set theory m n o

Set theory m n oFrom Wikipedia, the free encyclopedia

Contents

1 Admissible set 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Almost 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Benacerrafs identication problem 33.1 Historical motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 BIT predicate 54.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Private information retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.4 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.5 Construction of the Rado graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Cabal (set theory) 75.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6 Cantors diagonal argument 86.1 Uncountable set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6.1.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.1.2 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6.2 General sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2.1 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2.2 Version for Quines New Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Cantors rst uncountability proof 147.1 The article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 The proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Constructive or non-constructive nature of Cantors proof of the existence of transcendentals . . . . 177.4 The development of Cantors ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.5 Why Cantors article emphasizes the countability of the algebraic numbers . . . . . . . . . . . . . . 187.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Cantors paradise 248.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Cantors theorem 259.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.2 A detailed explanation of the proof when X is countably innite . . . . . . . . . . . . . . . . . . . 269.3 Related paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10 Cardinal assignment 3010.1 Cardinal assignment without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11 Cardinality of the continuum 3111.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11.1.1 Uncountability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.1.2 Cardinal equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.1.3 Alternative explanation for c = 2@0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11.2 Beth numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.3 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.4 Sets with cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.5 Sets with greater cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12 Categorical set theory 3612.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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12.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13 Changs conjecture 3713.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Class (set theory) 3814.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.3 Classes in formal set theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

15 Class logic 4015.1 Class logic in the strict sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

16 Club lter 4216.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17 Club set 4317.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.2 The closed unbounded lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18 Clubsuit 4518.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.2 and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

19 Code (set theory) 4619.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4619.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

20 Conality 4720.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.3 Conality of ordinals and other well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.4 Regular and singular ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.5 Conality of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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21 Condensation lemma 5021.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

22 Continuous function (set theory) 5122.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

23 Continuum (set theory) 5223.1 Linear continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5223.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5223.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

24 Controversy over Cantors theory 5324.1 Cantors argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5324.2 Reception of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5424.3 Objection to the axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5424.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5524.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

25 Deductive closure 5725.1 Epistemic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5725.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

26 Denable real number 5826.1 General facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5826.2 Notion does not exhaust unambiguously described numbers . . . . . . . . . . . . . . . . . . . . 5926.3 Other notions of denability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

26.3.1 Denability in other languages or structures . . . . . . . . . . . . . . . . . . . . . . . . . 5926.3.2 Denability with ordinal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

26.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5926.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

27 Diaconescus theorem 6127.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6127.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

28 Diagonal intersection 6328.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6328.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

29 Diamond principle 6429.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6429.2 Properties and use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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29.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6529.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

30 Dimensional operator 6630.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6630.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6630.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

31 Eastons theorem 6731.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.2 No extension to singular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6831.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6831.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

32 Equaliser (mathematics) 6932.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6932.2 Dierence kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6932.3 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7032.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7032.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7132.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7132.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

33 ErdsRado theorem 7233.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7233.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

34 Extension (semantics) 7334.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7334.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7334.3 Metaphysical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7434.4 General semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7434.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7434.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

35 Extensionality 7535.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7535.2 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7535.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7635.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

36 Fodors lemma 7736.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7736.2 Fodors lemma for trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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36.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

37 Game-theoretic rough sets 7837.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

38 Goodsteins theorem 7938.1 Hereditary base-n notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7938.2 Goodstein sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8038.3 Proof of Goodsteins theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8038.4 Extended Goodsteins theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8138.5 Sequence length as a function of the starting value . . . . . . . . . . . . . . . . . . . . . . . . . . 8138.6 Application to computable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8138.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8238.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8238.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8238.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

39 Gdel logic 8339.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

40 Hartogs number 8440.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8440.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

41 Hausdor gap 8541.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8541.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8541.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

42 Hereditarily countable set 8742.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8742.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

43 Hereditarily nite set 8843.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8843.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8843.3 Ackermanns bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8943.4 Rado graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8943.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8943.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

44 Hereditary property 9044.1 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9044.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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44.2.1 Monotone property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9044.3 In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9144.4 In matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9144.5 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9144.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

45 Hereditary set 9345.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.2 In formulations of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

46 Humes principle 9446.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9446.2 Inuence on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9446.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9546.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

47 Ideal (set theory) 9647.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9647.2 Examples of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

47.2.1 General examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9647.2.2 Ideals on the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9647.2.3 Ideals on the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9747.2.4 Ideals on other sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

47.3 Operations on ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9747.4 Relationships among ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9747.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9747.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

48 Implementation of mathematics in set theory 9948.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9948.2 Empty set, singleton, unordered pairs and tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . 10048.3 Ordered pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10048.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

48.4.1 Related denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10148.4.2 Properties and kinds of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

48.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10248.5.1 Operations on functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10248.5.2 Special kinds of function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

48.6 Size of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10348.7 Finite sets and natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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48.8 Equivalence relations and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10448.9 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

48.9.1 Digression: von Neumann ordinals in NFU . . . . . . . . . . . . . . . . . . . . . . . . . 10648.10Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10648.11The Axiom of Counting and subversion of stratication . . . . . . . . . . . . . . . . . . . . . . . 107

48.11.1 Properties of strongly cantorian sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10748.12Familiar number systems: positive rationals, magnitudes, and reals . . . . . . . . . . . . . . . . . 10748.13Operations on indexed families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10848.14The cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10848.15See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11048.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11048.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

49 Innitary combinatorics 11149.1 Ramsey theory for innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11149.2 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11249.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11249.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

50 Information diagram 11350.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

51 Jensens covering theorem 11551.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

52 Jnsson function 11652.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

53 Kuratowskis free set theorem 11753.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

54 Laver function 11854.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11854.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11854.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

55 Limit cardinal 11955.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11955.2 Relationship with ordinal subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11955.3 The notion of inaccessibility and large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 12055.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12055.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12055.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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56 List of exceptional set concepts 121

57 List of set theory topics 12357.1 Articles on individual set theory topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12357.2 Lists related to set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12657.3 Set theorists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12657.4 Societies and organizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

