Ordinal Numbers

Ordinal numbersFrom 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 BachmannHoward ordinal 44.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5 Burali-Forti paradox 55.1 Stated in terms of von Neumann ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 Stated more generally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.3 Resolution of the paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 ChurchKleene ordinal 76.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

7 Club set 87.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87.2 The closed unbounded lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8 Conality 108.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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8.3 Conality of ordinals and other well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.4 Regular and singular ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.5 Conality of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9 Continuous function (set theory) 139.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

10 Diagonal intersection 1410.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

11 0 1511.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511.2 Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.3 Surreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

12 Even and odd ordinals 1912.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

13 FefermanSchtte ordinal 2013.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

14 First uncountable ordinal 2114.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

15 Fixed-point lemma for normal functions 2215.1 Background and formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3 Example application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

16 Kleenes O 2416.1 Kleenes O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2416.2 Basic properties of

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16.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

17 Large countable ordinal 2717.1 Generalities on recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

17.1.1 Ordinal notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2717.1.2 Relationship to systems of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

17.2 Specic recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2817.2.1 Predicative denitions and the Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . 2817.2.2 The FefermanSchtte ordinal and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . 2917.2.3 Impredicative ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2917.2.4 Unrecursable recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

17.3 Beyond recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3.1 The ChurchKleene ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3.2 Admissible ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3.3 Beyond admissible ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3.4 Unprovable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

17.4 A pseudo-well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

17.5.1 On recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.5.2 Beyond recursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.5.3 Both recursive and nonrecursive ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.5.4 Inline references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

18 Large Veblen ordinal 3218.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

19 Limit ordinal 3319.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3319.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3319.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

20 Normal function 3620.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3720.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

21 Order type 3821.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3821.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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21.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

22 Ordinal analysis 4022.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4022.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

22.2.1 Theories with proof theoretic ordinal 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4022.2.2 Theories with proof theoretic ordinal 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4022.2.3 Theories with proof theoretic ordinal n . . . . . . . . . . . . . . . . . . . . . . . . . . . 4122.2.4 Theories with proof theoretic ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . 4122.2.5 Theories with proof theoretic ordinal 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4122.2.6 Theories with proof theoretic ordinal the Feferman-Schtte ordinal 0 . . . . . . . . . . . 4122.2.7 Theories with proof theoretic ordinal the Bachmann-Howard ordinal . . . . . . . . . . . . 4122.2.8 Theories with larger proof theoretic ordinals . . . . . . . . . . . . . . . . . . . . . . . . . 41

22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4222.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

23 Ordinal arithmetic 4323.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4323.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4423.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4523.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4723.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4823.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4823.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4923.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5023.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5023.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

24 Ordinal collapsing function 5124.1 An example leading up to the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . . . . 51

24.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5124.1.2 Computation of values of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5224.1.3 Ordinal notations up to the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . 5324.1.4 Standard sequences for ordinal notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.1.5 A terminating process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

24.2 Variations on the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.2.1 Making the function less powerful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.2.2 Going beyond the Bachmann-Howard ordinal . . . . . . . . . . . . . . . . . . . . . . . . 5724.2.3 A normal variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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24.3 Collapsing large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25 Ordinal logic 6125.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

26 Ordinal notation 6226.1 A simplied example using a pairing function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

26.1.1 -notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6226.2 Systems of ordinal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

26.2.1 Cantor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2.2 Veblen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2.3 Ackermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2.4 Bachmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2.5 Takeuti (ordinal diagrams) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426.2.6 Fefermans functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426.2.7 Buchholz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426.2.8 Kleenes O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


27 Ordinal number 6627.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

27.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 6927.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

27.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7027.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

27.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7027.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7127.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7127.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7127.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

27.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7227.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

27.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7327.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

27.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7327.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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27.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7427.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7427.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7427.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7427.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

28 Primitive recursive set function 7628.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7628.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

29 Recursive ordinal 7729.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7729.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

30 Regular cardinal 7830.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7830.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7830.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7930.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

31 Small Veblen ordinal 8031.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

32 Stationary set 8132.1 Classical notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.2 Jechs notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.3 Generalized notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

33 Successor ordinal 8333.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.2 In Von Neumanns model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.3 Ordinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8433.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

34 Systems of Logic Based on Ordinals 8534.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8534.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

35 Transnite induction 8635.1 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS vii

35.2 Relationship to the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

36 Transnite number 9036.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9036.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9136.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

37 Veblen function 9237.1 The Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

37.1.1 Fundamental sequences for the Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . 9237.1.2 The function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

37.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9337.2.1 Finitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9337.2.2 Transnitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

37.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

38 Von Neumann cardinal assignment 9538.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9538.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9538.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

39 Well-order 9739.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9739.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

39.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9839.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9839.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

39.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9939.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9939.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10039.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

40 Zero-based numbering 10140.1 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

40.1.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10140.1.2 Usage in programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10240.1.3 Numerical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

40.2 Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10340.3 Other elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

viii CONTENTS


41 () 10541.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10541.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10541.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 106

41.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10641.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10841.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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

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.

4.1 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.

4.2 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)

4

Chapter 5

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.

5.1 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

< + 1 and + 1 < .

5.2 Stated more generallyThe version of the paradox above is anachronistic, because it presupposes the denition of the ordinals due to Johnvon Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time theparadox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate witheach well-ordering an object called its "order type" in an unspecied way (the order types are the ordinal numbers).The order types (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must havean order type . It is easily shown in nave set theory (and remains true in ZFC but not in New Foundations) thatthe order type of all ordinal numbers less than a xed is itself. So the order type of all ordinal numbers less than

is itself. But this means that , being the order type of a proper initial segment of the ordinals, is strictly lessthan the order type of all the ordinals, but the latter is itself by denition. This is a contradiction.If we use the von Neumann denition, under which each ordinal is identied as the set of all preceding ordinals, theparadox is unavoidable: the oending proposition that the order type of all ordinal numbers less than a xed is itself must be true. The collection of von Neumann ordinals, like the collection in the Russell paradox, cannot be aset in any set theory with classical logic. But the collection of order types in New Foundations (dened as equivalenceclasses of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of theordinals less than turns out not to be .

5.3 Resolution of the paradoxModern axiomatic set theory such as ZF and ZFC circumvents this antinomy by simply not allowing constructionof sets with unrestricted comprehension terms like all sets with the property P ", as it was for example possible in

5

6 CHAPTER 5. BURALI-FORTI PARADOX

Gottlob Frege's axiom system. New Foundations uses a dierent solution. Rosser (1942) showed that in the originalversion of Mathematical Logic (ML), an extension of New Foundations, it is possible to derive the Burali-Fortiparadox, showing that this system is contradictory.

