 280116.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280116.2Statements that imply the existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280116.3Statements equivalent to existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281116.4Consequences of existence and non-existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281116.5Other sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281116.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281116.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

117Zero-based numbering 283117.1Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

117.1.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283117.1.2 Usage in programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284117.1.3 Numerical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

117.2Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285117.3Other elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

CONTENTS xix

117.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286117.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

118 (set theory) 287118.1Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

119() 288119.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288119.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288119.3Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 289

119.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289119.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297119.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Chapter 1

Ackermann ordinal

In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. Theterm Ackermann ordinal is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal.Unfortunately there is no standard notation for ordinals beyond the FefermanSchtte ordinal 0. Most systems ofnotation use symbols such as (), (), (), some of which are modications of the Veblen functions to producecountable ordinals even for uncountable arguments, and some of which are "collapsing functions".The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by Ackermann (1951), andis sometimes denoted by 2(0) or (2) or (

2

) . Ackermanns system of notation is weaker than the systemintroduced much earlier by Veblen (1908), which he seems to have been unaware of.

1.1 References Ackermann, Wilhelm (1951), Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse,Math. Z. 53 (5): 403413, doi:10.1007/BF01175640, MR 0039669

Veblen, Oswald (1908), Continuous Increasing Functions of Finite and Transnite Ordinals, Transactions ofthe American Mathematical Society 9 (3): 280292, doi:10.2307/1988605

Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244

1

Chapter 2

Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal is any ordinal number that is not0 such that for any ; < , we have + < : The class of additively indecomposable ordinals (aka gammanumbers) is denoted H:From the continuity of addition in its right argument, we get that if < and is additively indecomposable, then + = :

Obviously 1 2 H , since 0 + 0 < 1: No nite ordinal other than 1 is in H: Also, ! 2 H , since the sum of two niteordinals is still nite. More generally, every innite cardinal is in H:H is closed and unbounded, so the enumerating function of H is normal. In fact, fH() = !:The derivative f 0H() (which enumerates xed points of fH) is written : Ordinals of this form (that is, xed pointsof fH ) are called epsilon numbers. The number 0 = !!

!

is therefore the rst xed point of the sequence!; !!; !!

!

; : : :

2.1 Multiplicatively indecomposableA similar notion can be dened for multiplication. The multiplicatively indecomposable ordinals (aka delta numbers)are those of the form !! for any ordinal . Every epsilon number is multiplicatively indecomposable; and everymultiplicatively indecomposable ordinal is additively indecomposable. The delta numbers are the same as the primeordinals that are limits.

2.2 See also Ordinal arithmetic

2.3 References Sierpiski, Wacaw (1958), Cardinal and ordinal numbers., Polska Akademia NaukMonograeMatematyczne34, Warsaw: Pastwowe Wydawnictwo Naukowe, MR 0095787

This article incorporates material from Additively indecomposable on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

2

Chapter 3

Admissible ordinal

In set theory, an ordinal number is an admissible ordinal if L is an admissible set (that is, a transitive model ofKripkePlatek set theory); in other words, is admissible when is a limit ordinal and L0-collection.[1][2]

The rst two admissible ordinals are and !CK1 (the least non-recursive ordinal, also called the ChurchKleeneordinal).[2] Any regular uncountable cardinal is an admissible ordinal.By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to theChurch-Kleene ordinal, but for Turing machines with oracles.[1] One sometimes writes !CK for the -th ordinalwhich is either admissible or a limit of admissibles; an ordinal which is both is called recursively inaccessible.[3] Thereexists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can denerecursively Mahlo cardinals, for example).[4] But all these ordinals are still countable. Therefore, admissible ordinalsseem to be the recursive analogue of regular cardinal numbers.Notice that is an admissible ordinal if and only if is a limit ordinal and there does not exist a

Chapter 4

Aleph number

Aleph One redirects here. For other uses, see Aleph One (disambiguation).In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the

Aleph-naught, the smallest innite cardinal number

4

4.1. ALEPH-NAUGHT 5

cardinality (or size) of innite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (@ ) (though in older mathematics books the letter aleph is sometimes printed upside down[1]).The cardinality of the natural numbers is @0 (read aleph-naught or aleph-zero; the German term aleph-null is alsosometimes used), the next larger cardinality is aleph-one @1 , then @2 and so on. Continuing in this manner, it ispossible to dene a cardinal number @ for every ordinal number , as described below.The concept goes back to Georg Cantor, who dened the notion of cardinality and realized that innite sets can havedierent cardinalities.The aleph numbers dier from the innity () commonly found in algebra and calculus. Alephs measure the sizesof sets; innity, on the other hand, is commonly dened as an extreme limit of the real number line (applied to afunction or sequence that "diverges to innity or increases without bound), or an extreme point of the extendedreal number line.