58 List of statements undecidable in ZFC 12858.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12858.2 Set theory of the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12958.3 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12958.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12958.5 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13058.6 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13058.7 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13058.8 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13058.9 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13158.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13158.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

59 Lvy hierarchy 13259.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13259.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

59.2.1 0=0=0 formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13259.2.2 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.3 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.4 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.5 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.6 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.7 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13359.2.8 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13459.2.9 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13459.2.10 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13459.2.11 4-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

59.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13459.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13459.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

60 Mathematical structure 13560.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13560.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13660.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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61 Mengenlehreuhr 13761.1 Telling the time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13761.2 Kryptos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13761.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13761.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

62 MilnerRado paradox 14162.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14162.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

63 Morass (set theory) 14263.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14263.2 Variants and equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14263.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

64 Mostowski model 14464.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

65 Multiplicity (mathematics) 14565.1 Multiplicity of a prime factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14565.2 Multiplicity of a root of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

65.2.1 Behavior of a polynomial function near a multiple root . . . . . . . . . . . . . . . . . . . 14565.3 Intersection multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14665.4 In complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14765.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

66 Naive set theory 14866.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

66.1.1 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14866.1.2 Cantors theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14966.1.3 Axiomatic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14966.1.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14966.1.5 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

66.2 Sets, membership and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15066.2.1 Note on consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15066.2.2 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15166.2.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15166.2.4 Empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

66.3 Specifying sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15166.4 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15266.5 Universal sets and absolute complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15266.6 Unions, intersections, and relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 15266.7 Ordered pairs and Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

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66.8 Some important sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15366.9 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15466.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15566.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15566.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15666.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

67 Normal function 15767.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15767.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15767.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15867.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

68 Ontological maximalism 15968.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

69 Open coloring axiom 16069.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16069.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

70 Ordinal arithmetic 16170.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16170.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16270.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16370.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16570.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16670.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16670.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16770.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16870.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16870.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

71 Ordinal denable set 16971.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

72 Set (mathematics) 17072.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17172.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17172.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

72.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17372.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

72.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17472.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17472.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

xii CONTENTS

72.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17572.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17672.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17672.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

72.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17972.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17972.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18072.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18072.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18172.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18172.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18172.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

73 Set theory 18273.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18373.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18473.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18573.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18573.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18673.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

73.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18773.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18773.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18773.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18773.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

73.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 18973.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18973.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18973.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19073.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

74 -logic 19174.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19174.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19274.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 193

74.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19874.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Chapter 1

Admissible set

In set theory, a discipline within mathematics, an admissible set is a transitive set A such that hA;2i is a model ofKripkePlatek set theory (Barwise 1975).The smallest example of an admissible set is the set of hereditarily nite sets. Another example is the set of hereditarilycountable sets.

1.1 See also Admissible ordinal

1.2 References Barwise, Jon (1975). Admissible Sets and Structures: An Approach to Denability Theory, Perspectives inMathematical Logic, Volume 7, Springer-Verlag. Electronic version on Project Euclid.

1

Chapter 2

Almost

For other uses, see Almost (disambiguation).

In set theory, when dealing with sets of innite size, the term almost or nearly is used to mean all the elements exceptfor nitely many.In other words, an innite set S that is a subset of another innite set L, is almost L if the subtracted set L\S is ofnite size.Examples:

The set S = fn 2 Njn kg is almost N for any k in N, because only nitely many natural numbers are lessthan k.

The set of prime numbers is not almost N because there are innitely many natural numbers that are not primenumbers.

This is conceptually similar to the almost everywhere concept of measure theory, but is not the same. For example,the Cantor set is uncountably innite, but has Lebesgue measure zero. So a real number in (0, 1) is a member of thecomplement of the Cantor set almost everywhere, but it is not true that the complement of the Cantor set is almostthe real numbers in (0, 1).

2.1 See also Almost all Almost surely

2

Chapter 3

Benacerrafs identication problem

Benacerrafs identication problem is a philosophical argument, developed by Paul Benacerraf, against set-theoreticPlatonism.[1] In 1965, Benacerraf published a paradigm changing article entitled What Numbers Could Not Be.[1][2]Historically, the work became a signicant catalyst in motivating the development of structuralism in the philosophyof mathematics.[3] The identication problem argues that there exists a fundamental problem in reducing naturalnumbers to pure sets. Since there exists an innite number of ways of identifying the natural numbers with pure sets,no particular set-theoretic method can be determined as the true reduction. Benacerraf infers that any attempt tomake such a choice of reduction immediately results in generating a meta-level, set-theoretic falsehood, namely inrelation to other elementarily-equivalent set-theories not identical to the one chosen.[1] The identication problemargues that this creates a fundamental problem for Platonism, which maintains that mathematical objects have a real,abstract existence. Benacerrafs dilemma to Platonic set-theory is arguing that the Platonic attempt to identify thetrue reduction of natural numbers to pure sets, as revealing the intrinsic properties of these abstract mathematicalobjects, is impossible.[1] As a result, the identication problem ultimately argues that the relation of set theory tonatural numbers cannot have an ontologically Platonic nature.[1]

3.1 Historical motivationsThe historical motivation for the development of Benacerrafs identication problem derives from a fundamentalproblem of ontology. Since Medieval times, philosophers have argued as to whether the ontology of mathematicscontains abstract objects. In the philosophy of mathematics, an abstract object is traditionally dened as an entity that:(1) exists independent of the mind; (2) exists independent of the empirical world; and (3) has eternal, unchangeableproperties.[4] Traditional mathematical Platonismmaintains that some set ofmathematical elementsnatural numbers,real numbers, functions, relations, systemsare such abstract objects. Contrarily, mathematical nominalism denies theexistence of any such abstract objects in the ontology of mathematics.In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These includedintuitionism, formalism, and predicativism. By the mid-20th century, however, these anti-Platonist theories had anumber of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historiccontext that the motivations for the identication problem developed.

