Mathematical logicFrom Wikipedia, the free encyclopedia

Contents

1 Abstract logic 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Abstract model theory 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Abstract structure 33.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Algebraic definition 44.1 Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5 Algebraic semantics (mathematical logic) 55.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6 Algebraic sentence 66.1 Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7 Algorithmic logic 77.1 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8 Automated proof checking 88.1 Field journals and conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9 Axiom of adjunction 109.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

10 Axiom of regularity 1110.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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10.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110.1.2 No infinite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 1110.1.3 Simpler set-theoretic definition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 1210.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 12

10.2 The axiom of dependent choice and no infinite descending sequence of sets implies regularity . . . . 1210.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210.4 Regularity and Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

10.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11 Barwise compactness theorem 1611.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

12 Baumgartner’s axiom 1712.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13 Bernays–Schönfinkel class 1813.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1813.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

14 Beth definability 1914.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

15 Binary decision 2015.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

16 Blake canonical form 2116.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

17 Boolean domain 2217.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2217.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2217.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

18 Boolean function 24

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18.1 Boolean functions in applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

19 Borel equivalence relation 2619.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.2 Kuratowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

20 Cantor–Dedekind axiom 2720.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2720.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

21 Centered set 2821.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

22 Chang’s conjecture 2922.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

23 Class logic 3023.1 Class logic in the strict sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3023.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

24 Classical mathematics 3224.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3224.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

25 Coherent space 3325.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

26 Complete theory 3426.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3426.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

27 Completeness of atomic initial sequents 3627.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

28 Computable isomorphism 3728.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

29 Computable measure theory 3829.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

30 Computable model theory 3930.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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30.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3930.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3930.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

31 Computable real function 4031.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

32 Conservativity theorem 4132.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

33 Constructive non-standard analysis 4233.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4233.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

34 Continuous function (set theory) 4334.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

35 Continuum (set theory) 4435.1 Linear continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4435.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4435.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

36 Countryman line 4536.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

37 Cyclic negation 4637.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

38 Dense order 4738.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

39 Diagonal intersection 4839.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4839.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

40 Double recursion 4940.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4940.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

41 Double turnstile 5041.1 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5041.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5041.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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42 Effective Polish space 5142.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5142.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

43 Elementary definition 5243.1 Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5243.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

44 Elementary diagram 5344.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5344.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

45 Elementary sentence 5445.1 Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5445.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

46 Elementary theory 5546.1 Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5546.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

47 End extension 56

48 Equisatisfiability 5748.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

49 Erasure (logic) 5849.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

50 Extension (predicate logic) 5950.1 Relationship with characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5950.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

51 Extensionality 6051.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6051.2 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6051.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6151.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

52 Finite character 6252.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6252.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

53 Fluent (artificial intelligence) 6353.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

54 Friedberg numbering 64

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54.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6454.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

55 Gabbay’s separation theorem 6555.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

56 Ground axiom 6656.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

57 Herbrand interpretation 6757.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

58 Herbrand structure 6858.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

59 Heyting arithmetic 6959.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6959.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6959.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6959.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6959.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6959.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

60 High (computability) 7160.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7160.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

61 Hilbert–Bernays provability conditions 7261.1 The conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7261.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

62 Honest leftmost branch 7362.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7362.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

63 Indiscernibles 7463.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7463.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7463.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7463.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7463.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

64 Institutional model theory 7564.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7564.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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64.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7564.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

65 Jensen’s covering theorem 7765.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

66 Joint embedding property 7866.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

67 Judgment (mathematical logic) 7967.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7967.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

68 Kanamori–McAloon theorem 8068.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8068.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

69 Kleene–Rosser paradox 8169.1 The paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8169.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8169.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

70 Knaster’s condition 8270.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

71 Least fixed point 8371.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8471.2 Greatest fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8471.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8471.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8471.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

72 LEGO (proof assistant) 8572.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

73 Lightface analytic game 86

74 Limited principle of omniscience 8774.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8774.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8774.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

75 Lindström’s theorem 8875.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8875.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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76 Linked set 8976.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

77 LOGCFL 9077.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9077.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

78 Logic for Computable Functions 9178.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

79 Logical assertion 92

80 Logical graph 9380.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9380.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

81 Logical machine 9481.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9481.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

82 Low (computability) 9582.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9582.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

83 Low basis theorem 9683.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

84 Lusin’s separation theorem 9784.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9784.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

85 Material nonimplication 9885.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

85.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

85.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

85.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9985.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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86 Maximal set 10086.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

87 Michael D. Morley 10187.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10187.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

88 Milner–Rado paradox 10388.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10388.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

89 Minimal logic 10489.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

90 Omega-categorical theory 10590.1 Equivalent conditions for omega-categoricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10590.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10590.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

91 Ordinal logic 10791.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

92 Paraconsistent mathematics 10892.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10892.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

93 Polyadic algebra 10993.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10993.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

94 Predicate logic 11094.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11094.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11094.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

95 Principle of distributivity 112

96 Proof compression 11396.1 Problem Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

96.1.1 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11396.2 Compression algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11496.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

97 Proof mining 11597.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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98 Pseudo-order 11698.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

99 Reduced product 11799.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

100Redundant proof 118100.1Local redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118100.2Global redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

100.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118100.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

101Richardson’s theorem 120101.1Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120101.2Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120101.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121101.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121101.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121101.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

102Robinson’s joint consistency theorem 122102.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

103Scattered order 123103.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

104Semicomputable function 124104.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124104.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

105Separating set 125105.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125105.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

106Set constraint 126106.1Relation to regular tree grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126106.2Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

106.2.1 Literature on negative constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128106.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

107Set function 129107.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129107.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129107.3Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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108Soft set 130108.1Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130108.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

109Strength (mathematical logic) 131109.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131109.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

110Subcountability 132110.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

111Successor function 133111.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133111.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

112Sudan function 134112.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134112.2Value Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134112.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

113Supernatural number 135113.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135113.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136113.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

114Superposition calculus 137114.1Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137114.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

115Switching circuit theory 138115.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138115.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

116Systems of Logic Based on Ordinals 140116.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140116.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

117Takeuti’s conjecture 141117.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141117.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141117.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

118Tarski–Kuratowski algorithm 142118.1Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142118.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

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119The Paradoxes of the Infinite 143119.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

120Theory of pure equality 144120.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

121Trichotomy (mathematics) 145121.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145121.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

122Truth-table reduction 147122.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147122.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

123Zero dagger 148123.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148123.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148123.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

124Łoś–Tarski preservation theorem 149124.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149124.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

125Ω-logic 150125.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150125.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151125.3Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 152

125.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152125.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157125.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Chapter 1

Abstract logic

For other uses of “Abstract logic”, see Abstract logic (disambiguation).

In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relationwith specific properties related to occurrence, expansion, isomorphism, renaming and quantification.[1]

Based on Lindström's characterization, first order logic is, up to equivalence, the only abstract logic which is countablycompact and has Löwenheim number ω.[2]

1.1 See also• Abstract algebraic logic

• Abstract model theory

• Löwenheim number

• Lindström’s theorem

• Universal logic

1.2 References[1] C. C. Chang and Jerome Keisler Model Theory, 1990 ISBN 0-444-88054-2 page 128

[2] C. C. Chang and Jerome Keisler Model Theory, 1990 ISBN 0-444-88054-2 page 132

1

Chapter 2

Abstract model theory

In mathematical logic, abstract model theory is a generalization of model theory which studies the general propertiesof extensions of first-order logic and their models.[1]

Abstract model theory provides an approach that allows us to step back and study a wide range of logics and theirrelationships.[2] The starting point for the study of abstract models, which resulted in good examples was Lindström’stheorem.[3]

In 1974 Jon Barwise provided an axiomatization of abstract model theory.[4]

2.1 See also• Lindström’s theorem

• Institution (computer science)

• Institutional model theory

2.2 Notes[1] Institution-independent model theory by Răzvan Diaconescu 2008 ISBN 3-7643-8707-6 page 3

[2] Handbook of mathematical logic by Jon Barwise 1989 ISBN 0-444-86388-5 page 45

[3] Jean-Yves Béziau Logica universalis: towards a general theory of logic 2005 ISBN 978-3-7643-7259-0 pages 20–25

[4] J. Barwise, 1974 Axioms for abstract model theory, Annals of Math. Logic 7:221–265

2.3 Further reading• Jon Barwise; Solomon Feferman (1985). Model-theoretic logics. Springer-Verlag. ISBN 978-0-387-90936-3.