4.1 Aleph-naught@0 (Aleph-naught, also known as Aleph-null) is the cardinality of the set of all natural numbers, and is an innitecardinal. The set of all nite ordinals, called or 0, has cardinality @0 . A set has cardinality @0 if and only ifit is countably innite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers.Examples of such sets are

the set of all square numbers, the set of all cubic numbers, the set of all fourth powers, ... the set of all perfect powers, the set of all prime powers, the set of all even numbers, the set of all odd numbers, the set of all prime numbers, the set of all composite numbers, the set of all integers, the set of all rational numbers, the set of all algebraic numbers, the set of all computable numbers, the set of all denable numbers, the set of all binary strings of nite length, and the set of all nite subsets of any given countably innite set.

These innite ordinals: , +1, 2, 2, and 0 are among the countably innite sets.[2] For example, the sequence(with ordinality 2) of all positive odd integers followed by all positive even integers

{1, 3, 5, 7, 9, ..., 2, 4, 6, 8, 10, ...}

is an ordering of the set (with cardinality @0 ) of positive integers.If the axiom of countable choice (a weaker version of the axiom of choice) holds, then @0 is smaller than any otherinnite cardinal.

4.2 Aleph-one@1 is the cardinality of the set of all countable ordinal numbers, called 1 or (sometimes) . This 1 is itself anordinal number larger than all countable ones, so it is an uncountable set. Therefore @1 is distinct from @0 . Thedenition of @1 implies (in ZF, ZermeloFraenkel set theory without the axiom of choice) that no cardinal number isbetween @0 and @1 . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers

6 CHAPTER 4. ALEPH NUMBER

is totally ordered, and thus @1 is the second-smallest innite cardinal number. Using AC we can show one of themost useful properties of the set 1: any countable subset of 1 has an upper bound in 1. (This follows from thefact that a countable union of countable sets is countable, one of the most common applications of AC.) This fact isanalogous to the situation in @0 : every nite set of natural numbers has a maximum which is also a natural number,and nite unions of nite sets are nite.1 is actually a useful concept, if somewhat exotic-sounding. An example application is closing with respect tocountable operations; e.g., trying to explicitly describe the -algebra generated by an arbitrary collection of subsets(see e. g. Borel hierarchy). This is harder than most explicit descriptions of generation in algebra (vector spaces,groups, etc.) because in those cases we only have to close with respect to nite operationssums, products, and thelike. The process involves dening, for each countable ordinal, via transnite induction, a set by throwing in allpossible countable unions and complements, and taking the union of all that over all of 1.

4.3 Continuum hypothesisMain article: Continuum hypothesisSee also: Beth number

The cardinality of the set of real numbers (cardinality of the continuum) is 2@0 . It cannot be determined from ZFC(ZermeloFraenkel set theory with the axiom of choice) where this number ts exactly in the aleph number hierarchy,but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity

2@0 = @1:The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. CHis independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided thatZFC is consistent). That CH is consistent with ZFC was demonstrated by Kurt Gdel in 1940 when he showed thatits negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen in 1963 whenhe showed, conversely, that the CH itself is not a theorem of ZFC by the (then novel) method of forcing.

4.4 Aleph-Conventionally the smallest innite ordinal is denoted , and the cardinal number @! is the least upper bound of

f@n : n 2 f 0; 1; 2; : : : g gamong alephs.Aleph- is the rst uncountable cardinal number that can be demonstrated within ZermeloFraenkel set theory notto be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that2@0 = @n , and moreover it is possible to assume 2@0 is as large as we like. We are only forced to avoid setting itto certain special cardinals with conality @0 , meaning there is an unbounded function from @0 to it (see Eastonstheorem).