3.2 DescriptionThe identication problem begins by evidencing some set of elementarily-equivalent, set-theoretic models of thenatural numbers.[1] Benacerraf considers two such set-theoretic methods:

Set-theoretic method I0 = 1 = {}2 = {{}}3 = {{{}}}

3

4 CHAPTER 3. BENACERRAFS IDENTIFICATION PROBLEM

...

Set-theoretic method II0 = 1 = {}2 = {, {}}3 = {, {}, {, {}}}...

As Benacerraf demonstrates, both method I and II reduce natural numbers to sets.[1] Benacerraf formulates thedilemma as a question: which of these set-theoretic methods uniquely provides the true identity statements, whichelucidates the true ontological nature of the natural numbers?[1] Either method I or II could be used to dene thenatural numbers and subsequently generate true arithmetical statements to form a mathematical system. In theirrelation, the elements of such mathematical systems are isomorphic in their structure. However, the problem ariseswhen these isomorphic structures are related together on the meta-level. The denitions and arithmetical statementsfrom system I are not identical to the denitions and arithmetical statements from system II. For example, the twosystems dier in their answer to whether 0 2, insofar as is not an element of {{}}. Thus, in terms of failing thetransitivity of identity, the search for true identity statements similarly fails.[1] By attempting to reduce the naturalnumbers to sets, this renders a set-theoretic falsehood between the isomorphic structures of dierent mathematicalsystems. This is the essence of the identication problem.According to Benacerraf, the philosophical ramications of this identication problem result in Platonic approachesfailing the ontological test.[1] The argument is used to demonstrate the impossibility for Platonism to reduce numbersto sets that reveals the existence of abstract objects.

3.3 See also Philosophy of mathematics Structuralism (philosophy of mathematics) Paul Benacerraf

3.4 References[1] Paul Benacerraf (1965), What Numbers Could Not Be, Philosophical Review Vol. 74, pp. 4773.[2] Bob Hale and Crispin Wright (2002) Benacerrafs Dilemma Revisited European Journal of Philosophy, Issue 10:1.[3] Stewart Shapiro (1997) Philosophy of Mathematics: Structure and Ontology New York: Oxford University Press, p. 37.

ISBN 0195139305[4] Michael Loux (2006) Metaphysics: A Contemporary Introduction (Routledge Contemporary Introductions to Philosophy),

London: Routledge. ISBN 0415401348

3.5 Bibliography Benacerraf, Paul (1965) What Numbers Could Not Be Philosophical Review Vol. 74, pp. 4773. Benacerraf, Paul (1973) Mathematical Truth, in Benacerraf & Putnam Philosophy ofMathematics: SelectedReadings, Cambridge: Cambridge University Press, 2nd edition. 1983, pp. 403420.

Hale, Bob (1987) Abstract Objects. Oxford: Basil Blackwell. ISBN 0631145931 Hale, Bob and Wright, Crispin (2002) Benacerrafs Dilemma Revisited European Journal of Philosophy,Issue 10:1.

Shapiro, Stewart (1997) Philosophy of Mathematics: Structure and Ontology New York: Oxford UniversityPress. ISBN 0195139305

Chapter 4

BIT predicate

In mathematics and computer science, the BIT predicate or Ackermann coding, sometimes written BIT(i, j), is apredicate which tests whether the jth bit of the number i is 1, when i is written in binary.

4.1 History

The BIT predicate was rst introduced as the encoding of hereditarily nite sets as natural numbers by WilhelmAckermann in his 1937 paper[1][2] (The Consistency of General Set Theory).Each natural number encodes a nite set and each nite set is represented by a natural number. This mapping uses thebinary numeral system. If the number n encodes a nite set A and the ith binary digit of n is 1 then the set encodedby i is element of A. The Ackermann coding is a primitive recursive function.[3]

4.2 Implementation

In programming languages such as C, C++, Java, or Python that provide a right shift operator >> and a bitwiseBoolean and operator &, the BIT predicate BIT(i, j) can be implemented by the expression (i>>j)&1. Here the bitsof i are numbered from the low order bits to high order bits in the binary representation of i, with the ones bit beingnumbered as bit 0.[4]

4.3 Private information retrieval

In the mathematical study of computer security, the private information retrieval problem can be modeled as one inwhich a client, communicating with a collection of servers that store a binary number i, wishes to determine the resultof a BIT predicate BIT(i, j) without divulging the value of j to the servers. Chor et al. (1998) describe a method forreplicating i across two servers in such a way that the client can solve the private information retrieval problem usinga substantially smaller amount of communication than would be necessary to recover the complete value of i.[5]

4.4 Mathematical logic

The BIT predicate is often examined in the context of rst-order logic, where we can examine the system resultingfrom adding the BIT predicate to rst-order logic. In descriptive complexity, the complexity class FO + BIT resultingfrom adding the BIT predicate to FO results in a more robust complexity class.[6] The class FO + BIT, of rst-orderlogic with the BIT predicate, is the same as the class FO + PLUS + TIMES, of rst-order logic with addition andmultiplication predicates.[7]

5

6 CHAPTER 4. BIT PREDICATE

4.5 Construction of the Rado graphAckermann in 1937 and Richard Rado in 1964 used this predicate to construct the innite Rado graph. In theirconstruction, the vertices of this graph correspond to the non-negative integers, written in binary, and there is anundirected edge from vertex i to vertex j, for i < j, when BIT(j,i) is nonzero.[8]

4.6 References[1] Ackermann, Wilhelm (1937). Die Widerspruchsfreiheit der allgemeinen Mengenlehre. Mathematische Annalen 114:

305315. doi:10.1007/bf01594179. Retrieved 2012-01-09.

[2] Kirby, Laurence (2009). Finitary Set Theory. Notre Dame Journal of Formal Logic 50 (3): 227244. doi:10.1215/00294527-2009-009. Retrieved 31 May 2011.

[3] Rautenberg,Wolfgang (2010). AConcise Introduction toMathematical Logic (3rd ed.). NewYork: Springer Science+BusinessMedia. p. 261. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6.