2

Chapter 3

Abstract structure

An abstract structure in mathematics is a formal object that is defined by a set of laws, properties, and relationshipsin a way that is logically if not always historically independent of the structure of contingent experiences, for example,those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fieldsthat apply them, as computer science, and in the studies that reflect on them, such as philosophy and especially thephilosophy of mathematics. Indeed, modern mathematics has been defined in a very general sense as the study ofabstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure).An abstract structure may be represented (perhaps with some degree of approximation) by one or more physicalobjects — this is called an implementation or instantiation of the abstract structure. But the abstract structure itselfis defined in a way that is not dependent on the properties of any particular implementation.An abstract structure has a richer structure than a concept or an idea. An abstract structure must include preciserules of behaviour which can be used to determine whether a candidate implementation actually matches the abstractstructure in question. Thus we may debate how well a particular government fits the concept of democracy, but thereis no room for debate over whether a given sequence of moves is or is not a valid game of chess.

3.1 Examples

A sorting algorithm is an abstract structure, but a recipe is not, because it depends on the properties and quantities ofits ingredients.A simple melody is an abstract structure, but an orchestration is not, because it depends on the properties of particularinstruments.Euclidean geometry is an abstract structure, but the theory of continental drift is not, because it depends on thegeology of the Earth.A formal language is an abstract structure, but a natural language is not, because its rules of grammar and syntax areopen to debate and interpretation.

3.2 See also• Abstraction in computer science• Abstraction in general• Abstraction in mathematics• Abstract object• Deductive apparatus• Formal sciences• Mathematical structure

3

Chapter 4

Algebraic definition

In mathematical logic, an algebraic definition is one that can be given using only equations between terms with freevariables. Inequalities and quantifiers are specifically disallowed.Saying that a definition is algebraic is a stronger condition than saying it is elementary.

4.1 Related• Algebraic sentence

• Algebraic theory

• Algebraic expression.

• Algebraic equation.

4

Chapter 5

Algebraic semantics (mathematical logic)

In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraiclogic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, booleanalgebras with an interior operator. Other modal logics are characterized by various other algebras with operators. Theclass of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositionalintuitionistic logic.

5.1 Further reading• Josep Maria Font; Ramón Jansana (1996). A general algebraic semantics for sentential logics. Springer-Verlag.ISBN 9783540616993. (2nd published by ASL in 2009) open access at Project Euclid

• W.J. Blok; Don Pigozzi (1989). Algebraizable logics. American Mathematical Society. ISBN 0821824597.

• Janusz Czelakowski (2001). Protoalgebraic logics. Springer. ISBN 9780792369400.

• J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford UniversityPress. ISBN 9780198531920. Good introduction for readers with prior exposure to non-classical logics butwithout much background in order theory and/or universal algebra; the book covers these prerequisites atlength. The book however has been criticized for poor and sometimes incorrect presentation of AAL results.

5

Chapter 6

Algebraic sentence

In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with freevariables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logicinvolving only algebraic sentences.Saying that a sentence is algebraic is a stronger condition than saying it is elementary.

6.1 Related• Algebraic theory

• Algebraic definition

• Algebraic expression

6

Chapter 7

Algorithmic logic

Algorithmic logic is a calculus which allows the expression of semantical properties of programs by appropriatelogical formulas. It provides a framework that permits proving the formulas from the axioms of program constructssuch as assignment, iteration and composition instructions and from the axioms of the data structures in question seeMirkowska & Salwicki (1987), Banachowski et al. (1977).Another logic of programs is dynamic logic, see dynamic logic, Harel, Kozen & Tiuryn (2000).

7.1 Footnotes

7.2 Bibliography1. [Mirkowska & Salwicki] |Mirkowska, Grażyna; Salwicki, Andrzej (1987). Algorithmic Logic. Warszawa &

Boston: PWN & D. Reidel Publ. p. 372. ISBN 8301068590.

2. [Banachowski et al.] |Banachowski, Lech; Kreczmar, Antoni; Mirkowska, Grażyna; Rasiowa, Helena; Sal-wicki, Andrzej (1977). An introduction to Algorithmic Logic - Metamathematical Investigations of Theory ofPrograms. Banach Center Publications 2. Warszawa: PWN. pp. 7–99.

3. Harel, David; Kozen, Dexter; Tiuryn, Jerzy (2000). Dynamic Logic. Cambridge Massachusetts: MIT Press.p. 459.

7

Chapter 8

Automated proof checking

Automated proof checking is the process of using software for checking proofs for correctness. It is one of themost developed fields in automated reasoning.Automated proof checking differs from automated theorem proving in that automated proof checking simply me-chanically checks the formal workings of an existing proof, instead of trying to develop new proofs or theorems itself.Because of this, the task of automated proof verification is much simpler than that of automated theorem proving,allowing automated proof checking software to be much simpler than automated theorem proving software.Because of this small size, some automated proof checking systems can have less than a thousand lines of core code,and are thus themselves amenable to both hand-checking and automated software verification.The Mizar system, HOL Light, and Metamath are examples of automated proof checking systems.Automated proof checking can be done either as a batch operation, or interactively, as part of an interactive theoremproving system.

8.1 Field journals and conferences

• Intelligent Computer Mathematics

• Journal of Formalized Reasoning

• Interactive Theorem Proving

• Formalized Mathematics

• Studies in Logic, Grammar and Rhetoric

8.2 See also

• Computer-aided proof

• Formal verification

• Proof assistant

• QED manifesto

8.3 External links

• Julie Rehmeyer (November 14, 2008). “How to (really) trust a mathematical proof”. ScienceNews. Retrieved2008-11-14.

8

8.3. EXTERNAL LINKS 9

• Metamath: a proof checking system with an extensive collection of machine-readable proofs covering a con-siderable range of mathematical fields

• Digimath: Freek Wiedijk’s alphabetic list of systems

• MathSystem: Mathematical Software systems

Chapter 9

Axiom of adjunction

In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} givenby “adjoining” the set y to the set x.

∀x ∀y ∃w ∀z [z ∈ w ↔ (z ∈ x ∨ z = y)].

Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of settheory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as generalset theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursiveset functions.Tarski and Smielew showed that Robinson arithmetic can be interpreted in a weak set theory whose axioms areextensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).

9.1 References• Bernays, Paul (1937), “A System ofAxiomatic Set Theory--Part I”, The Journal of Symbolic Logic (Associationfor Symbolic Logic) 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862

• Kirby, Laurence (2009), “Finitary Set Theory”, Notre Dame J. Formal Logic 50 (3): 227–244, MR 2572972

• Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amster-dam: North-Holland Publishing Company, MR 0058532

• Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMSColloquium Publications, v. 41.

10

Chapter 10

Axiom of regularity

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkelset theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic theaxiom reads:

∀x (x = ∅ → ∃y ∈ x (y ∩ x = ∅))

The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is anelement of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), thisresult can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, the axiomof regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downwardinfinite membership chains.The axiom of regularity was introduced by von Neumann (1925); it was adopted in a formulation closer to the onefound in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics basedon set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makessome properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but alsoon proper classes that are well-founded relational structures such as the lexicographical ordering on {(n, α)|n ∈ω ∧ α ordinal an is } .Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induc-tion. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones thatdo not accept the law of the excluded middle), where the two axioms are not equivalent.In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of setsthat are elements of themselves.