4.5 Aleph- for general To dene @ for arbitrary ordinal number , we must dene the successor cardinal operation, which assigns toany cardinal number the next larger well-ordered cardinal + (if the axiom of choice holds, this is the next largercardinal).We can then dene the aleph numbers as follows:

@0 = !

4.6. FIXED POINTS OF OMEGA 7

@+1 = @+and for , an innite limit ordinal,

@ =[

8 CHAPTER 4. ALEPH NUMBER

4.9 ReferencesNotes

[1] For example, in (Sierpinski 1958, p.402) the letter aleph appears both the right way up and upside down.

[2] Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag

[3] aleph numbers at PlanetMath.org.

[4] Harris, Kenneth (April 6, 2009). Math 582 Intro to Set Theory, Lecture 31 (PDF). Department of Mathematics, Uni-versity of Michigan. Retrieved September 1, 2012.

Sierpiski, Wacaw (1958), Cardinal and ordinal numbers., Polska Akademia NaukMonograeMatematyczne34, Warsaw: Pastwowe Wydawnictwo Naukowe, MR 0095787

4.10 External links Hazewinkel, Michiel, ed. (2001), Aleph-zero, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Aleph-0, MathWorld.

Chapter 5

Axiom of determinacy

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

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

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

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

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

9

10 CHAPTER 5. AXIOM OF DETERMINACY

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

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

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

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

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

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

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

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

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

5.6. REFERENCES 11

5.6 References Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

Kanamori, Akihiro (2000). The Higher Innite (2nd ed.). Springer. ISBN 3-540-00384-3. Martin, Donald A.; Steel, John R. (Jan 1989). A Proof of Projective Determinacy. Journal of the AmericanMathematical Society 2 (1): 71125. doi:10.2307/1990913. JSTOR 1990913.

Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Mycielski, Jan; Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice. Bulletinde l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques 10: 13.ISSN 0001-4117. MR 0140430.

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

5.7 Further reading Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, Universityof Bonn, Germany, 2001

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

Chapter 6

BachmannHoward ordinal

In mathematics, the BachmannHoward ordinal (or Howard ordinal) is a large countable ordinal. It is the prooftheoretic ordinal of several mathematical theories, such as KripkePlatek set theory (with the axiom of innity) andthe system CZF of constructive set theory. It is named after William Alvin Howard and Heinz Bachmann.

6.1 DenitionThe BachmannHoward ordinal is dened using an ordinal collapsing function (with more details given in the relevantarticle):

enumerates the epsilon numbers, the ordinals such that = . = 1 is the rst uncountable ordinal. is the rst epsilon number after = . (0) is dened to be the smallest ordinal that cannot be constructed by starting with 0, 1, and , andrepeatedly applying ordinal addition, multiplication and exponentiation.

() is dened in the same way, except that it also allows applications of to previously constructed ordinalsless than .

The BachmannHoward ordinal is ().

The BachmannHoward ordinal can also be dened as "+1(0) for an extension of the Veblen functions touncountable ; this extension is not completely straightforward.

6.2 References Bachmann, Heinz (1950), Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ord-nungszahlen, Vierteljschr. Naturforsch. Ges. Zrich 95: 115147, MR 0036806

Howard, W. A. (1972), A system of abstract constructive ordinals., J. Symbolic Logic (Association for Sym-bolic Logic) 37 (2): 355374, doi:10.2307/2272979, JSTOR 2272979, MR 0329869

Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics 1407, Berlin: Springer-Verlag, ISBN3-540-51842-8, MR 1026933

Rathjen, Michael (August 2005). Proof Theory: Part III, Kripke-Platek Set Theory. Retrieved 2008-04-17.(slides of a talk given at Fischbachau)

12

Chapter 7

Beth number

In mathematics, the innite cardinal numbers are represented by the Hebrew letter @ (aleph) indexed with a subscriptthat runs over the ordinal numbers (see aleph number). The second Hebrew letter i (beth) is used in a related way,but does not necessarily index all of the numbers indexed by @ .