[4] Venugopal, K. R. (1997). Mastering C++. Muhammadali Shaduli. p. 123. ISBN 9780074634547..

[5] Chor, Benny; Kushilevitz, Eyal; Goldreich, Oded; Sudan, Madhu (1998). Private information retrieval. Journal of theACM 45 (6): 965981. doi:10.1145/293347.293350..

[6] Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag. ISBN 0-387-98600-6.

[7] Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag. pp. 1416. ISBN 0-387-98600-6.

[8] Rado, Richard (1964). Universal graphs and universal functions. Acta Arith. 9: 331340..

Chapter 5

Cabal (set theory)

The Cabal was, or perhaps is, a grouping of set theorists in Southern California, particularly at UCLA and Caltech,but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is dicult to say whetherit still exists or exactly who has been a member, but it has included such notable gures as Donald A. Martin, YiannisN. Moschovakis, John R. Steel, and Alexander S. Kechris. Others who have published in the proceedings of theCabal seminar include Robert M. Solovay, W. Hugh Woodin, Matthew Foreman, and Steve Jackson.The work of the group is characterized by free use of large cardinal axioms, and research into the descriptive settheoretic behavior of sets of reals if such assumptions hold.Some of the philosophical views of the Cabal seminar were described in Maddy 1988a and Maddy 1988b.

5.1 Publications Kechris, A. S. et al. (1978). Cabal Seminar 76-77: Proceedings. Caltech-UCLA Logic Seminar 1976-77.Springer. ISBN 0-387-09086-X.

Kechris, A. S. (editor) (1983). Cabal Seminar 79-81: Proc Caltech-UCLA Logic Seminar 1979-81 (LectureNotes in Mathematics). Springer. ISBN 0-387-12688-0.

Martin, D. A., A. S. Kechris, J. R. Steel (1988). Cabal Seminar 81-85: Proceedings Caltech UCLA LogicSeminar (Lecture Notes in Mathematics, No 1333). Springer. ISBN 0-387-50020-0.

Alexander S. Kechris, Benedikt Lwe, John R. Steel (2008). Games, Scales, and Suslin cardinals: The CabalSeminar Volume I: Lecture Notes in Logic. CUP. ISBN 9780521899512.

5.2 References Maddy, Penelope (1988). Believing the Axioms I (PDF). The Journal of Symbolic Logic 53 (2): 481511. Maddy, Penelope (1988). Believing the Axioms II (PDF). The Journal of Symbolic Logic 53 (3): 736764.

7

Chapter 6

Cantors diagonal argument

In set theory, Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argu-ment or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are innitesets which cannot be put into one-to-one correspondence with the innite set of natural numbers.[1][2][3] Such sets arenow known as uncountable sets, and the size of innite sets is now treated by the theory of cardinal numbers whichCantor began.The diagonal argument was not Cantors rst proof of the uncountability of the real numbers; it was actually pub-lished much later than his rst proof, which appeared in 1874.[4][5] However, it demonstrates a powerful and generaltechnique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy withthe argument used in this proof. The most famous examples are perhaps Russells paradox, the rst of Gdelsincompleteness theorems, and Turings answer to the Entscheidungsproblem.

6.1 Uncountable setIn his 1891 article, Cantor considered the set T of all innite sequences of binary digits (i.e. consisting only of zeroesand ones). He begins with a constructive proof of the following theorem:

If s1, s2, , sn, is any enumeration of elements from T, then there is always an element s of T whichcorresponds to no sn in the enumeration.

To prove this, given an enumeration of arbitrary members from T, like e.g.

he constructs the sequence s by choosing its nth digit as complementary to the nth digit of sn, for every n. In theexample, this yields:

By construction, s diers from each sn, since their nth digits dier (highlighted in the example). Hence, s cannotoccur in the enumeration.Based on this theorem, Cantor then uses an indirect argument to show that:

The set T is uncountable.

He assumes for contradiction that T was countable. Then (all) its elements could be written as an enumeration s1,s2, , sn, . Applying the previous theorem to this enumeration would produce a sequence s not belonging tothe enumeration. However, s was an element of T and should therefore be in the enumeration. This contradicts theoriginal assumption, so T must be uncountable.

8

6.1. UNCOUNTABLE SET 9

An illustration of Cantors diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannotoccur anywhere in the enumeration of sequences above.

6.1.1 Interpretation

The interpretation of Cantors result will depend upon ones view of mathematics. To constructivists, the argumentshows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibilitythat the latter are subcountable. In the context of classical mathematics, this is impossible, and the diagonal argumentestablishes that, although both sets are innite, there are actually more innite sequences of ones and zeros than thereare natural numbers.

10 CHAPTER 6. CANTORS DIAGONAL ARGUMENT

YX123

x

246

2x. .

. .

An innite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to theeven numbers demonstrates. Nevertheless, innite sets of dierent cardinalities exist, as Cantors diagonal argument shows.

6.1.2 Real numbers

The uncountability of the real numbers was already established by Cantors rst uncountability proof, but it also fol-lows from the above result. To see this, we will build a one-to-one correspondence between the set T of innite binarystrings and a subset of R (the set of real numbers). Since T is uncountable, this subset of R must be uncountable.Hence R is uncountable.To build this one-to-one correspondence (or bijection), observe that the string t = 0111 appears after the binarypoint in the binary expansion 0.0111. This suggests dening the function f(t) = 0.t, where t is a string in T.Unfortunately, f(1000) = 0.1000 = 1/2, and f(0111) = 0.0111 = 1/4 + 1/8 + 1/16 + = 1/2. So thisfunction is not a bijection since two strings correspond to one numbera number having two binary expansions.However, modifying this function produces a bijection from T to the interval (0, 1)that is, the real numbers > 0and < 1. The idea is to remove the problem elements from T and (0, 1), and handle them separately. From (0, 1),remove the numbers having two binary expansions. Put these numbers in a sequence: a = (1/2, 1/4, 3/4, 1/8, 3/8,5/8, 7/8, ). From T, remove the strings appearing after the binary point in the binary expansions of 0, 1, and thenumbers in sequence a. Put these eventually-constant strings in a sequence: b = (000, 111, 1000, 0111,01000, 11000, 00111, 10111, ...). A bijection g(t) from T to (0, 1) is dened by: If t is the nth string insequence b, let g(t) be the nth number in sequence a; otherwise, let g(t) = 0.t.To build a bijection from T to R: start with the tangent function tan(x), which provides a bijection from (/2, /2)to R; see right picture. Next observe that the linear function h(x) = x - /2 provides a bijection from (0, 1) to(/2, /2); see left picture. The composite function tan(h(x)) = tan(x - /2) provides a bijection from (0, 1) toR. Compose this function with g(t) to obtain tan(h(g(t))) = tan(g(t) - /2), which is a bijection from T to R. Thismeans that T and R have the same cardinalitythis cardinality is called the "cardinality of the continuum.