10.1 Elementary implications of regularity

10.1.1 No set is an element of itself

Let A be a set, and apply the axiom of regularity to {A}, which is a set by the axiom of pairing. We see that theremust be an element of {A} which is disjoint from {A}. Since the only element of {A} is A, it must be that A is disjointfrom {A}. So, since A ∈ {A}, we cannot have A ∈ A (by the definition of disjoint).

10.1.2 No infinite descending sequence of sets exists

Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for eachn. Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema ofreplacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the definitionof S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an

11

12 CHAPTER 10. AXIOM OF REGULARITY

element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets. Sinceour supposition led to a contradiction, there must not be any such function, f.The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.Notice that this argument only applies to functions f that can be represented as sets as opposed to undefinable classes.The hereditarily finite sets, Vω, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom ofinfinity). So if one forms a non-trivial ultrapower of Vω, then it will also satisfy the axiom of regularity. The resultingmodel will contain elements, called non-standard natural numbers, that satisfy the definition of natural numbers inthat model but are not really natural numbers. They are fake natural numbers which are “larger” than any actualnatural number. This model will contain infinite descending sequences of elements. For example, suppose n is anon-standard natural number, then (n − 1) ∈ n and (n − 2) ∈ (n − 1) , and so on. For any actual natural numberk, (n− k− 1) ∈ (n− k) . This is an unending descending sequence of elements. But this sequence is not definablein the model and thus not a set. So no contradiction to regularity can be proved.

10.1.3 Simpler set-theoretic definition of the ordered pair

The axiom of regularity enables defining the ordered pair (a,b) as {a,{a,b}}. See ordered pair for specifics. Thisdefinition eliminates one pair of braces from the canonical Kuratowski definition (a,b) = {{a},{a,b}}.

10.1.4 Every set has an ordinal rank

This was actually the original form of von Neumann’s axiomatization.

10.1.5 For every two sets, only one can be an element of the other

Let X and Y be sets. Then apply the axiom of regularity to the set {X,Y}. We see there must be an element of {X,Y}which is also disjoint from it. It must be either X or Y. By the definition of disjoint then, we must have either Y is notan element of X or vice versa.

10.2 The axiom of dependent choice and no infinite descending sequenceof sets implies regularity

Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-emptyintersection with S. We define a binary relation R on S by aRb :⇔ b ∈ S ∩ a , which is entire by assumption. Thus,by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is aninfinite descending chain, we arrive at a contradiction and so, no such S exists.

10.3 Regularity and the rest of ZF(C) axioms

Regularity was shown to be relatively consistent with the rest of ZF by von Neumann (1929), meaning that if ZFwithout regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation seeVaught (2001, §10.1) for instance.The axiom of regularity was also shown to be independent from the other axioms of ZF(C), assuming they areconsistent. The result was announced by Paul Bernays in 1941, although he did not publish a proof until 1954. Theproof involves (and led to the study of) Rieger-Bernays permutation models (or method), which were used for otherproofs of independence for non-well-founded systems (Rathjen 2004, p. 193 and Forster 2003, pp. 210–212).

10.4 Regularity and Russell’s paradox

Naive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent dueto Russell’s paradox. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension

10.5. REGULARITY, THE CUMULATIVE HIERARCHY, AND TYPES 13

with the much weaker axiom schema of separation. However, this makes set theory too weak. So some of thepower of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset,replacement, and infinity) which may be regarded as special cases of comprehension. So far, these axioms do notseem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added toexclude models with some undesirable properties. These two axioms are known to be relatively consistent.In the presence of the axiom schema of separation, Russell’s paradox becomes a proof that there is no set of all sets.The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition isredundant when added to the rest of ZF. If the ZF axioms without regularity were already inconsistent, then addingregularity would not make them consistent.The existence of Quine atoms (sets that satisfy the formula equation x = {x}, i.e. have themselves as their only ele-ments) is consistent with the theory obtained by removing the axiom of regularity fromZFC. Various non-wellfoundedset theories allow “safe” circular sets, such as Quine atoms, without becoming inconsistent by means of Russell’sparadox.(Rieger 2011, pp. 175,178)

10.5 Regularity, the cumulative hierarchy, and types

In ZF it can be proven that the class∪

α Vα (see cumulative hierarchy) is equal to the class of all sets. This statementis even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which doesnot satisfy axiom of regularity, a model which satisfies it can be constructed by taking only sets in

∪α Vα .

Herbert Enderton (1977, p. 206) wrote that “The idea of rank is a descendant of Russell’s concept of type". Com-paring ZF with type theory, Alasdair Urquhart wrote that “Zermelo’s system has the notational advantage of notcontaining any explicitly typed variables, although in fact it can be seen as having an implicit type structure built intoit, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [Zermelo 1930],and again in a well-known article of George Boolos [Boolos 1971].” Urquhart (2003, p. 305)Dana Scott (1974) went further and claimed that:

The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of someform of the theory of types. That was at the basis of both Russell’s and Zermelo’s intuitions. Indeed thebest way to regard Zermelo’s theory is as a simplification and extension of Russell’s. (We mean Russell’ssimple theory of types, of course.) The simplification was to make the types cumulative. Thus mixing oftypes is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate theearlier ones, we can then easily imagine extending the types into the transfinite—just how far we want togo must necessarily be left open. Now Russell made his types explicit in his notation and Zermelo leftthem implicit. [emphasis in original]

In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchyturns out to be equivalent to ZF, including regularity. (Lévy 2002, p. 73)

10.6 History

The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. Lévy(2002, p. 68) and Hallett (1986, §4.4, esp. p. 186, 188). Mirimanoff called a set x “regular” (French: “ordinaire”) ifevery descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (andwell-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 62–63); in later papers Mirimanoffalso explored what are now called non-well-founded sets (“extraordinaire” in Mirimanoff’s terminology) (Sangiorgi2011, pp. 17–19, 26).According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as “superfluous” (on p. 404 invan Heijenoort 's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) whichexcludes some, but not all, non-well-founded sets (Rieger 2011, p. 179). In a subsequent publication, von Neumann(1928) gave the following axiom (rendered in modern notation by A. Rieger):

∀x (x = ∅ → ∃y ∈ x (y ∩ x = ∅))

14 CHAPTER 10. AXIOM OF REGULARITY

10.7 See also

• Non-well-founded set theory

10.8 References

• Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, ISBN 3-540-44085-2

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

• Boolos, George (1971), “The iterative conception of set”, Journal of Philosophy 68: 215–231, doi:10.2307/2025204reprinted in Boolos, George (1998), Logic, Logic and Logic, Harvard University Press, pp. 13–29

• Enderton, Herbert B. (1977), Elements of Set Theory, Academic Press

• Urquhart, Alasdair (2003), “The Theory of Types”, in Griffin, Nicholas, The Cambridge Companion to BertrandRussell, Cambridge University Press

• Halbeisen, Lorenz J. (2012), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer

• Sangiorgi, Davide (2011), “Origins of bisimulation and coinduction”, in Sangiorgi, Davide; Rutten, Jan, Ad-vanced Topics in Bisimulation and Coinduction, Cambridge University Press

• Lévy, Azriel (2002) [first published in 1979], Basic set theory, Dover Publications, ISBN 0-486-42079-5

• Hallett, Michael (1996) [first published 1984], Cantorian set theory and limitation of size, Oxford UniversityPress, ISBN 0-19-853283-0

• Rathjen, M. (2004), “Predicativity, Circularity, and Anti-Foundation”, in Link, Godehard, One Hundred Yearsof Russell s Paradox: Mathematics, Logic, Philosophy (PDF), Walter de Gruyter, ISBN 978-3-11-019968-0

• Forster, T. (2003), Logic, induction and sets, Cambridge University Press

• Rieger, Adam (2011), “Paradox, ZF, and the Axiom of Foundation”, in David DeVidi, Michael Hallett, PeterClark, Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell., pp. 171–187,doi:10.1007/978-94-007-0214-1_9, ISBN 978-94-007-0213-4

• Vaught, Robert L. (2001), Set Theory: An Introduction (2nd ed.), Springer, ISBN 978-0-8176-4256-3