5.4 References Burali-Forti, Cesare (1897), Una questione sui numeri transniti, Rendiconti del Circolo Matematico diPalermo 11: 154164, doi:10.1007/BF03015911

Moore, Gregory H; Garciadiego, Alejandro (1981), Burali-Fortis paradox: A reappraisal of its origins,Historia Mathematica 8 (3): 319350, doi:10.1016/0315-0860(81)90070-7

Rosser, Barkley (1942), The Burali-Forti paradox, J. Symbolic Logic 7: 117, MR 0006327

5.5 External links Stanford Encyclopedia of Philosophy: "Paradoxes and Contemporary Logic" -- by Andrea Cantini.

Chapter 6

ChurchKleene ordinal

Inmathematics, theChurchKleene ordinal,!CK1 , named after AlonzoChurch and S. C. Kleene, is a large countableordinal. It is the set of all recursive ordinals and the smallest non-recursive ordinal. It is also the rst ordinal whichis not hyperarithmetical, and the rst admissible ordinal after .

6.1 References Church, Alonzo; Kleene, S. C. (1937), Formal denitions in the theory of ordinal numbers., Fundamentamathematicae, Warszawa, 28: 1121, JFM 63.0029.02

Church, Alonzo (1938), The constructive second number class, Bull. Amer. Math. Soc. 44 (4): 224232,doi:10.1090/S0002-9904-1938-06720-1

Kleene, S. C. (1938), On Notation for Ordinal Numbers, The Journal of Symbolic Logic (The Journal ofSymbolic Logic, Vol. 3, No. 4) 3 (4): 150155, doi:10.2307/2267778, JSTOR 2267778

Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Eective Computability, First MIT presspaperback edition, ISBN 978-0-262-68052-3

7

Chapter 7

Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal whichis closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is acontraction of closed and unbounded.

7.1 Formal denition

Formally, if is a limit ordinal, then a setC is closed in if and only if for every < , if sup(C\) = 6= 0, then 2 C . Thus, if the limit of some sequence from C is less than , then the limit is also in C .If is a limit ordinal and C then C is unbounded in if for any < , there is some 2 C such that < .If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every properclass of ordinals is unbounded in the class of all ordinals).For example, the set of all countable limit ordinals is a club set with respect to the rst uncountable ordinal; but it isnot a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded. The set of all limitordinals < is closed unbounded in ( regular). In fact a club set is nothing else but the range of a normalfunction (i.e. increasing and continuous).More generally, if X is a nonempty set and is a cardinal, then C [X] is club if every union of a subset of C isin C and every subset of X of cardinality less than is contained in some element of C (see stationary set).

7.2 The closed unbounded lter

Let be a limit ordinal of uncountable conality : For some < , let hC : < i be a sequence of closedunbounded subsets of : Then T

7.3. SEE ALSO 9

7.3 See also Club lter Stationary set Clubsuit

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

Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002,Dover. ISBN 0-486-42079-5

This article incorporates material from Club on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 8

Conality

Not to be confused with coniteness.

In mathematics, especially in order theory, the conality cf(A) of a partially ordered set A is the least of thecardinalities of the conal subsets of A.This denition of conality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinalnumbers has a least member. The conality of a partially ordered set A can alternatively be dened as the leastordinal x such that there is a function from x to A with conal image. This second denition makes sense withoutthe axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the twodenitions are equivalent.Conality can be similarly dened for a directed set and is used to generalize the notion of a subsequence in a net.

8.1 Examples The conality of a partially ordered set with greatest element is 1 as the set consisting only of the greatestelement is conal (and must be contained in every other conal subset).

In particular, the conality of any nonzero nite ordinal, or indeed any nite directed set, is 1, since suchsets have a greatest element.

Every conal subset of a partially ordered set must contain all maximal elements of that set. Thus the conalityof a nite partially ordered set is equal to the number of its maximal elements.

In particular, let A be a set of size n, and consider the set of subsets of A containing no more than melements. This is partially ordered under inclusion and the subsets with m elements are maximal. Thusthe conality of this poset is n choose m.

A subset of the natural numbers N is conal in N if and only if it is innite, and therefore the conality of 0is 0. Thus 0 is a regular cardinal.

The conality of the real numbers with their usual ordering is 0, since N is conal in R. The usual ordering ofR is not order isomorphic to c, the cardinality of the real numbers, which has conality strictly greater than 0.This demonstrates that the conality depends on the order; dierent orders on the same set may have dierentconality.

8.2 PropertiesIf A admits a totally ordered conal subset, then we can nd a subset B which is well-ordered and conal in A. Anysubset of B is also well-ordered. If two conal subsets of B have minimal cardinality (i.e. their cardinality is theconality of B), then they are order isomorphic to each other.

10

8.3. COFINALITY OF ORDINALS AND OTHER WELL-ORDERED SETS 11

8.3 Conality of ordinals and other well-ordered setsThe conality of an ordinal is the smallest ordinal which is the order type of a conal subset of . The conalityof a set of ordinals or any other well-ordered set is the conality of the order type of that set.Thus for a limit ordinal, there exists a -indexed strictly increasing sequence with limit . For example, the conalityof is , because the sequence m (where m ranges over the natural numbers) tends to ; but, more generally,any countable limit ordinal has conality . An uncountable limit ordinal may have either conality as does or an uncountable conality.The conality of 0 is 0. The conality of any successor ordinal is 1. The conality of any nonzero limit ordinal is aninnite regular cardinal.

8.4 Regular and singular ordinalsA regular ordinal is an ordinal which is equal to its conality. A singular ordinal is any ordinal which is not regular.Every regular ordinal is the initial ordinal of a cardinal. Any limit of regular ordinals is a limit of initial ordinals andthus is also initial but need not be regular. Assuming the Axiom of choice, !+1 is regular for each . In this case,the ordinals 0, 1, ! , !1 , and !2 are regular, whereas 2, 3, !! , and are initial ordinals which are not regular.The conality of any ordinal is a regular ordinal, i.e. the conality of the conality of is the same as the conalityof . So the conality operation is idempotent.