6.2. GENERAL SETS 11

6.2 General sets

Illustration of the generalized diagonal argument: The set T = {n: nf(n)} at the bottom cannot occur anywhere in the range off:P(). The example mapping f happens to correspond to the example enumeration s in the above picture.

A generalized form of the diagonal argument was used by Cantor to prove Cantors theorem: for every set S the powerset of S; that is, the set of all subsets of S (here written as P(S)), has a larger cardinality than S itself. This proofproceeds as follows:Let f be any function from S to P(S). It suces to prove f cannot be surjective. That means that some member T ofP(S), i.e. some subset of S, is not in the image of f. As a candidate consider the set:

T = { s S: s f(s) }.

For every s in S, either s is in T or not. If s is in T, then by denition of T, s is not in f(s), so T is not equal to f(s).On the other hand, if s is not in T, then by denition of T, s is in f(s), so again T is not equal to f(s); cf. picture. Fora more complete account of this proof, see Cantors theorem.

6.2.1 Consequences

This result implies that the notion of the set of all sets is an inconsistent notion. If S were the set of all sets then P(S)would at the same time be bigger than S and a subset of S.Russells Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contra-dictory. Note that there is a similarity between the construction of T and the set in Russells paradox. Therefore,

12 CHAPTER 6. CANTORS DIAGONAL ARGUMENT

depending on how we modify the axiom scheme of comprehension in order to avoid Russells paradox, argumentssuch as the non-existence of a set of all sets may or may not remain valid.The diagonal argument shows that the set of real numbers is bigger than the set of natural numbers (and therefore,the integers and rationals as well). Therefore, we can ask if there is a set whose cardinality is between that ofthe integers and that of the reals. This question leads to the famous continuum hypothesis. Similarly, the questionof whether there exists a set whose cardinality is between |S| and |P(S)| for some innite S leads to the generalizedcontinuum hypothesis.Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certainobjects. For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonalargument. Also, diagonalization was originally used to show the existence of arbitrarily hard complexity classes andplayed a key role in early attempts to prove P does not equal NP.

6.2.2 Version for Quines New Foundations

The above proof fails for W. V. Quine's "New Foundations" set theory (NF). In NF, the naive axiom scheme ofcomprehension is modied to avoid the paradoxes by introducing a kind of local type theory. In this axiom scheme,

{ s S: s f(s) }

is not a set i.e., does not satisfy the axiom scheme. On the other hand, we might try to create a modied diagonalargument by noticing that

{ s S: s f({s}) }

is a set in NF. In which case, if P1(S) is the set of one-element subsets of S and f is a proposed bijection from P1(S)to P(S), one is able to use proof by contradiction to prove that |P1(S)| < |P(S)|.The proof follows by the fact that if f were indeed a map onto P(S), then we could nd r in S, such that f({r})coincides with the modied diagonal set, above. We would conclude that if r is not in f({r}), then r is in f({r}) andvice versa.It is not possible to put P1(S) in a one-to-one relation with S, as the two have dierent types, and so any function sodened would violate the typing rules for the comprehension scheme.

6.3 See also Cantors rst uncountability proof

Controversy over Cantors theory

6.4 References[1] Georg Cantor (1892). Ueber eine elementare Frage der Mannigfaltigkeitslehre (PDF). Jahresbericht der Deutschen

Mathematiker-Vereinigung 18901891 1: 7578 (8487 in pdf le). (in german)

[2] Keith Simmons (30 July 1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. CambridgeUniversity Press. pp. 20. ISBN 978-0-521-43069-2.

[3] Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. p. 30. ISBN 0070856133.

[4] Gray, Robert (1994), Georg Cantor and Transcendental Numbers (PDF), American Mathematical Monthly 101: 819832, doi:10.2307/2975129

[5] Bloch, Ethan D. (2011). The Real Numbers and Real Analysis. New York: Springer. p. 429. ISBN 978-0-387-72176-7.

6.5. EXTERNAL LINKS 13

6.5 External links Cantors Diagonal Proof at MathPages Weisstein, Eric W., Cantor Diagonal Method, MathWorld.

Chapter 7

Cantors rst uncountability proof

Georg Cantors rst proof of uncountability demonstrates that the set of all real numbers is uncountably, ratherthan countably, innite. This proof diers from the more familiar proof that uses his diagonal argument. Cantorsrst uncountability proof was published in 1874, in an article that also contains a proof that the set of real algebraicnumbers is countable, and a proof of the existence of transcendental numbers.[1]

Two points about which not all authors writing about Cantors article have agreed are these:

Is Cantors proof of the existence of transcendental numbers constructive or non-constructive?[2]

Why did Cantor emphasize the countability of the real algebraic numbers rather than the uncountability of thereal numbers?[3]

In 1891 Cantor published his diagonal argument,[4] which produces an uncountability proof that is generally con-sidered simpler and more elegant than his rst proof. Both uncountability proofs contain ideas that can be usedelsewhere. The diagonal argument is a general technique that is useful in mathematical logic and theoretical com-puter science, while Cantors rst uncountability proof can be generalized to any ordered set with the same orderproperties as the real numbers.[5]