10.8.1 Primary sources

• Mirimanoff, D. (1917), “Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theoriedes ensembles”, L'Enseignement Mathématique 19: 37–52

• von Neumann, J. (1925), “Eine axiomatiserung der Mengenlehre”, Journal für die reine und angewandte Math-ematik 154: 219–240; translation in van Heijenoort, Jean (1967), From Frege to Gödel: A Source Book inMathematical Logic, 1879–1931, pp. 393–413

• von Neumann, J. (1928), "Über die Definition durch transfinite Induktion und verwandte Fragen der allge-meinen Mengenlehre”, Mathematische Annalen 99: 373–391, doi:10.1007/BF01459102

• von Neumann, J. (1929), “Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre”, Journalfur die reine und angewandte Mathematik 160: 227–241, doi:10.1515/crll.1929.160.227

• Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagender Mengenlehre.” (PDF), Fundamenta Mathematicae 16: 29–47; translation in Ewald, W.B., ed. (1996),From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2, Clarendon Press, pp. 1219–33

• Bernays, P. (1941), “A system of axiomatic set theory. Part II”, The Journal of Symbolic Logic 6: 1–17,doi:10.2307/2267281

10.9. EXTERNAL LINKS 15

• Bernays, P. (1954), “A system of axiomatic set theory. Part VII”, The Journal of Symbolic Logic 19: 81–96,doi:10.2307/2268864

• Riegger, L. (1957), “A contribution to Gödel’s axiomatic set theory” (PDF), Czechoslovak Mathematical Jour-nal 7: 323–357

• Scott, D. (1974), “Axiomatizing set theory”,Axiomatic set theory. Proceedings of Symposia in PureMathematicsVolume 13, Part II, pp. 207–214

10.9 External links• http://www.trinity.edu/cbrown/topics_in_logic/sets/sets.html contains an informative description of the axiomof regularity under the section on Zermelo-Fraenkel set theory.

• Axiom of Foundation at PlanetMath.org.

Chapter 11

Barwise compactness theorem

In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usualcompactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwisein 1967.

11.1 Statement of the theorem

Let A be a countable admissible set. Let L be an A -finite relational language. Suppose Γ is a set of LA -sentences,where Γ is a Σ1 set with parameters from A , and every A -finite subset of Γ is satisfiable. Then Γ is satisfiable.

11.2 References• Barwise, J. (1967). Infinitary Logic and Admissible Sets (Ph. D. Thesis). Stanford University.

• C. J. Ash; Knight, J. (2000). Computable Structures and the Hyperarithmetic Hierarchy. Elsevier. p. 366.ISBN 0-444-50072-3.

• Jon Barwise; Solomon Feferman; John T. Baldwin (1985). Model-theoretic logics. Springer-Verlag. p. 295.ISBN 3-540-90936-2.

11.3 External links• Stanford Encyclopedia of Philosophy, “Infinitary Logic”, Section 5, “Sublanguages of L(ω1,ω) and the BarwiseCompactness Theorem”

16

Chapter 12

Baumgartner’s axiom

In mathematical set theory, Baumgartner’s axiom (BA) can be one of three different axioms introduced by JamesEarl Baumgartner.An axiom introduced by Baumgartner (1973) states that any two ℵ1-dense subsets of the real line are isomorphic.Another axiom introduced by Baumgartner (1975) states that Martin’s axiom for partially ordered sets MAP(κ) istrue for all posets P that are countable closed, well met and ℵ1-linked and all cardinals κ less than 2ℵ1 .Baumgartner’s axiom A is an axiom for posets introduced in (Baumgartner 1983, section 7). A partial order (P, ≤)is said to satisfy axiom A if there is a family ≤n of partial orderings on P for n = 0, 1, 2, ... such that

1. ≤0 is the same as ≤

2. If p ≤n₊₁q then p ≤nq

3. If there is a sequence pn with pn₊₁ ≤n pn then there is a q with q ≤n pn for all n.

4. If I is a pairwise incompatible subset of P then for all p and for all natural numbers n there is a q such that q≤n p and the number of elements of I compatible with q is countable.

12.1 References• Baumgartner, James E. (1973), “All ℵ1-dense sets of reals can be isomorphic”, Fund. Math. 79 (2): 101–106,MR 0317934

• Baumgartner, James E. (1975), Generalizing Martin’s axiom, unpublished manuscript

• Baumgartner, James E. (1983), “Iterated forcing”, in Mathias, A. R. D., Surveys in set theory, London Math.Soc. Lecture Note Ser. 87, Cambridge: Cambridge Univ. Press, pp. 1–59, ISBN 0-521-27733-7, MR0823775

• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001

17

Chapter 13

Bernays–Schönfinkel class

The Bernays–Schönfinkel class (also known as Bernays–Schönfinkel-Ramsey class) of formulas, named afterPaul Bernays and Moses Schönfinkel (and Frank P. Ramsey), is a decidable fragment of first-order logic formulas.It is the set of satisfiable formulas which, when written in prenex normal form, have an ∃∗∀∗ quantifier prefix and donot contain any function symbols.This class of logic formulas is also sometimes referred as effectively propositional (EPR) since it can be effectivelytranslated into propositional logic formulas by a process of grounding or instantiation.The decision problem for this class is NEXPTIME-complete.[1]

13.1 See also• Prenex normal form

13.2 References[1] Harry R. Lewis, Complexity Results for Classes of Quantificational Formulas, J. Computer and System Sciences, 21, 317-

353 (1980) doi:10.1016/0022-0000(80)90027-6

• Ramsey, F. (1930), “On a problem in formal logic”, Proc. LondonMath. Soc. 30: 264–286, doi:10.1112/plms/s2-30.1.264

• Piskac, R.; deMoura, L.; Bjorner, N. (December 2008), “Deciding Effectively Propositional Logic with Equal-ity”, Microsoft Research Technical Report (2008-181)

18

Chapter 14

Beth definability

In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicitdefinability, specifically the theorem states that the two senses of definability are equivalent.

14.1 Statement

The theorem states that, given any two models A and B of a first-order theory T in the language L' ⊇ L such that A|L= B|L (where A|L is the reduct of A to L), it is the case that A ⊨ φ[a] if and only if B ⊨ φ[a] (for φ a formula in L'and for all tuples a of A) only if it is also the case that φ is equivalent modulo T to a formula ψ in L. Less formally: aproperty is implicitly definable in a theory in language L (via introduction of a new symbol φ of an extended languageL') only if that property is explicitly definable in that theory (by formula ψ in the original language L).Clearly the converse holds as well, so that we have an equivalence between implicit and explicit definability. That is,a “property” is implicitly definable with respect to a theory if and only if it is explicitly definable.The theorem does not hold if the condition is restricted to finite models. We may have A ⊨ φ[a] if and only if B ⊨φ[a] for all pairs A,B of finite models without there being any L-formula ψ equivalent to φ modulo T.The result was first proven by Evert Willem Beth.

14.2 Sources• Hodges W. A Shorter Model Theory. Cambridge University Press, 1997.

19

Chapter 15

Binary decision

A binary decision is a choice between two alternatives, for instance between taking some specific action or not takingit.[1]

Binary decisions are basic to many fields. Examples include:

• Truth values in mathematical logic, and the corresponding Boolean data type in computer science, representinga value which may be chosen to be either true or false.[2]

• Conditional statements (if-then or if-then-else) in computer science, binary decisions about which piece ofcode to execute next.[3]

• Decision trees and binary decision diagrams, representations for sequences of binary decisions.[4]

• Binary choice, a statistical model for the outcome of a binary decision.[5]

15.1 References[1] Snow, Roberta M.; Phillips, Paul H. (2007),Making Critical Decisions: A Practical Guide for Nonprofit Organizations, John

Wiley & Sons, p. 44, ISBN 9780470185032.

[2] Dixit, J. B. (2009), Computer Fundamentals and Programming in C, Firewall Media, p. 61, ISBN 9788170088820.