8.5 Conality of cardinalsIf is an innite cardinal number, then cf() is the least cardinal such that there is an unbounded function from cf()to ; cf() is also the cardinality of the smallest set of strictly smaller cardinals whose sum is ; more precisely

cf() = min(card(I) j =

Xi2I

i and (8i)(i < ))

That the set above is nonempty comes from the fact that

=[i2

fig

i.e. the disjoint union of singleton sets. This implies immediately that cf() . The conality of any totallyordered set is regular, so one has cf() = cf(cf()).Using Knigs theorem, one can prove < cf() and < cf(2) for any innite cardinal .The last inequality implies that the conality of the cardinality of the continuum must be uncountable. On the otherhand,

@! =[n@n

the ordinal number being the rst innite ordinal, so that the conality of @! is card() = @0 . (In particular, @!is singular.) Therefore,

2@0 6= @!:(Compare to the continuum hypothesis, which states 2@0 = @1 .)Generalizing this argument, one can prove that for a limit ordinal

12 CHAPTER 8. COFINALITY

cf(@) = cf()

8.6 See also Initial ordinal

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

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

Chapter 9

Continuous function (set theory)

In mathematics, specically set theory, a continuous function is a sequence of ordinals such that the values assumedat limit stages are the limits (limit suprema and limit inma) of all values at previous stages. More formally, let bean ordinal, and s := hsj < i be a -sequence of ordinals. Then s is continuous if at every limit ordinal < ,

s = lim supfsj < g = inffsupfsj < gj < g

and

s = lim inffsj < g = supfinffsj < gj < g :

Alternatively, s is continuous if s: range(s) is a continuous function when the sets are each equipped with theorder topology. These continuous functions are often used in conalities and cardinal numbers.

9.1 References Thomas Jech. Set Theory, 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer, ISBN3-540-44085-2

13

Chapter 10

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If is an ordinal number and hX j < i is a sequence of subsets of , then the diagonal intersection, denoted by

Chapter 11

0

This article is about an ordinal in mathematics. For the physical constant 0, see vacuum permittivity.

In mathematics, the epsilon numbers are a collection of transnite numbers whose dening property is that they arexed points of an exponential map. Consequently, they are not reachable from 0 via a nite series of applicationsof the chosen exponential map and of weaker operations like addition and multiplication. The original epsilonnumbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers thatsatisfy the equation

" = !";

in which is the smallest innite ordinal. Any solution to this equation has Cantor normal form " = !" .The least such ordinal is 0 (pronounced epsilon nought or epsilon zero), which can be viewed as the limit obtainedby transnite recursion from a sequence of smaller limit ordinals:

"0 = !!!

= supf!; !!; !!! ; !!!!

; : : : gLarger ordinal xed points of the exponential map are indexed by ordinal subscripts, resulting in "1; "2; : : : ; "!; "!+1; : : : ; ""0 ; : : : ; ""1 ; : : : ; """ ; : : :. The ordinal 0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals,and uncountable epsilon numbers whose index is an uncountable ordinal).The smallest epsilon number 0 is very important in many induction proofs, because for many purposes, transniteinduction is only required up to 0 (as in Gentzens consistency proof and the proof of Goodsteins theorem). Itsuse by Gentzen to prove the consistency of Peano arithmetic, along with Gdels second incompleteness theorem,show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with thisproperty, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peanoarithmetic).Many larger epsilon numbers can be dened using the Veblen function.A more general class of epsilon numbers has been identied by John Horton Conway and Donald Knuth in the surrealnumber system, consisting of all surreals that are xed points of the base exponential map x x.Hessenberg (1906) dened gamma numbers (see additively indecomposable ordinal) to be numbers >0 such that+= whenever 1 such that =whenever 0

16 CHAPTER 11. 0

0 = 1 ; +1 = ; = lim sup 1, the mapping 7! is a normal function, so it hasarbitrarily large xed points by the xed-point lemma for normal functions. When = ! , these xed points areprecisely the ordinal epsilon numbers. The smallest of these, , is the supremum of the sequence

0; !0 = 1; !1 = !; !!; !!!

; : : : ; ! "" k; : : :

in which every element is the image of its predecessor under the mapping 7! ! . (The general term is given usingKnuths up-arrow notation; the "" operator is equivalent to tetration.) Just as is dened as the supremum of { k} for natural numbers k, the smallest ordinal epsilon number may also be denoted ! "" ! ; this notation is muchless common than .The next epsilon number after "0 is

"1 = supf"0 + 1; !"0+1; !!"0+1 ; !!!"0+1

; : : : g;

in which the sequence is again constructed by repeated base exponentiation but starts at "0 + 1 instead of at 0.Notice

!"0+1 = !"0 !1 = "0 ! ;

!!"0+1

= !("0!) = (!"0)! = "!0 ;

!!!"0+1

= !"0!

= !"01+!

= !("0"0!) = (!"0)

"0!

= "0"0!

:

A dierent sequence with the same supremum, "1 , is obtained by starting from 0 and exponentiating with base instead:

"1 = supf0; 1; "0; "0"0 ; "0"0"0 ; : : :g;

The epsilon number "+1 indexed by any successor ordinal +1 is constructed similarly, by base exponentiationstarting from " + 1 (or by base " exponentiation starting from 0).

"+1 = supf" + 1; !"+1; !!"+1 ; : : : g = supf0; 1; "; "" ; """ ; : : : g

An epsilon number indexed by a limit ordinal is constructed dierently. The number " is the supremum of the setof epsilon numbers f" ; < g . The rst such number is "! . Whether or not the index is a limit ordinal, " isa xed point not only of base exponentiation but also of base exponentiation for all ordinals 1 < < " .Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinalnumbers themselves. For any ordinal number , " is the least epsilon number (xed point of the exponential map)not already in the set f" ; < g . It might appear that this is the non-constructive equivalent of the constructivedenition using iterated exponentiation; but the two denitions are equally non-constructive at steps indexed by limitordinals, which represent transnite recursion of a higher order than taking the supremum of an exponential series.The following facts about epsilon numbers are very straightforward to prove:

Although it is quite a large number, "0 is still countable, being a countable union of countable ordinals; in fact," is countable if and only if is countable.

The union (or supremum) of any nonempty set of epsilon numbers is an epsilon number; so for instance

11.2. VEBLEN HIERARCHY 17

"! = supf"0; "1; "2; : : :g

is an epsilon number. Thus, the mapping n 7! "n is a normal function.

Every uncountable cardinal number is an epsilon number.