7.1 DenitionTo dene the beth numbers, start by letting

i0 = @0

be the cardinality of any countably innite set; for concreteness, take the set N of natural numbers to be a typicalcase. Denote by P(A) the power set of A; i.e., the set of all subsets of A. Then dene

i+1 = 2i ;

which is the cardinality of the power set of A if i is the cardinality of A.Given this denition,

i0; i1; i2; i3; : : :

are respectively the cardinalities of

N; P (N); P (P (N)); P (P (P (N))); : : : :

so that the second beth number i1 is equal to c , the cardinality of the continuum, and the third beth number i2 isthe cardinality of the power set of the continuum.Because of Cantors theorem each set in the preceding sequence has cardinality strictly greater than the one precedingit. For innite limit ordinals the corresponding beth number is dened as the supremum of the beth numbers forall ordinals strictly smaller than :

i = supfi : < g:

One can also show that the von Neumann universes V!+have cardinality i .

13

14 CHAPTER 7. BETH NUMBER

7.2 Relation to the aleph numbersAssuming the axiom of choice, innite cardinalities are linearly ordered; no two cardinalities can fail to be comparable.Thus, since by denition no innite cardinalities are between @0 and @1 , it follows that

i1 @1:

Repeating this argument (see transnite induction) yields i @ for all ordinals .The continuum hypothesis is equivalent to

i1 = @1:

The generalized continuum hypothesis says the sequence of beth numbers thus dened is the same as the sequenceof aleph numbers, i.e., i = @ for all ordinals .

7.3 Specic cardinals

7.3.1 Beth nullSince this is dened to be @0 or aleph null then sets with cardinality i0 include:

the natural numbers N the rational numbers Q the algebraic numbers the computable numbers and computable sets the set of nite sets of integers

7.3.2 Beth oneMain article: cardinality of the continuum

Sets with cardinality i1 include:

the transcendental numbers the irrational numbers the real numbers R the complex numbers C Euclidean space Rn

the power set of the natural numbers (the set of all subsets of the natural numbers) the set of sequences of integers (i.e. all functions N Z, often denoted ZN) the set of sequences of real numbers, RN

the set of all continuous functions from R to R the set of nite subsets of real numbers

7.4. GENERALIZATION 15

7.3.3 Beth twoi2 (pronounced beth two) is also referred to as 2c (pronounced two to the power of c).Sets with cardinality i2 include:

The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of setsof real numbers

The power set of the power set of the set of natural numbers The set of all functions from R to R (RR) The set of all functions from Rm to Rn

The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets ofsequences of natural numbers

The Stoneech compactications of R, Q, and N

7.3.4 Beth omegai! (pronounced beth omega) is the smallest uncountable strong limit cardinal.

7.4 GeneralizationThe more general symbol i() , for ordinals and cardinals , is occasionally used. It is dened by:

i0() = ;

i+1() = 2i();i() = supfi() : < gSo

i = i(@0):In ZF, for any cardinals and , there is an ordinal such that:

i():And in ZF, for any cardinal and ordinals and :

i(i()) = i+():

Consequently, in ZermeloFraenkel set theory absent ur-elements with or without the axiom of choice, for any car-dinals and , the equality

i() = i()

holds for all suciently large ordinals (that is, there is an ordinal such that the equality holds for every ordinal ).This also holds in ZermeloFraenkel set theory with ur-elements with or without the axiom of choice provided theur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements).If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.

16 CHAPTER 7. BETH NUMBER

7.5 References T. E. Forster, Set Theory with a Universal Set: Exploring an Untyped Universe, Oxford University Press, 1995 Beth number is dened on page 5.

Bell, John Lane; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974edition ed.). Dover Publications. ISBN 0-486-44979-3. See pages 6 and 204205 for beth numbers.

Roitman, Judith (2011). Introduction to Modern Set Theory. Virginia Commonwealth University. ISBN 978-0-9824062-4-3. See page 109 for beth numbers.

Chapter 8

Burali-Forti paradox

In set theory, a eld of mathematics, the Burali-Forti paradox demonstrates that navely constructing the set of allordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.It is named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him,contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, andwhen he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him byBurali-Fortis paper, with the result that it came to be known by Burali-Fortis name.

8.1 Stated in terms of von Neumann ordinalsLet be the set of all ordinals. Since carries all properties of an ordinal number, it is an ordinal number itself.We can therefore construct its successor +1 , which is strictly greater than . However, this ordinal number mustbe an element of , since contains all ordinal numbers. Finally, we arrive at

large cardinals

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