[3] Yourdon, Edward (March 19, 1975), “Clear thinking vital: Nested IFs not evil plot leading to program bugs”,Computerworld:15.

[4] Clarke, E. M.; Grumberg, Orna; Peled, Doron (1999), Model Checking, MIT Press, p. 51, ISBN 9780262032704.

[5] Ben-Akiva, Moshe E.; Lerman, Steven R. (1985), Discrete Choice Analysis: Theory and Application to Travel Demand,Transportation Studies 9, MIT Press, p. 59, ISBN 9780262022170.

20

Chapter 16

Blake canonical form

In Boolean logic, a formula for a Boolean function f is in Blake canonical form, also called the complete sum ofprime implicants,[1] the complete sum,[2] or the disjunctive prime form,[3] when it is a disjunction of all the primeimplicants of f.[4] Blake canonical form is a disjunctive normal form.The Blake canonical form is not necessarily minimal, however all the terms of a minimal sum are contained in theBlake canonical form.[2]

It was introduced in 1937 by Archie Blake, who called it the “simplified canonical form";[5] it was named in honorof Blake by Frank Markham Brown in 1990.[4]

Blake discussed three methods for calculating the canonical form: exhaustion of implicants, iterated consensus, andmultiplication. The iterated consensus method was rediscovered by Samson and Mills, Quine, and Bing.[4]

16.1 See also• Horn clause

16.2 Notes[1] Tsutomu Sasao, “Ternary Decision Diagrams and their Applications”, in Tsutomu Sasao, Masahira Fujita, eds., Represen-

tations of Discrete Functions ISBN 0792397207, 1996, p. 278

[2] Abraham Kandel, Foundations of Digital Logic Design, p. 177

[3] Donald E. Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, Part 1, 2011, p. 54

[4] Frank Markham Brown, “The Blake Canonical Form”, chapter 4 of Boolean Reasoning: The Logic of Boolean Equations,ISBN 0486427854, 2nd edition, 2012, p. 77ff (first edition, 1990)

[5] “Canonical expressions in Boolean algebra”, Dissertation, Dept. of Mathematics, U. of Chicago, 1937, reviewed in J. C.C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634

21

Chapter 17

Boolean domain

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpre-tations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usuallywritten as {0, 1},[1][2][3] {false, true}, {F, T},[4] {⊥,⊤} [5] or B. [6][7]

The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. Theinitial object in the category of bounded lattices is a Boolean domain.In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programminglanguages feature reserved words or symbols for the elements of the Boolean domain, for example false and true.However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, forexample, falsity is represented by the number 0 and truth is represented by the number 1 or −1 respectively, and allvariables that can take these values can also take any other numerical values.

17.1 Generalizations

The Boolean domain {0, 1} can be replaced by the unit interval [0,1], in which case rather than only taking values 0or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1−x,conjunction (AND) is replaced with multiplication ( xy ), and disjunction (OR) is defined via De Morgan’s law to be1− (1− x)(1− y) .Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logicand probabilistic logic. In these interpretations, a value is interpreted as the “degree” of truth – to what extent aproposition is true, or the probability that the proposition is true.

17.2 See also

• Boolean-valued function

17.3 Notes[1] Dirk van Dalen, Logic and Structure. Springer (2004), page 15.

[2] David Makinson, Sets, Logic and Maths for Computing. Springer (2008), page 13.

[3] George S. Boolos and Richard C. Jeffrey, Computability and Logic. Cambridge University Press (1980), page 99.

[4] Elliott Mendelson, Introduction to Mathematical Logic (4th. ed.). Chapman & Hall/CRC (1997), page 11.

[5] Eric C. R. Hehner, A Practical Theory of Programming. Springer (1993, 2010), page 3.

[6] Ian Parberry (1994). Circuit Complexity and Neural Networks. MIT Press. p. 65. ISBN 978-0-262-16148-0.

22

17.3. NOTES 23

[7] Jordi Cortadella et al. (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Science & BusinessMedia. p. 73. ISBN 978-3-540-43152-7.

Chapter 18

Boolean function

Not to be confused with Binary function.

In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ : Bk →B, where B = {0, 1} is a Boolean domain and k is a non-negative integer called the arity of the function. In the casewhere k = 0, the “function” is essentially a constant element of B.Every k-ary Boolean function can be expressed as a propositional formula in k variables x1, …, xk, and two propo-sitional formulas are logically equivalent if and only if they express the same Boolean function. There are 22k k-aryfunctions for every k.

18.1 Boolean functions in applications

A Boolean function describes how to determine a Boolean value output based on some logical calculation fromBoolean inputs. Such functions play a basic role in questions of complexity theory as well as the design of circuitsand chips for digital computers. The properties of Boolean functions play a critical role in cryptography, particularlyin the design of symmetric key algorithms (see substitution box).Boolean functions are often represented by sentences in propositional logic, and sometimes asmultivariate polynomialsover GF(2), but more efficient representations are binary decision diagrams (BDD), negation normal forms, andpropositional directed acyclic graphs (PDAG).In cooperative game theory, monotone Boolean functions are called simple games (voting games); this notion isapplied to solve problems in social choice theory.

18.2 See also

• Algebra of sets

• Boolean algebra

• Boolean algebra topics

• Boolean domain

• Boolean-valued function

• Logical connective

• Truth function

• Truth table

• Symmetric Boolean function

24

18.3. REFERENCES 25

• Decision tree model

• Evasive Boolean function

• Indicator function

• Balanced boolean function

• 3-ary Boolean functions

18.3 References• Crama, Y; Hammer, P. L. (2011), Boolean Functions, Cambridge University Press.

• Hazewinkel, Michiel, ed. (2001), “Boolean function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Janković, Dragan; Stanković, Radomir S.; Moraga, Claudio (November 2003). “Arithmetic expressions opti-misation using dual polarity property” (PDF). Serbian Journal of Electrical Engineering 1 (71 - 80, number 1).Retrieved 2015-06-07.

• Mano, M. M.; Ciletti, M. D. (2013), Digital Design, Pearson.

Chapter 19

Borel equivalence relation

In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borelsubset of X × X (in the product topology).

19.1 Formal definition

Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducibleto F, in symbols E ≤B F, if and only if there is a Borel function

Θ : X → Y

such that for all x,x' ∈ X, one has

xEx' ⇔ Θ(x)FΘ(x' ).

Conceptually, if E is Borel reducible to F, then E is “not more complicated” than F, and the quotient space X/E hasa lesser or equal “Borel cardinality” than Y/F, where “Borel cardinality” is like cardinality except for a definabilityrestriction on the witnessing mapping.

19.2 Kuratowski’s theorem

A measure space X is called a standard Borel space if it is Borel-isomorphic to a Borel subset of a Polish space.Kuratowski’s theorem then states that two standard Borel spaces X and Y are Borel-isomorphic iff |X| = |Y |.

19.3 References• Harrington, L. A., A. S. Kechris, A. Louveau (Oct 1990). “A Glimm-Effros Dichotomy for Borel equivalencerelations”. Journal of the American Mathematical Society (Journal of the American Mathematical Society, Vol.3, No. 4) 3 (2): 903–928. doi:10.2307/1990906. JSTOR 1990906.

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

• Silver, Jack H. (1980). “Counting the number of equivalence classes of Borel and coanalytic equivalencerelations”. Annals of Mathematical Logic 18 (1): 1–28. doi:10.1016/0003-4843(80)90002-9.

• Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44.American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3

26

Chapter 20

Cantor–Dedekind axiom

In mathematical logic, the phraseCantor–Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to onecorrespondence between real numbers and points on a line.This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartesexplicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or planeinto a conceptual metaphor. This is sometimes referred to as the real number line blend:[1]

A consequence of this axiom is that Alfred Tarski’s proof of the decidability of the ordered real field could be seenas an algorithm to solve any problem in Euclidean geometry.

20.1 Notes[1] George Lakoff and Rafael E. Núñez (2000). Where Mathematics Comes From: How the embodied mind brings mathematics

into being. Basic Books. ISBN 0-465-03770-4.