1 ! "! = ! :

11.2 Veblen hierarchyMain article: Veblen function

The xed points of the epsilon mapping x 7! "x form a normal function, whose xed points form a normal function,whose ; this is known as the Veblen hierarchy (the Veblen functions with base 0() = ). In the notation of theVeblen hierarchy, the epsilon mapping is 1, and its xed points are enumerated by 2.Continuing in this vein, one can dene maps for progressively larger ordinals (including, by this rareed formof transnite recursion, limit ordinals), with progressively larger least xed points (0). The least ordinal notreachable from 0 by this procedurei. e., the least ordinal for which (0)=, or equivalently the rst xed pointof the map ! (0)is the FefermanSchtte ordinal 0. In a set theory where such an ordinal can be provento exist, one has a map that enumerates the xed points 0, 1, 2, ... of ! (0) ; these are all still epsilonnumbers, as they lie in the image of for every 0, including of the map 1 that enumerates epsilon numbers.

11.3 Surreal numbersIn On Numbers and Games, the classic exposition on surreal numbers, John Horton Conway provided a number ofexamples of concepts that had natural extensions from the ordinals to the surreals. One such function is the ! -mapn 7! !n ; this mapping generalises naturally to include all surreal numbers in its domain, which in turn provides anatural generalisation of the Cantor normal form for surreal numbers.It is natural to consider any xed point of this expanded map to be an epsilon number, whether or not it happens tobe strictly an ordinal number. Some examples of non-ordinal epsilon numbers are

"1 = f0; 1; !; !!; : : : j "0 1; !"01; : : :g

and

" 12= f"0 + 1; !"0+1; : : : j "1 1; !"11; : : :g:

There is a natural way to dene "n for every surreal number n, and the map remains order-preserving. Conwaygoes on to dene a broader class of irreducible surreal numbers that includes the epsilon numbers as a particularly-interesting subclass.

11.4 See also Ordinal arithmetic Large countable ordinal

18 CHAPTER 11. 0

11.5 References J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6 Section XIV.20 of Sierpiski, Wacaw (1965), Cardinal and ordinal numbers (Second revised ed.), PWN Polish Scientic Publishers

11.6 External links Fusible numbers

Chapter 12

Even and odd ordinals

Inmathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers.They are useful in some transnite induction proofs.The literature contains a few equivalent denitions of the parity of an ordinal :

Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa.[1][2]

Let = + n, where is a limit ordinal and n is a natural number. The parity of is the parity of n.[3]

Let n be the nite term of the Cantor normal form of . The parity of is the parity of n.[4]

Let = + n, where n is a natural number. The parity of is the parity of n.[5]

If = 2, then is even. Otherwise = 2 + 1 and is odd.[5][6]

Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form 2= + . Ordinal multiplication is not commutative, so in general 2 2. In fact, the even ordinal + 4 cannot beexpressed as + , and the ordinal number

( + 3)2 = ( + 3) + ( + 3) = + (3 + ) + 3 = + + 3 = 2 + 3

is not even.A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem).Given an innite cardinal , or generally any limit ordinal , is order-isomorphic to both its subset of even ordinalsand its subset of odd ordinals. Hence one has the cardinal sum + = .[2][7]

12.1 References[1] Bruckner, AndrewM., Judith B. Bruckner, and Brian S. Thomson (1997). Real Analysis. pp. p. 37. ISBN 0-13-458886-X.

[2] Salzmann, H., T. Grundhfer, H. Hhl, and R. Lwen (2007). The Classical Fields: Structural Features of the Real andRational Numbers. Cambridge University Press. pp. p. 168. ISBN 0-521-86516-6.

[3] Foran, James (1991). Fundamentals of Real Analysis. CRC Press. pp. p. 110. ISBN 0-8247-8453-7.

[4] Harzheim, Egbert (2005). Ordered Sets. Springer. pp. p. 296. ISBN 0-387-24219-8.

[5] Kamke, Erich (1950). Theory of Sets. Courier Dover. pp. p. 96. ISBN 0-486-60141-2.

[6] Hausdor, Felix (1978). Set Theory. American Mathematical Society. pp. p. 99. ISBN 0-8284-0119-5.

[7] Roitman, Judith (1990). Introduction to Modern Set Theory. Wiley-IEEE. pp. p. 88. ISBN 0-471-63519-7.

19

Chapter 13

FefermanSchtte ordinal

In mathematics, the FefermanSchtte ordinal 0 is a large countable ordinal. It is the proof theoretic ordinal ofseveral mathematical theories, such as arithmetical transnite recursion. It is named after Solomon Feferman andKurt Schtte.It is sometimes said to be the rst impredicative ordinal, though this is controversial, partly because there is nogenerally accepted precise denition of predicative. Sometimes an ordinal is said to be predicative if it is less than0.Unfortunately there is no standard notation for ordinals at and beyond the FefermanSchtte ordinal, so there areseveral ways of representing it, some of which use ordinal collapsing functions: (

) , () or (0)

13.1 DenitionThe FefermanSchtte ordinal can be dened as the smallest ordinal that cannot be obtained by starting with 0 andusing the operations of ordinal addition and the Veblen functions (). That is, it is the smallest such that (0)= .

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


20

Chapter 14

First uncountable ordinal

In mathematics, the rst uncountable ordinal, traditionally denoted by1 or sometimes by, is the smallest ordinalnumber that, considered as a set, is uncountable. It is the supremum of all countable ordinals. The elements of 1are the countable ordinals, of which there are uncountably many.Like any ordinal number (in von Neumanns approach), 1 is a well-ordered set, with set membership ("") servingas the order relation. 1 is a limit ordinal, i.e. there is no ordinal with + 1 = 1.The cardinality of the set 1 is the rst uncountable cardinal number, 1 (aleph-one). The ordinal 1 is thus theinitial ordinal of 1. Indeed, in most constructions 1 and 1 are equal as sets. To generalize: if is an arbitraryordinal we dene as the initial ordinal of the cardinal .The existence of 1 can be proven without the axiom of choice. (See Hartogs number.)

14.1 Topological propertiesAny ordinal number can be turned into a topological space by using the order topology. When viewed as a topologicalspace, 1 is often written as [0,1) to emphasize that it is the space consisting of all ordinals smaller than 1.Every increasing -sequence of elements of [0,1) converges to a limit in [0,1). The reason is that the union(=supremum) of every countable set of countable ordinals is another countable ordinal.The topological space [0,1) is sequentially compact but not compact. As a consequence, it is not metrizable. It ishowever countably compact and thus not Lindelf. In terms of axioms of countability, [0,1) is rst countable butnot separable nor second countable.The space [0, 1] = 1 + 1 is compact and not rst countable. 1 is used to dene the long line and the Tychonoplank, two important counterexamples in topology.