7.1 The articleCantors article[6] begins with a discussion of the real algebraic numbers, and a statement of his rst theorem: Thecollection of real algebraic numbers can be put into one-to-one correspondence with the collection of positive integers.Cantor restates this theorem in terms more familiar to mathematicians of his time: The collection of real algebraicnumbers can be written as an innite sequence in which each number appears only once.Next Cantor states his second theorem: Given any sequence of real numbers x1, x2, x3, and any interval [a, b],[7]one can determine numbers in [a, b] that are not contained in the given sequence.Cantor observes that combining his two theorems yields a new proof of the theorem: Every interval [a, b] containsinnitely many transcendental numbers. This theorem was rst proved by Joseph Liouville.[8]

He then remarks that his second theorem is:

the reason why collections of real numbers forming a so-called continuum (such as, all real numberswhich are 0 and 1) cannot correspond one-to-one with the collection () [the collection of all positiveintegers]; thus I have found the clear dierence between a so-called continuum and a collection like thetotality of real algebraic numbers.[9]

The rst half of this remark is Cantors uncountability theorem. Cantor does not explicitly prove this theorem, whichfollows easily from his second theorem. To prove it, use proof by contradiction. Assume that the interval [a, b] canbe put into one-to-one correspondence with the set of positive integers, or equivalently: The real numbers in [a, b]can be written as a sequence in which each real number appears only once. Applying Cantors second theorem to

14

7.1. THE ARTICLE 15

Georg Cantor

this sequence and [a, b] produces a real number in [a, b] that does not belong to the sequence. This contradicts ouroriginal assumption, and proves the uncountability theorem.Cantors second theorem is constructive and thereby separates the constructive content of his work from the proof bycontradiction needed to establish uncountability.[10]

16 CHAPTER 7. CANTORS FIRST UNCOUNTABILITY PROOF

7.2 The proofs

Algebraic numbers on the complex plane colored by polynomial degree. (red = 1, green = 2, blue = 3, yellow = 4). Points becomessmaller as the integer polynomial coecients become larger.

To prove that the set of real algebraic numbers is countable, Cantor starts by dening the height of a polynomial ofdegree n to be: n 1 + |a0| + |a1| + + |an|, where a0, a1, , an are the (integer) coecients of the polynomial.Then Cantor orders the polynomials by their height, and orders the real roots of polynomials of the same height bynumeric order. Since there are only a nite number of roots of polynomials of a given height, Cantors orderings putthe real algebraic numbers into a sequence.[11]

Next Cantor proves his second theorem: Given any sequence of real numbers x1, x2, x3, and any interval [a, b],one can determine a number in [a, b] that is not contained in the given sequence.[12]

To nd such a number, Cantor builds two sequences of real numbers as follows: Find the rst two numbers of thegiven sequence x1, x2, x3, that belong to the interior of the interval [a, b].[13] Designate the smaller of these twonumbers by a1, and the larger by b1. Similarly, nd the rst two numbers of the given sequence belonging to theinterior of the interval [a1, b1]. Designate the smaller by a2 and the larger by b2. Continuing this procedure generatesa sequence of intervals [a1, b1], [a2, b2], such that each interval in the sequence contains all succeeding intervals.This implies the sequence a1, a2, a3, is increasing, the sequence b1, b2, b3, is decreasing, and every memberof the rst sequence is smaller than every member of the second sequence.Cantor now breaks the proof into two cases: Either the number of intervals generated is nite or innite. If nite, let[aN, bN] be the last interval. Since at most one xn can belong to the interior of [aN, bN], any number belonging tothis interior besides xn is not contained in the given sequence.If the number of intervals is innite, let a = limn an.[14] At this point, Cantor could nish his proof by notingthat a is not contained in the given sequence since for every n, a belongs to the interior of [an, bn] but xn doesnot.[15]

Instead Cantor analyzes the situation further. He lets b= limn bn,[16] and then breaks the proof into two cases:a = b and a < b. In the rst case, as mentioned above, a is not contained in the given sequence. In thesecond case, any real number in [a, b] is not contained in the given sequence. Cantor observes that the sequenceof real algebraic numbers falls into the rst case, thus indicating how his proof handles this particular sequence.[17]

7.3. CONSTRUCTIVEORNON-CONSTRUCTIVENATUREOFCANTORS PROOFOFTHEEXISTENCEOFTRANSCENDENTALS17

7.3 Constructive or non-constructive nature of Cantors proof of the exis-tence of transcendentals

Some mathematicians claim that Cantors proof of the existence of transcendental numbers is constructivethat is,it provides a method of constructing a transcendental number. For example, Irving Kaplansky writes:

It is often said that Cantors proof is not constructive, and so does not yield a tangible transcendentalnumber. This remark is not justied. If we set up a denite listing of all algebraic numbers and thenapply the diagonal procedure , we get a perfectly denite transcendental number (it could be computedto any number of decimal places) (I owe these remarks to R. M. Robinson.)[18]

Other mathematicians claim that Cantors proof is non-constructive. According to Ian Stewart:

The set of real numbers is uncountable. There is an innity bigger than the innity of natural numbers!The proof is highly original. Roughly, the idea is to assume that the reals are countable, and argue for acontradiction. Building on this, Cantor was able to give a dramatic proof that transcendental numbersmust exist. Cantor showed that the set of algebraic numbers is countable. Since the full set of reals isuncountable, there must exist numbers that are not algebraic. End of proof (which is basically a triviality);collapse of audience in incredulity. In fact Cantors argument shows more: it shows that there must beuncountably many transcendentals! There are more transcendental numbers than algebraic ones; and youcan prove it without ever exhibiting a single example of either.[19]