20.2 References• Ehrlich, P. (1994). “General introduction”. Real Numbers, Generalizations of the Reals, and Theories of

Continua, vi–xxxii. Edited by P. Ehrlich, Kluwer Academic Publishers, Dordrecht

• Bruce E. Meserve (1953) Fundamental Concepts of Algebra, p. 32, at Google Books

• B.E. Meserve (1955) Fundamental Concepts of Geometry, p. 86, at Google Books

27

Chapter 21

Centered set

In mathematics, in the area of order theory, an upwards centered set S is a subset of a partially ordered set, P, suchthat any finite subset of S has an upper bound in P. Similarly, any finite subset of a downwards centered set has alower bound. An upwards centered set can also be called a consistent set. Note that any directed set is necessarilycentered, and any centered set is linked.A subset B of a partial order is said to be σ-centered if it is a countable union of centered sets.

21.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

• Davey, B. A.; Priestley, Hilary A. (2002), “9.1”, Introduction to Lattices and Order (2nd ed.), CambridgeUniversity Press, p. 201, ISBN 978-0-521-78451-1, Zbl 1002.06001.

28

Chapter 22

Chang’s conjecture

In model theory, a branch of mathematical logic, Chang’s conjecture, attributed to Chen Chung Chang by Vaught(1963, p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type(ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinalityβ. The usual notation is (ω2, ω1) ↠ (ω1, ω) .The axiom of constructibility implies that Chang’s conjecture fails. Silver proved the consistency of Chang’s con-jecture from the consistency of an ω1-Erdős cardinal. Hans-Dieter Donder showed the reverse implication: if CCholds, then ω2 is ω1-Erdős in K.More generally, Chang’s conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β)for a countable language has an elementary submodel of type (γ,δ). The consistency of (ω3, ω2) ↠ (ω2, ω1) wasshown by Laver from the consistency of a huge cardinal.

22.1 References• Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations ofMathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3

• Vaught, R. L. (1963), “Models of complete theories”, Bulletin of the American Mathematical Society 69: 299–313, doi:10.1090/S0002-9904-1963-10903-9, ISSN 0002-9904, MR 0147396

29

Chapter 23

Class logic

Class logic is a logic in its broad sense, whose objects are called classes. In a narrower sense, one speaks of a classof logic only if classes are described by a property of their elements. This class logic is thus a generalization of settheory, which allows only a limited consideration of classes.

23.1 Class logic in the strict sense

The first class logic in the strict sense was created by Giuseppe Peano in 1889 as the basis for his arithmetic (PeanoAxioms). He introduced the class term, which formally correctly describes classes through a property of their ele-ments. Today the class term is denoted in the form {x|A(x)}, where A(x) is an arbitrary statement, which all classmembers x meet. Peano axiomatized the class term for the first time and used it fully. Gottlob Frege also triedestablishing the arithmetic logic with class terms in 1893; Bertrand Russell discovered a conflict in it in 1902 whichbecame known as Russell’s paradox. As a result, it became generally known that you can not safely use class terms.To solve the problem, Russell developed his type theory from 1903 to 1908, which allowed only a verymuch restricteduse of class terms. In the long term she not prevailed but, but more comfortable and more powerful, 1907 initiated byErnst Zermelo set theory. Not a class logic in the narrower sense, but in its present form (ZF or NBG) because it doesnot axiomatize the class term, but used only in practice as a useful notation. Willard Van Orman Quine described aset theory New Foundations (NF) in 1937, oriented not at Cantor, or Zermelo-Fraenkel, but on the theory of types.In 1940 Quine advanced NF to Mathematical Logic (ML). Since the antinomy of Burali-Forti was derived in the firstversion of ML,[1] Quine clarified ML, retaining the widespread use of classes, and took up a proposal by HaoWang[2]introducing in 1963 in his theory of {x|A(x)} as a virtual class, so that classes are although not yet full-fledged terms,but sub-terms in defined contexts.[3]

After Quine, Arnold Oberschelp developed the first fully functional modern axiomatic class logic starting in 1974. Itis a consistent extension of predicate logic and allows the unrestricted use of class terms (such as Peano).[4] It uses allclasses that produce antinomies of naive set theory as a term. This is possible because the theory assumes no existenceaxioms for classes. It presupposes in particular any number of axioms, but can also take those and syntactically correctto formulate in the traditionally simple design with class terms. For example, the Oberschelp set theory developedthe Zermelo–Fraenkel set theory within the framework of class logic.[5] Three principles guarantee that cumbersomeZF formulas are translatable into convenient classes formulas; guarantee a class logical increase in the ZF languagethey form without quantities axioms together with the axioms of predicate logic an axiom system for a simple logicof general class.[6]

The principle of abstraction (Abstraktionsprinzip) states that classes describe their elements via a logical property:

∀y : (y ∈ {x | A(x)} ⇐⇒ A(y))

The principle of extensionality (Extensionalitätsprinzip ) describes the equality of classes by matching their elementsand eliminates the axiom of extensionality in ZF:

30

23.2. BIBLIOGRAPHY 31

A = B ⇐⇒ ∀x : (x ∈ A ⇐⇒ x ∈ B)

The principle of comprehension (Komprehensionsprinzip) determines the existence of a class as an element:

{x | A(x)} ∈ B ⇐⇒ ∃y : (y = {x | A(x)} ∧ y ∈ B)

23.2 Bibliography• Giuseppe Peano: Arithmetices principia. Nova methodo exposita. Corso, Torino u. a. 1889 (Auch in: GiuseppePeano: Opere scelte. Band 2. Cremonese, Rom 1958, S. 20–55).

• G. Frege: Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet. Band 1. Pohle, Jena 1893.

• Willard Van Orman Quine: New Foundations for Mathematical Logic, in: American Mathematical Monthly44 (1937), S. 70-80.

• WillardVanOrmanQuine: Set Theory and its Logic. HarvardUniversity Press, CambridgeMA1963 (DeutscheÜbersetzung: Mengenlehre und ihre Logik (= Logik und Grundlagen der Mathematik. Bd. 10). Vieweg, Braun-schweig 1973, ISBN 3-548-03532-9).

• Arnold Oberschelp: Elementare Logik und Mengenlehre (= BI-Hochschultaschenbücher 407–408). 2 Bände.Bibliographisches Institut, Mannheim u. a. 1974–1978, ISBN 3-411-00407-X (Bd. 1), ISBN 3-411-00408-8(Bd. 2).

• Albert Menne Grundriß der formalen Logik (= Uni-Taschenbücher 59 UTB für Wissenschaft). Schöningh,Paderborn 1983, ISBN 3-506-99153-1 (Renamed Grundriß der Logistik starting with 5th Edition – The bookshows, among other calcului, a possible application of calculus to class logic, based on the propositional andpredicate calculus and carried the basic terms of formal systems to class logic. It also discusses briefly theparadoxes and type theory).

• Jürgen-Michael Glubrecht, ArnoldOberschelp, Günter Todt: Klassenlogik. Bibliographisches Institut, Mannheimu. a. 1983, ISBN 3-411-01634-5.

• Arnold Oberschelp: Allgemeine Mengenlehre. BI-Wissenschafts-Verlag, Mannheim u. a. 1994, ISBN 3-411-17271-1.

23.3 References[1] John Barkley Rosser: Burali-Forti paradox. In: Journal of Symbolic Logic, Band 7, 1942, p. 1-17

[2] Hao Wang: A formal system for logic. In: Journal of Symbolic Logic, Band 15, 1950, p. 25-32

[3] Willard Van Orman Quine: Mengenlehre und ihre Logik. 1973, S. 12.

[4] Arnold Oberschelp: Allgemeine Mengenlehre. 1994, p. 75 f.

[5] The advantages of the class logic are shown in a comparison of ZFC in class logic and predicate logic form in: ArnoldOberschelp: Allgemeine Mengenlehre. 1994, p. 261.