14.2 See also Ordinal arithmetic Large countable ordinal Continuum hypothesis

14.3 References Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN3-540-44085-2.

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York,1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

21

Chapter 15

Fixed-point lemma for normal functions

The xed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal functionhas arbitrarily large xed points (Levy 1979: p. 117). It was rst proved by Oswald Veblen in 1908.

15.1 Background and formal statementA normal function is a class function f from the class Ord of ordinal numbers to itself such that:

f is strictly increasing: f() < f() whenever < . f is continuous: for every limit ordinal (i.e. is neither zero nor a successor), f() = sup { f() : < }.

It can be shown that if f is normal then f commutes with suprema; for any nonempty set A of ordinals,

f(sup A) = sup {f() : A }.

Indeed, if sup A is a successor ordinal then sup A is an element of A and the equality follows from the increasingproperty of f. If sup A is a limit ordinal then the equality follows from the continuous property of f.A xed point of a normal function is an ordinal such that f() = .The xed point lemma states that the class of xed points of any normal function is nonempty and in fact is unbounded:given any ordinal , there exists an ordinal such that and f() = .The continuity of the normal function implies the class of xed points is closed (the supremum of any subset of theclass of xed points is again a xed point). Thus the xed point lemma is equivalent to the statement that the xedpoints of a normal function form a closed and unbounded class.

15.2 ProofThe rst step of the proof is to verify that f() for all ordinals and that f commutes with suprema. Given theseresults, inductively dene an increasing sequence (n < ) by setting 0 = , and n = f(n) for n . Let = sup {n : n }, so . Moreover, because f commutes with suprema,

f() = f(sup {n : n < })= sup {f(n) : n < }= sup {n : n < }= .

The last equality follows from the fact that the sequence increases.

22

15.3. EXAMPLE APPLICATION 23

15.3 Example applicationThe function f : Ord Ord, f() = is normal (see initial ordinal). Thus, there exists an ordinal such that =. In fact, the lemma shows that there is a closed, unbounded class of such .

15.4 References Levy, A. (1979). Basic Set Theory. Springer. ISBN 0-387-08417-7. Republished, Dover, 2002. ISBN 0-486-42079-5.

Veblen, O. (1908). Continuous increasing functions of nite and transnite ordinals. Trans. Amer. MathSoc. (American Mathematical Society) 9 (3): 280292. doi:10.2307/1988605. ISSN 0002-9947. JSTOR1988605. Available via JSTOR.

Chapter 16

Kleenes O

In set theory and computability theory, Kleene's O is a canonical subset of the natural numbers when regardedas ordinal notations. It contains ordinal notations for every recursive ordinal, that is, ordinals below ChurchKleeneordinal, !CK1 . Since !CK1 is the rst ordinal not representable in a computable system of ordinal notations theelements of O can be regarded as the canonical ordinal notations.Kleene (1938) described a system of notation for all recursive ordinals (those less than the ChurchKleene ordinal).It uses a subset of the natural numbers instead of nite strings of symbols. Unfortunately, there is in general noeective way to tell whether some natural number represents an ordinal, or whether two numbers represent the sameordinal. However, one can eectively nd notations which represent the ordinal sum, product, and power (see ordinalarithmetic) of any two given notations in Kleenes O ; and given any notation for an ordinal, there is a recursivelyenumerable set of notations which contains one element for each smaller ordinal and is eectively ordered.

16.1 Kleenes OThe basic idea of Kleenes system of ordinal notations is to build up ordinals in an eective manner. For members pof O , the ordinal for which p is a notation is jpj . The standard denition proceeds via transnite induction and theordering

16.3. PROPERTIES OF PATHS IN O 25

Since every recursive function has countably many indices, each innite ordinal receives countably many no-tations; the nite ordinals have unique notations, n usually denoted nO .

The rst ordinal that doesn't receive a notation is called the ChurchKleene ordinal and is denoted by !CK1 .Since there are only countably many recursive functions, the ordinal !CK1 is evidently countable.

The ordinals with a notation in Kleenes O are exactly the recursive ordinals. (The fact that every recursiveordinal has a notation follows from the closure of this system of ordinal notations under successor and eectivelimits.)

26 CHAPTER 16. KLEENES O

16.5 References Church, Alonzo (1938), The constructive second number class, Bull. Amer. Math. Soc. 44 (4): 224232,doi:10.1090/S0002-9904-1938-06720-1

Kleene, S. C. (1938), On Notation for Ordinal Numbers, The Journal of Symbolic Logic (Association forSymbolic Logic) 3 (4): 150155, doi:10.2307/2267778, JSTOR 2267778

Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Eective Computability, First MIT presspaperback edition, ISBN 978-0-262-68052-3

Feferman, Solomon; Spector, Cliord (1962), Incompleteness along paths in progressions of theories, Jour-nal of Symbolic Logic 27 (4): 383390, doi:10.2307/2964544

Chapter 17

Large countable ordinal

Main article: Ordinal number

In the mathematical discipline of set theory, there are many ways of describing specic countable ordinals. Thesmallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, manyordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decideeectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to theunsolvability of the halting problem); various more-concrete ways of dening ordinals that denitely have notationsare available.Since there are only countably many notations, all ordinals with notations are exhausted well below the rst uncount-able ordinal 1; their supremum is called ChurchKleene 1 or 1CK (not to be confused with the rst uncountableordinal, 1), described below. Ordinal numbers below 1CK are the recursive ordinals (see below). Countableordinals larger than this may still be dened, but do not have notations.Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. Theordinals described here are not as large as the ones described in large cardinals, but they are large among those thathave constructive notations (descriptions). Larger and larger ordinals can be dened, but they become more and moredicult to describe.