The above quotations show that these two groups of mathematicians are discussing dierent but related proofsoneproof is constructive while the other is non-constructive. Both proofs use a construction that takes a sequence of realnumbers and produces a real number not belonging to this sequence. This construction is either the one in Cantors1874 article, or it uses his diagonal method. The proofs dier in how they use this construction.The constructive proof applies it to the sequence of real algebraic numbers, thus producing a transcendental number.Cantor gave this proof in his article (see "The article").The non-constructive proof starts by assuming that the set of real numbers is countable, or equivalently: the realnumbers can be written as a sequence. Applying the construction to this sequence produces a real number not inthe sequence, which contradicts the assumption that this sequence contains all real numbers. Hence, the set of realnumbers is uncountable. Since the set of algebraic numbers is countable, transcendental numbers must exist. Thisproof does not construct a single transcendental number.Cantors constructions have been used to write computer programs that generate transcendental numbers.[20] Theseprograms show that his constructions produce computable numbers (as long as one starts with a computable sequenceof computable numbers). The program that uses Cantors diagonal method computes the digits of a transcendentalnumber in polynomial time,[10] while the program that uses his 1874 construction requires at least sub-exponentialtime.[21]

The constructive nature of Cantors work is easily demonstrated by using his two methods to produce irrationalnumbers. Both constructions start with the same sequence of rational numbers between 0 and 1. This sequence isformed by ordering these rational numbers by increasing denominators, and ordering those with the same denominatorby increasing numerators.The table below constructs an irrational number x by using Cantors diagonal method. The strategy is to constructthe decimal representation of a number that diers from the decimal representation of every rational number in thesequence. We choose the n-th digit of x so that it diers from the n-th digit of the n-th member of the sequence. Ifthe latter digit is between 0 and 7, add 1 to obtain the n-th digit of x; otherwise, let the n-th digit of x be 0. So thedecimal representation for x diers from every decimal in the sequence. Also, x is between 0 and 1, and its decimalrepresentation does not contain the digit 9.[22] Hence, x is irrational.The next table constructs an irrational number by using Cantors 1874 construction. The strategy is to construct asequence of nested intervals such that every rational number is excluded from the interior of some interval. Cantorsconstruction starts by nding the rst two numbers in the sequence that belong to the interior of the starting interval[0, 1]. These numbers are 1/2 and 1/3, and they form the interval [1/3, 1/2]. Next we nd the next two numbersin the sequence that belong to the interior of [1/3, 1/2]. Continuing this process generates a sequence of nestedintervals. This sequence does not terminate since we can always nd two rational numbers belonging to the interiorof an interval.[23]


In the table, the rst column contains the interval, and the last column lists the rationals excluded in the search for therst two rationals belonging to this intervals interior. These excluded rationals are in the same order as the originalsequence with one exceptionnamely, one of the endpoints of the next interval. For example, the exception in the rstrow is 2/5, and it is the rst number excluded in the next row. Every rational number is excluded from some intervalsinterior because the sequence of intervals does not terminate and the interior of every interval excludes at least tworational numbers (the intervals endpoints). Thus, a real number belonging to the interior of every interval is irrational.In his proof, Cantor constructs such a real number by taking the limits of the endpoints of the intervals.[24][25]

7.4 The development of Cantors ideasThe development leading to Cantors article appears in the correspondence between Cantor and his fellow mathe-matician Richard Dedekind.[26] On November 29, 1873, Cantor asked Dedekind whether the collection of positiveintegers and the collection of positive real numbers can be corresponded so that each individual of one collectioncorresponds to one and only one of the other?" Cantor added that collections having such a correspondence includethe collection of positive rational numbers, and collections of the form (an1, n2, , n) where n1, n2,, n, and are positive integers.[27]

Dedekind replied that he was unable to answer Cantors question, and said that it did not deserve too much eortbecause it has no particular practical interest. Dedekind also sent Cantor a proof that the set of algebraic numbersis countable.[28]

On December 2, Cantor pointed out that his question does have interest: It would be nice if it could be answered;for example, provided that it could be answered no, one would have a new proof of Liouvilles theorem that there aretranscendental numbers.[29]

On December 7, Cantor sent Dedekind an intricate proof by contradiction that the set of real numbers is uncountable.This proof uses innitely many sequences of real numbers while the published proof uses only two sequences.[30]Taken together, the letters of December 2 and 7 provide a non-constructive proof of the existence of transcendentalnumbers.On December 9, Cantor announced the theorem that allows him to construct transcendental numbers as well as provethe uncountability of the set of real numbers:

I show directly that if I start with a sequence(I) 1, 2, , n, I can determine, in every given interval [, ], a number that is not included in (I).[31]

This theorem is the second theorem in Cantors article.After the publication, Dedekind wrote a letter to Cantor, in which he pointed out that the proofs in the article weretaken from the earlier letters sent by Dedekind to Cantor. According to Dedekind, Cantors paper was based on atranscription of two of his letters, one written on November 30 or December 1, the other on December 8, 1873, bothof which are now lost. In his personal notes, Dedekind wrote: after a short time, this theorem and its proof werereproduced almost literally, including the use of the technical term 'height' [Hhe], in Cantors article.[32]

7.5 WhyCantors article emphasizes the countability of the algebraic num-bers

During the Christmas holidays, Cantor visited Berlin and showed his work to his former professor Karl Weierstrass.On December 25, Cantor wrote to Dedekind about his decision to publish:

Although I did not yet wish to publish the subject I recently for the rst time discussed with you, I havenevertheless unexpectedly been caused to do so. I communicated my results to Mr. Weierstrass on the 22nd; on the 23rd I had the pleasure of a visit from him, at which I could communicate the proofs to him.He was of the opinion that I must publish the thing at least in so far as it concerns the algebraic numbers.So I wrote a short paper with the title: On a property of the set of real algebraic numbers, and sent it toProfessor Borchardt[33] to be considered for the Journal fr Math [Crelles Journal].[34]

7.5. WHY CANTORS ARTICLE EMPHASIZES THE COUNTABILITY OF THE ALGEBRAIC NUMBERS 19

In a letter to Philip Jourdain, Cantor provided more details of Weierstrass reaction:

With Mr. Weierstrass I had good relations. Of the conception of enumerability [countability] of whichhe heard from me at Berlin on Christmas holydays 1873, he became at rst quite amazed, but [after] oneor two days passed over, it became his own and helped him to an unexpected development of his wonderfultheory of functions.[35]