[6] Arnold Oberschelp, p. 262, 41.7. The axiomatization is much more complicated, but here is reduced to a book-end to theessentials.

Chapter 24

Classical mathematics

In the foundations ofmathematics, classicalmathematics refers generally to themainstream approach tomathematics,which is based on classical logic and ZFC set theory.[1] It stands in contrast to other types of mathematics such asconstructive mathematics or predicative mathematics. In practice, the most common non-classical systems are usedin constructive mathematics.[2]

Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist and other objections tothe logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost allmathematics, however, is done in the classical tradition, or in ways compatible with it.Defenders of classical mathematics, such as David Hilbert, have argued that it is easier to work in, and is most fruitful;although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematicscould not (or could not so easily) attain, they argue that on the whole, it is the other way round.In terms of the philosophy and history of mathematics, the very existence of non-classical mathematics raises thequestion of the extent to which the foundational mathematical choices humanity has made arise from their “superi-ority” rather than from, say, expedience-driven concentrations of effort on particular aspects.

24.1 See also• Constructivism (mathematics)

• Finitism

• Intuitionism

• Non-classical analysis

• Traditional mathematics

• Ultrafinitism

• Philosophy of Mathematics

24.2 References[1] Stewart Shapiro, ed. (2005). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press,

USA. ISBN 978-0-19-514877-0.

[2] Torkel Franzén (1987). Provability and Truth. Almqvist & Wiksell International. ISBN 91-22-01158-7.

32

Chapter 25

Coherent space

In proof theory, a coherent space is a concept introduced in the semantic study of linear logic.Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. Fora family of C-sets (i.e., F ⊆ ℘(C)), the dual of F, written F ⊥, is defined as the set of all C-sets S such that for everyT ∈ F, S ⊥ T. A coherent space F over C is a family C-sets for which F = (F ⊥) ⊥.In topology, a coherent space is another name for spectral space. A continuous map between coherent spaces iscalled coherent if it is spectral.In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the Frenchoriginal they were espaces cohérents, the coherence space translation was used because spectral spaces are sometimescalled coherent spaces.

25.1 References• Girard, J.-Y.; Lafont, Y.; Taylor, P. (1989), Proofs and types, Cambridge University Press.

• Girard, J.-Y. (2004), “Between logic and quantic: a tract”, in Ehrhard; Girard; Ruet et al., Linear logic incomputer science (PDF), Cambridge University Press .

• Johnstone, Peter (1982), “II.3 Coherent locales”, Stone Spaces, Cambridge University Press, pp. 62–69, ISBN978-0-521-33779-3.

33

Chapter 26

Complete theory

In mathematical logic, a theory is complete if it is amaximal consistent set of sentences, i.e., if it is consistent, andnone of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this isequivalent to asking that for every sentence φ in the language of the theory it contains either φ itself or its negation¬φ.Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to beformulated cannot be complete, as demonstrated by Gödel’s incompleteness theorem.This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can beformulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of “seman-tically valid”). Gödel’s completeness theorem is about this latter kind of completeness.Complete theories are closed under a number of conditions internally modelling the T-schema:

• For a set S : A ∧B ∈ S if and only if A ∈ S and B ∈ S ,

• For a set S : A ∨B ∈ S if and only if A ∈ S or B ∈ S .

Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existencein a given case is usually a straightforward consequence of Zorn’s lemma, based on the idea that a contradictioninvolves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent setsextending a theory T (closed under the necessitation rule) can be given the structure of a model of T, called thecanonical model.

26.1 Examples

Some examples of complete theories are:

• Presburger arithmetic

• Tarski’s axioms for Euclidean geometry

• The theory of dense linear orders

• The theory of algebraically closed fields of a given characteristic

• The theory of real closed fields

• Every uncountably categorical countable theory

• Every countably categorical countable theory

34

26.2. REFERENCES 35

26.2 References• Mendelson, Elliott (1997). Introduction to Mathematical Logic (Fourth ed.). Chapman & Hall. p. 86. ISBN978-0-412-80830-2.

Chapter 27

Completeness of atomic initial sequents

In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A isan arbitrary formula) can be derived from only atomic initial sequents p ⊢ p (where p is an atomic formula). Thistheorem plays a role analogous to eta expansion in lambda calculus, and dual to cut-elimination and beta reduction.Typically it can be established by induction on the structure of A, much more easily than cut-elimination.

27.1 References• Gaisi Takeuti. Proof theory. Volume 81 of Studies in Logic and the Foundation ofMathematics. North-Holland,Amsterdam, 1975.

• Anne Sjerp Troelstra and Helmut Schwichtenberg. Basic Proof Theory. Edition: 2, illustrated, revised. Pub-lished by Cambridge University Press, 2000.

36

Chapter 28

Computable isomorphism

In computability theory two sets A;B ⊆ N of natural numbers are computably isomorphic or recursively isomor-phic if there exists a total bijective computable function f : N → N with f(A) = B . By the theorem of Myhill,[1]the relation of computable isomorphism coincides with the relation of one-one reduction.Two numberings ν and µ are called computably isomorphic if there exists a computable bijection f so that ν = µ◦fComputably isomorphic numberings induce the same notion of computability on a set.

28.1 References[1] Theorem 7.VI, Hartley Rogers, Jr., Theory of recursive functions and effective computability

• Rogers, Hartley, Jr. (1987), Theory of recursive functions and effective computability (2nd ed.), Cambridge,MA: MIT Press, ISBN 0-262-68052-1, MR 886890.

37

Chapter 29

Computable measure theory

In mathematics, computable measure theory is the part of computable analysis that deals with effective versions ofmeasure theory.

29.1 References• Jeremy Avigad (2012), “Inverting the Furstenberg correspondence”, Discrete and Continuous Dynamical Sys-

tems, Series A, 32, pp. 3421–3431.

• Abbas Edalat (2009), “A computable approach to measure and integration theory”, Information and Compu-tation 207:5, pp. 642–659.

• Stephen G. Simpson (2009), Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, CambridgeUniversity Press. ISBN 978-0-521-88439-6

38

Chapter 30

Computable model theory

Computable model theory is a branch of model theory which deals with questions of computability as they applyto model-theoretical structures.

30.1 History

It was developed almost simultaneously by mathematicians in the West, primarily located in the United States andAustralia, and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little com-munication between these two groups and so a number of important results were discovered independently.

30.2 Introduction

Computable model theory introduces the ideas of computable and decidable models and theories and one of the basicproblems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditionscan be shown to exist.

30.3 See also• Vaught conjecture

30.4 References• Harizanov, V. S. (1998), “Pure Computable Model Theory”, in Ershov, Iurii Leonidovich, Handbook of Re-

cursive Mathematics, Volume 1: Recursive Model Theory, Studies in Logic and the Foundations of Mathematics138, North Holland, pp. 3–114, ISBN 978-0-444-50003-8, MR 1673621.

39

Chapter 31

Computable real function

In mathematical logic, specifically computability theory, a function f : R → R is sequentially computable if, for everycomputable sequence {xi}∞i=1 of real numbers, the sequence {f(xi)}∞i=1 is also computable.A function f : R → R is effectively uniformly continuous if there exists a recursive function d : N → N such that, if|x− y| < 1

d(n)

then|f(x)− f(y)| < 1

n

A real function is computable if it is both sequentially computable and effectively uniformly continuous,[1]

These definitions can be generalized to functions of more than one variable or functions only defined on a subset ofRn. The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:LetD be a subset ofRn.A function f : D → R is sequentially computable if, for everyn -tuplet ({xi 1}∞i=1, . . . {xi n}∞i=1)of computable sequences of real numbers such that(∀i) (xi 1, . . . xi n) ∈ D ,

the sequence {f(xi)}∞i=1 is also computable.This article incorporates material from Computable real function on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

31.1 References[1] see Grzegorczyk, Andrzej (1957), “On the Definitions of Computable Real Continuous Functions” (PDF), Fundamenta

Mathematicae 44: 61–77

40

Chapter 32

Conservativity theorem

In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula

∃x1 . . .∃xm φ(x1, . . . , xm)

is a theorem of a first-order theory T . Let T1 be a theory obtained from T by extending its language with newconstants

a1, . . . , am

and adding a new axiom

φ(a1, . . . , am)

Then T1 is a conservative extension of T , which means that the theory T1 has the same set of theorems in the originallanguage (i.e., without constants ai ) as the theory T .In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by intro-ducing a new functional symbol:

Suppose that a closed formula ∀y ∃xφ(x, y) is a theorem of a first-order theory T , where we denotey := (y1, . . . , yn) . Let T1 be a theory obtained from T by extending its language with new functionalsymbol f (of arity n ) and adding a new axiom ∀y φ(f(y), y) . Then T1 is a conservative extension ofT , i.e. the theories T and T1 prove the same theorems not involving the functional symbol f ).