17.1 Generalities on recursive ordinalsMain article: Recursive ordinal

17.1.1 Ordinal notationsMain article: Ordinal notation

Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those represented by acomputable function. There are several equivalent denitions of this: the simplest is to say that a computable ordinalis the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinalis recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) canmanipulate them (and, essentially, compare them).A dierent denition uses Kleene's system of ordinal notations. Briey, an ordinal notation is either the name zero(describing the ordinal 0), or the successor of an ordinal notation (describing the successor of the ordinal described bythat notation), or a Turing machine (computable function) that produces an increasing sequence of ordinal notations(that describe the ordinal that is the limit of the sequence), and ordinal notations are (partially) ordered so as to makethe successor of o greater than o and to make the limit greater than any term of the sequence (this order is computable;however, the setO of ordinal notations itself is highly non-recursive, owing to the impossibility of deciding whether agiven Turing machine does indeed produce a sequence of notations); a recursive ordinal is then an ordinal described

27

28 CHAPTER 17. LARGE COUNTABLE ORDINAL

by some ordinal notation.Any ordinal smaller than a recursive ordinal is itself recursive, so the set of all recursive ordinals forms a certain(countable) ordinal, the Church-Kleene ordinal (see below).It is tempting to forget about ordinal notations, and only speak of the recursive ordinals themselves: and some state-ments are made about recursive ordinals which, in fact, concern the notations for these ordinals. This leads to di-culties, however, as even the smallest innite ordinal, , has many notations, some of which cannot be proven to beequivalent to the obvious notation (the limit of the simplest program that enumerates all natural numbers).

17.1.2 Relationship to systems of arithmetic

There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least areasonable fragment of Peano arithmetic).Certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formalsystem might not be suciently powerful to show that o is, indeed, an ordinal notation: the system does not showtransnite induction for such large ordinals.For example, the usual rst-order Peano axioms do not prove transnite induction for (or beyond) 0: while theordinal 0 can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to showthat it is indeed an ordinal; in fact, transnite induction on 0 proves the consistency of Peanos axioms (a theorem byGentzen), so by Gdels second incompleteness theorem, Peanos axioms cannot formalize that reasoning. (This is atthe basis of the KirbyParis theorem on Goodstein sequences.) We say that 0 measures the proof-theoretic strengthof Peanos axioms.But we can do this for systems far beyond Peanos axioms. For example, the proof-theoretic strength of KripkePlatekset theory is the Bachmann-Howard ordinal (see below), and, in fact, merely adding to Peanos axioms the axiomsthat state the well-ordering of all ordinals below the BachmannHoward ordinal is sucient to obtain all arithmeticalconsequences of KripkePlatek set theory.

17.2 Specic recursive ordinals

17.2.1 Predicative denitions and the Veblen hierarchy

Main article: Veblen function

We have already mentioned (see Cantor normal form) the ordinal 0, which is the smallest satisfying the equation! = , so it is the limit of the sequence 0, 1, ! , !! , !!! , etc. The next ordinal satisfying this equation is called1: it is the limit of the sequence

"0 + 1; !"0+1 = "0 !; !!"0+1 = ("0)!; etc.

More generally, the -th ordinal such that ! = is called " . We could dene 0 as the smallest ordinal suchthat " = , but since the Greek alphabet does not have transnitely many letters it is better to use a more robustnotation: dene ordinals '() by transnite induction as follows: let '0() = ! and let '+1() be the -thxed point of ' (i.e., the -th ordinal such that '() = ; so for example, '1() = " ), and when is a limitordinal, dene '() as the -th common xed point of the ' for all < . This family of functions is known asthe Veblen hierarchy. (There are inessential variations in the denition, such as letting, for a limit ordinal, '()be the limit of the '() for < : this essentially just shifts the indices by 1, which is harmless.) ' is called the

th Veblen function (to the base ! ).Ordering: '() < '() if and only if either ( = and < ) or ( < and < '() ) or ( > and'() < ).

17.2. SPECIFIC RECURSIVE ORDINALS 29

17.2.2 The FefermanSchtte ordinal and beyond

The smallest ordinal such that '(0) = is known as the FefermanSchtte ordinal and generally written 0 . Itcan be described as the set of all ordinals that can be written as nite expressions, starting from zero, using only theVeblen hierarchy and addition. The Feferman-Schtte ordinal is important because, in a sense that is complicated tomake precise, it is the smallest (innite) ordinal that cannot be (predicatively) described using smaller ordinals. Itmeasures the strength of such systems as arithmetical transnite recursion.More generally, enumerates the ordinals that cannot be obtained from smaller ordinals using addition and theVeblen functions.It is, of course, possible to describe ordinals beyond the Feferman-Schtte ordinal. One could continue to seek xedpoints in more and more complicated manner: enumerate the xed points of 7! , then enumerate the xedpoints of that, and so on, and then look for the rst ordinal such that is obtained in steps of this process, andcontinue diagonalizing in this ad hoc manner. This leads to the denition of the small and large Veblen ordinals.

17.2.3 Impredicative ordinals

Main article: Ordinal collapsing function

To go far beyond the Feferman-Schtte ordinal, one needs to introduce new methods. Unfortunately there is notyet any standard way to do this: every author in the subject seems to have invented their own system of notation,and it is quite hard to translate between the dierent systems. The rst such system was introduced by Bachmannin 1950 (in an ad hoc manner), and dierent extensions and variations of it were described by Buchholz, Takeuti(ordinal diagrams), Feferman ( systems), Aczel, Bridge, Schtte, and Pohlers. However most systems use the samebasic idea, of constructing new countable ordinals by using the existence of certain uncountable ordinals. Here is anexample of such a denition, described in much greater detail in the article on ordinal collapsing function:

() is dened to be the smallest ordinal that cannot be constructed by starting with 0, 1, and , and repeat-edly applying addition, multiplication and exponentiation, and to previously constructed ordinals (exceptthat can only be applied to arguments less than , to ensure that it is well dened).

Here = 1 is the rst uncountable ordinal. It is put in because otherwise the function gets stuck at the smallestordinal such that =: in particular ()= for any ordinal satisfying . However the fact that we included allows us to get past this point: (+1) is greater than . The key property of that we used is that it is greaterthan any ordinal produced by .To construct still larger ordinals, we can extend the denition of by throwing in more ways of constructing un-countable ordinals. There are several ways to do this, described to some extent in the article on ordinal collapsingfunction.The Bachmann-Howard ordinal (sometimes just called the Howard ordinal, () with the notation above) isan important one, because it describes the proof-theoretic strength of Kripke-Platek set theory. Indeed, the mainimportance of these large ordinals, and the reason to describe them, is their relation to certain formal systems asexplained above. However, such powerful formal systems as full second-order arithmetic, let alone Zermelo-Fraenkelset theory, seem beyond reach for the moment.