Weierstrass probably urged Cantor to publish because he found the countability of the set of algebraic numbers bothsurprising and useful.[36] On December 27, Cantor told Dedekind more about his article, and mentioned its quickacceptance (only four days after submission):[37]

The restriction which I have imposed on the published version of my investigations is caused in part bylocal [Berlin] circumstances (about which I shall perhaps later speak with you orally) and in part becauseI believe that it is important to apply my ideas at rst to a single case (such as that of the real algebraicnumbers) [34]

As Mr. Borchardt has already responded to me today, he will have the kindness to include this article soonin the Math. Journal.[38]

Cantor gave two reasons for restricting his article: local circumstances and the importance of applying my ideasat rst to a single case. Cantor never told Dedekind what the local circumstances were.[39] This has led to acontroversy: Who inuenced Cantor so that his article emphasizes the countability of the set of algebraic numbersrather than the uncountability of the set of real numbers? This controversy is also fueled by Cantors earlier letters,which indicate that he was most interested in the set of real numbers.Cantor biographer Joseph Dauben argues that local circumstances refers to the inuence of Leopold Kronecker,Weierstrass colleague at the University of Berlin. Dauben states that publishing in Crelles Journal could be dif-cult because Kronecker, a member of the journals editorial board, had a restricted view of what was acceptablein mathematics.[40] Dauben argues that to avoid publication problems,[41] Cantor wrote his article to emphasize thecountability of the set of real algebraic numbers.Dauben uses examples from Cantors article to show Kroneckers inuence.[42] For example, Cantor did not prove theexistence of the limits used in the proof of his second theorem.[43] Cantor did this despite using Dedekinds versionof the proof. In his private notes, Dedekind wrote:

[my] version is carried over almost word-for-word in Cantors article (Crelles Journal, 77); of coursemy use of the principle of continuity is avoided at the relevant place [44]

The principle of continuity requires a general theory of the irrationals, such as Cantors or Dedekinds constructionof the real numbers from the rationals. Kronecker accepted neither theory.[45]

In his history of set theory, Jos Ferreirs analyzes the situation in Berlin and arrives at a dierent conclusion. Fer-reirs emphasizes Weierstrass inuence: Weierstrass was interested in the countability of the set of real algebraicnumbers because he could use it to build interesting functions.[46] Also, Ferreirs suspects that in 1873 Weierstrassmight not have accepted the idea that innite sets can have dierent sizes. The following year, Weierstrass statedthat two 'innitely great magnitudes are not comparable and can always be regarded as equal.[47] Weierstrass opin-ion on innite sets may have led him to advise Cantor to omit his remark on the essential dierence between thecollections of real numbers and real algebraic numbers.[48] (This remark appears above in "The article.) Cantormentions Weierstrass advice in his December 27 letter:

The remark on the essential dierence of the collections, which I could have very well included, wasomitted on the advice of Mr. Weierstrass; but [he also advised that I] could add it later as a marginal noteduring proofreading.[49]

Ferreirs strongest statement about the local circumstancesmentions both Kronecker andWeierstrass: HadCantoremphasized it [the uncountability result], as he had in the correspondence with Dedekind, there is no doubt thatKronecker and Weierstrass would have reacted negatively.[50] Ferreirs also mentions another aspect of the localsituation: Cantor, thinking of his future career in mathematics, desired to maintain good relations with the Berlinmathematicians.[51] This desire could havemotivated Cantor to create an article that appealed toWeierstrass interests,and did not antagonize Kronecker.[52]


7.6 See also Cantors theorem Cantors diagonal argument

7.7 Notes[1] Cantor 1874. English translation: Ewald 1996, pp. 840843.

[2] Gray 1994.

[3] Dauben 1979, pp. 6670; Ferreirs 2007, pp. 183186.

[4] Cantor 1891. English translation: Ewald 1996, pp. 920922.

[5] The diagonal argument cannot be applied to sets that only have an ordering. Applying the diagonal argument to the realnumbers requires a numeral systemsuch as the decimal representation systemand a numeral system uses the additionand multiplication properties of the real numbers.

[6] Cantor 1874. English translation: Ewald 1996, pp. 840843.

[7] The notation [a, b] denotes the set of real numbers that are a and b.

[8] Liouville proved this theorem by constructing what are now known as Liouville numbers, and then proving that thesenumbers are transcendental.

[9] Cantor 1874, p. 259. English translation: Gray 1994, p. 820.

[10] Gray 1994, p. 823.

[11] Using this ordering and placing only the rst occurrence of a real algebraic number in the sequence produces a sequencewithout duplicates. Cantor obtains the same sequence by using irreducible polynomials:

[12] Although Cantors proof determines a single number, he states: and consequently innitely many such numbers canbe determined. (Cantor 1874, p. 260. English translation: Ewald 1996, p. 841.) To see this, consider (for example) theinterval [0, 1]. By dividing it into the subintervals [0, 1/2], [1/2, 3/4], [3/4, 7/8], and then applying the construction inCantors proof to each subinterval, we obtain innitely many numbers that are not contained in the given sequence.

[13] The interior of the interval [a, b] consists of all numbers in the interval except the endpoints a and b.

[14] Cantor does not state why this limit exists. It exists because the completeness property of the real numbers implies thatevery increasing sequence that is bounded above has a limit. The sequence an is increasing and it is bounded above by b(that is, for all n, an < b). The easiest proof that such a sequence has a limit uses the least upper bound property: Everynonempty set of real numbers that has an upper bound also has a least upper bound. The least upper bound property is oneof several equivalent ways to express completeness (see the article "Real number").

[15] Cantor states (without proof) that xn does not belong to the interior of [an, bn]. To prove this, we use an inductive proofto prove the stronger result: x1, x2, , xn do not belong to the interior of [an, bn]. By Cantors construction, a1 and b1are the rst two numbers in the given sequence xn that belong to the interior of [a, b]. If x1 is one of these numbers, thenit does not belong to the interior of [a1, b1]. However, if x1 is not one of these numbers, then it does not even belong tothe interior of the

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