17.2.4 Unrecursable recursive ordinals

By dropping the requirement of having a useful description, even larger recursive countable ordinals can be obtainedas the ordinals measuring the strengths of various strong theories; roughly speaking, these ordinals are the smallestordinals that the theories cannot prove are well ordered. By taking stronger and stronger theories such as second-order arithmetic, Zermelo set theory, Zermelo-Fraenkel set theory, or Zermelo-Fraenkel set theory with various largecardinal axioms, one gets some extremely large recursive ordinals. (Strictly speaking it is not known that all of thesereally are ordinals: by construction, the ordinal strength of a theory can only be proven to be an ordinal from an evenstronger theory. So for the large cardinal axioms this becomes quite unclear.)

30 CHAPTER 17. LARGE COUNTABLE ORDINAL

17.3 Beyond recursive ordinals

17.3.1 The ChurchKleene ordinal

The set of recursive ordinals is an ordinal that is the smallest ordinal that cannot be described in a recursive way.(It is not the order type of any recursive well-ordering of the integers.) That ordinal is a countable ordinal calledthe ChurchKleene ordinal, !CK1 . Thus, !CK1 is the smallest non-recursive ordinal, and there is no hope of preciselydescribing any ordinals from this point onwe can only dene them. But it is still far less than the rst uncountableordinal, !1 . However, as its symbol suggests, it behaves in many ways rather like !1 .

17.3.2 Admissible ordinals

Main article: Admissible ordinal

The Church-Kleene ordinal is again related to Kripke-Platek set theory, but now in a dierent way: whereas theBachmann-Howard ordinal (described above) was the smallest ordinal for which KP does not prove transnite induc-tion, the Church-Kleene ordinal is the smallest such that the construction of the Gdel universe, L, up to stage ,yields a model L of KP. Such ordinals are called admissible, thus !CK1 is the smallest admissible ordinal (beyond in case the axiom of innity is not included in KP).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. One sometimes writes !CK for the -th ordinal that iseither admissible or a limit of admissible.

17.3.3 Beyond admissible ordinals

An ordinal that is both admissible and a limit of admissibles, or equivalently such that is the -th admissibleordinal, is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallelto that of (small) large cardinals. For example, we can dene recursively Mahlo ordinals: these are the such thatevery -recursive closed unbounded subset of contains an admissible ordinal (a recursive analog of the denitionof a Mahlo cardinal). But note that we are still talking about possibly countable ordinals here. (While the existenceof inaccessible or Mahlo cardinals cannot be proved in Zermelo-Fraenkel set theory, that of recursively inaccessibleor recursively Mahlo ordinals is a theorem of ZFC: in fact, any regular cardinal is recursively Mahlo and more, buteven if we limit ourselves to countable ordinals, ZFC proves the existence of recursively Mahlo ordinals. They are,however, beyond the reach of Kripke-Platek set theory.)An admissible ordinal is called nonprojectible if there is no total -recursive injective function mapping into asmaller ordinal. (This is trivially true for regular cardinals; however, we are mainly interested in countable ordinals.)Being nonprojectible is a much stronger condition than being admissible, recursively inaccessible, or even recursivelyMahlo. It is equivalent to the statement that the Gdel universe, L, up to stage , yields a model L of KP + 1-separation.

17.3.4 Unprovable ordinals

We can imagine even larger ordinals that are still countable. For example, if ZFC has a transitive model (a hypothesisstronger than the mere hypothesis of consistency, and implied by the existence of an inaccessible cardinal), then thereexists a countable such that L is a model of ZFC. Such ordinals are beyond the strength of ZFC in the sense thatit cannot (by construction) prove their existence.Even larger countable ordinals, called the stable ordinals, can be dened by indescribability conditions or as those such that L is a 1-elementary submodel of L; the existence of these ordinals can be proven in ZFC,[1] and they areclosely related to the nonprojectible ordinals.

17.4. A PSEUDO-WELL-ORDERING 31

17.4 A pseudo-well-orderingWithin the scheme of notations of Kleene some represent ordinals and some do not. One can dene a recursive totalordering that is a subset of the Kleene notations and has an initial segment which is well-ordered with order-type !CK1. Every recursively enumerable (or even hyperarithmetic) nonempty subset of this total ordering has a least element.So it resembles a well-ordering in some respects. For example, one can dene the arithmetic operations on it. Yetit is not possible to eectively determine exactly where the initial well-ordered part ends and the part lacking a leastelement begins.

17.5 ReferencesMost books describing large countable ordinals are on proof theory, and unfortunately tend to be out of print.

17.5.1 On recursive ordinals Wolfram Pohlers, Proof theory, Springer 1989 ISBN 0-387-51842-8 (for Veblen hierarchy and some impred-icative ordinals). This is probably the most readable book on large countable ordinals (which is not sayingmuch).

Gaisi Takeuti, Proof theory, 2nd edition 1987 ISBN 0-444-10492-5 (for ordinal diagrams) Kurt Schtte, Proof theory, Springer 1977 ISBN 0-387-07911-4 (for Veblen hierarchy and some impredicativeordinals)

Craig Smorynski, The varieties of arboreal experienceMath. Intelligencer 4 (1982), no. 4, 182189; containsan informal description of the Veblen hierarchy.

Hartley Rogers, Jr., Theory of Recursive Functions and Eective Computability McGraw-Hill (1967) ISBN0-262-68052-1 (describes recursive ordinals and the ChurchKleene ordinal)

LarryW. Miller, Normal Functions and Constructive Ordinal Notations, The Journal of Symbolic Logic, volume41, number 2, June 1976, pages 439 to 459, JSTOR 2272243,

Hilbert Levitz, Transnite Ordinals and Their Notations: For The Uninitiated, expository article (8 pages, inPostScript)

Herman Ruge Jervell, Truth and provability, manuscript in progress.

17.5.2 Beyond recursive ordinals Barwise, Jon (1976). Admissible Sets and Structures: an Approach to Denability Theory. Perspectives inMathematical Logic. Springer-Verlag. ISBN 3-540-07451-1.

Hinman, Peter G. (1978). Recursion-theoretic hierarchies. Perspectives in Mathematical Logic. Springer-Verlag.

17.5.3 Both recursive and nonrecursive ordinals Michael Rathjen, The realm of ordinal analysis. in S. Cooper and J. Truss (eds.): Sets and Proofs. (CambridgeUniversity Press, 1999) 219279. At Postscript le.

17.5.4 Inline references[1] Barwise (1976), theorem 7.2.

Chapter 18

Large Veblen ordinal

In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen.There is no standard notation for ordinals beyond the FefermanSchtte ordinal 0. Most systems of notation usesymbols such as (), (), (), some of which are modications of the Veblen functions to produce countableordinals even for uncountable arguments, and some of which are collapsing functions.The large Veblen ordinal is sometimes denoted by

(0) or (

) or (

) . It was constructed by Veblenusing an extension of Veblen functions allowing innitely many arguments.

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


32

Chapter 19

Limit ordinal

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, anordinal is a limit ordinal if and only if there is an ordinal less than , and whenever is an ordinal less than , thenthere exists an ordinal such that < < . Every ordinal number is either zero, or a successor ordinal, or a limitordinal.For example, , the smallest ordinal greater than every natural number is a limit ordinal because for any smallerordinal (i.e., for any natural number) n we can nd another natural number larger than it (e.g. n+1), but still less than.Using the Von Neumann denition of ordinals, every ordinal is the well-ordered set of all smaller ordinals. The unionof a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using Von Neumann cardinalassignment, every innite cardinal number is also a limit ordinal.

19.1 Alternative denitionsVarious other ways to dene limit ordinal are:

It is equal to the supremum of all the ordinals below it, but is not zero. (Compare with a successor ordinal: theset of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)

It is not zero and has no maximum element. It can be written in the form for > 0. That is, in the Cantor normal form there is no nite number as lastterm, and the ordinal is nonzero.

It is a limit point of the class of ordinal numbers, with respect to the order topology. (The other ordinals areisolated points.)

Some contention exists on whether or not 0 should be classied as a limit ordinal, as it does not have an immediatepredecessor; some textbooks include 0 in the class of limit ordinals[1] while others exclude it.[2]

19.2 ExamplesBecause the class of ordinal numbers is well-ordered, there is a smallest innite limit ordinal; denoted by (omega).The ordinal is also the smallest innite ordinal (disregarding limit), as it is the least upper bound of the naturalnumbers. Hence represents the order type of the natural numbers. The next limit ordinal above the rst is + = 2, which generalizes to n for any natural number n. Taking the union (the supremum operation on any set ofordinals) of all the n, we get = 2, which generalizes to n for any natural number n. This process can befurther iterated as follows to produce:

!3; !4; : : : ; !!; !!!

; : : : ; 0 = !!!

; : : :

33

34 CHAPTER 19. LIMIT ORDINAL

0

12

3

+

1+

2+3

2

3

2+

1

2+2

4

+1

+2

+

+22

34

+

+5

4

5

+4

2

2+3

Representation of the ordinal numbers up to . Each turn of the spiral represents one power of . Limit ordinals are those thatare non-zero and have no predecessor, such as or 2

In general, all of these recursive denitions via multiplication, exponentiation, repeated exponentiation, etc. yieldlimit ordinals. All of the ordinals discussed so far are still countable ordinals. However, there is no recursivelyenumerable scheme for systematically naming all ordinals less than the ChurchKleene ordinal, which is a countableordinal.Beyond the countable, the rst uncountable ordinal is usually denoted 1. It is also a limit ordinal.Continuing, one can obtain the following (all of which are now increasing in cardinality):

!2; !3; : : : ; !!; !!+1; : : : ; !!! ; : : :

19.3. PROPERTIES 35

In general, we always get a limit ordinal when taking the union of a nonempty set of ordinals that has no maximumelement.The ordinals of the form , for > 0, are limits of limits, etc.

19.3 PropertiesThe classes of successor ordinals and limit ordinals (of various conalities) as well as zero exhaust the entire class ofordinals, so these cases are often used in proofs by transnite induction or denitions by transnite recursion. Limitordinals represent a sort of turning point in such procedures, in which one must use limiting operations such astaking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking theunion is continuous in the order topology and this is usually desirable.If we use the Von Neumann cardinal assignment, every innite cardinal number is also a limit ordinal (and this is atting observation, as cardinal derives from the Latin cardo meaning hinge or turning point): the proof of this fact isdone by simply showing that every innite successor ordinal is equinumerous to a limit ordinal via the Hotel Innityargument.Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).

19.4 See also Ordinal arithmetic Limit cardinal Fundamental sequence (ordinals)

19.5 References[1] for example, Thomas Jech, Set Theory. Third Millennium edition. Springer.

[2] for example, Kenneth Kunen, Set Theory. An introduction to independence proofs. North-Holland.

19.6 Further reading Cantor, G., (1897), Beitrage zur Begrundung der transnitenMengenlehre. II (tr.: Contributions to the Foundingof the Theory of Transnite Numbers II), Mathematische Annalen 49, 207-246 English translation.

Conway, J. H. and Guy, R. K. Cantors Ordinal Numbers. In The Book of Numbers. New York: Springer-Verlag, pp. 266267 and 274, 1996.

Sierpiski, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Pastwowe WydawnictwoNaukowe. Also denes ordinal operations in terms of the Cantor Normal Form.

Chapter 20

Normal function

In axiomatic set theory, a function f : Ord Ord is called normal (or a normal function) i it is continuous (withrespect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:

1. For every limit ordinal (i.e. is neither zero nor a successor), f() = sup {f() : < }.2. For all ordinals < , f() < f().

20.1 ExamplesA simple normal function is given by f() = 1 + (see ordinal arithmetic). But f() = + 1 is not normal. If is axed ordinal, then the functions f() = + , f() = (for 1), and f() = (for 2) are all normal.More important examples of normal functions are given by the aleph numbers f() = @ which connect ordinal andcardinal numbers, and by the beth numbers f() = i .

20.2 PropertiesIf f is normal, then for any ordinal ,

f() .[1]

Proof: If not, choose minimal such that f() < . Since f is strictly monotonically increasing, f(f()) < f(),contradicting minimality of .Furthermore, for any non-empty set S of ordinals, we have

f(sup S) = sup f(S).

Proof: "" follows from the monotonicity of f and the denition of the supremum. For "", set = sup S andconsider three cases:

if = 0, then S = {0} and sup f(S) = f(0); if = + 1 is a successor, then there exists s in S with < s, so that s. Therefore, f() f(s), whichimplies f() sup f(S);

if is a nonzero limit, pick any < , and an s in S such that < s (possible since = sup S). Therefore f() . Also, 1 is the smallest uncountable ordinal (to see that it exists, considerthe set of equivalence classes of well-orderings of the natural numbers; each such well-ordering denes a countableordinal, and 1 is the order type of that set), 2 is the smallest ordinal whose cardinality is greater than 1, and soon, and is the limit of n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed therst cardinal after all the n).Innite initial ordinals are limit ordinals. Using ordinal arithmetic, < implies + = , and 1 < implies =

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