32.1 References• Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.

• J.R. Shoenfield (1967). Mathematical Logic. Addison-Wesley Publishing Company.

41

Chapter 33

Constructive non-standard analysis

In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's non-standard analysis,developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote:

The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998,2001). Themodel of constructive nonstandard analysis studied there is an extension ofMoerdijk’s (1995)model for constructive nonstandard arithmetic.

33.1 See also• Smooth infinitesimal analysis

• John Lane Bell

33.2 References• Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73(1995), pp. 37–51.

“Abstract: This paper provides an explicit description of a model for intuitionistic non-standard arith-metic, which can be formalized in a constructive metatheory without the axiom of choice.”

• Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bull. Symbolic Logic Volume 4, Number3 (1998), 233–272.

“Abstract: We develop a constructive version of nonstandard analysis, extending Bishop's constructiveanalysis with infinitesimal methods. ...”

• Juha Ruokolainen 2004, Constructive Nonstandard Analysis Without Actual Infinity

42

Chapter 34

Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumedat limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More formally, let γ bean ordinal, and s := ⟨sα|α < γ⟩ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,

sβ = lim sup{sα|α < β} = inf{sup{sα|δ ≤ α < β}|δ < β}

and

sβ = lim inf{sα|α < β} = sup{inf{sα|δ ≤ α < β}|δ < β} .

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

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

43

Chapter 35

Continuum (set theory)

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite)cardinal number, c . Georg Cantor proved that the cardinality c is larger than the smallest infinity, namely, ℵ0 . Healso proved that c equals 2ℵ0 , the cardinality of the power set of the natural numbers.The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes statedby saying that no cardinality lies between that of the continuum and that of the natural numbers, ℵ0 .

35.1 Linear continuum

Main article: Linear continuum

According to Raymond Wilder (1965) there are four axioms that make a set C and the relation < into a linearcontinuum:

• C is simply ordered with respect to <.

• If [A,B] is a cut of C, then either A has a last element or B has a first element. (compare Dedekind cut)

• There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ Ssuch that x < z < y. (separability axiom)

• C has no first element and no last element. (Unboundedness axiom)

These axioms characterize the order type of the real number line.

35.2 See also• Suslin’s problem

35.3 References• Raymond L. Wilder (1965) The Foundations of Mathematics, 2nd ed., page 150, John Wiley & Sons.

44

Chapter 36

Countryman line

ACountryman line is an uncountable linear ordering whose square is the union of countably many chains. The exis-tence of Countryman lines was first proven by Shelah. Shelah also conjectured that, assuming PFA, every Aronszajnline contains a Countryman line. This conjecture, which remained open for three decades, was proven by JustinMoore.

36.1 References• Shelah, Saharon (1976). “Decomposing uncountable squares to countably many chains”. Journal of Combi-

natorial Theory Series A 21 (1): 110–114. doi:10.1016/0097-3165(76)90053-4.

• Moore, Justin (2006). “A five element basis for the uncountable linear orders”. Annals of Mathematics. SecondSeries 163 (2): 669–688. doi:10.4007/annals.2006.163.669.

45

Chapter 37

Cyclic negation

In many-valued logic with linearly ordered truth values, cyclic negation is a unary truth function that takes a truthvalue n and returns n − 1 as value if n isn't the lowest value; otherwise it returns the highest value.For example, let the set of truth values be {0,1,2}, let ~ denote negation, and let p be a variable ranging over truthvalues. For these choices, if p = 0 then ~p = 2; and if p = 1 then ~p = 0.Cyclic negation was originally introduced by the logician and mathematician Emil Post.

37.1 References• Mares, Edwin (2011), “Negation”, in Horsten, Leon; Pettigrew, Richard, The Continuum Companion to Philo-

sophical Logic, Continuum International Publishing, pp. 180–215, ISBN 9781441154231. See in particularpp. 188–189.

46

Chapter 38

Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z inX such that x < z < y.

38.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

38.2 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

38.3 See also• Dense set

• Dense-in-itself

• Kripke semantics

38.4 References• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

47

Chapter 39

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If δ is an ordinal number and ⟨Xα | α < δ⟩ is a sequence of subsets of δ , then the diagonal intersection, denoted by

∆α<δXα,

is defined to be

{β < δ | β ∈∩α<β

Xα}.

That is, an ordinal β is in the diagonal intersection ∆α<δXα if and only if it is contained in the first β members ofthe sequence. This is the same as

∩α<δ

([0, α] ∪Xα),

where the closed interval from 0 to α is used to avoid restricting the range of the intersection.

39.1 See also• Fodor’s lemma

• Club set

• Club filter

39.2 References• Thomas Jech, Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003,page 92.

• Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.

This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

48

Chapter 40

Double recursion

In recursive function theory, double recursion is an extension of primitive recursion which allows the definition ofnon-primitive recursive functions like the Ackermann function.Raphael M. Robinson called functions of two natural number variables G(n, x) double recursive with respect to givenfunctions, if

• G(0, x) is a given function of x.

• G(n + 1, 0) is obtained by substitution from the function G(n, ·) and given functions.

• G(n + 1, x + 1) is obtained by substitution from G(n + 1, x), the function G(n, ·) and given functions.[1]

Robinson goes on to provide a specific double recursive function (originally defined by Rózsa Péter)

• G(0, x) = x + 1

• G(n + 1, 0) = G(n, 1)

• G(n + 1, x + 1) = G(n, G(n + 1, x))

where the given functions are primitive recursive, butG is not primitive recursive. In fact, this is precisely the functionnow known as the Ackermann function.

40.1 See also• Primitive recursion

• Ackermann function

40.2 References[1] Raphael M. Robinson (1948). “Recursion and Double Recursion”. Bulletin of the American Mathematical Society 54:

987–93. doi:10.1090/S0002-9904-1948-09121-2.

49

Chapter 41

Double turnstile

Not to be confused with .

In logic, the symbol ⊨, ⊨ or |= is called the double turnstile. It is closely related to the turnstile symbol ⊢ , which hasa single bar across the middle. It is often read as "entails", "models", “is a semantic consequence of” or “is strongerthan”.[1] In TeX, the turnstile symbols ⊨ and |= are obtained from the commands \vDash and \models respectively.In Unicode it is encoded at U+22A8 ⊨ true (HTML &#8872;)In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and iscapable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial onusing this package.

41.1 Meaning

The double turnstile is a binary relation. It has several different meanings in different contexts:

• To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denotethat if every sentence on the left is true, the sentence on the right must be true, e.g. Γ ⊨ φ . This usage isclosely related to the single-barred turnstile symbol which denotes syntactic consequence.

• To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denotethat the structure is a model for (or satisfies) the set of sentences, e.g. A |= Γ .

• To denote a tautology, ⊨ φ . which is to say that the expression φ is a semantic consequence of the empty set.

41.2 See also• List of logic symbols

• List of mathematical symbols

41.3 References[1] Nederpelt, Rob (2004). “Chapter 7: Strengthening and weakening”. Logical Reasoning: A First Course (3rd revised ed.).

King’s College Publications. p. 62. ISBN 0-9543006-7-X.

50

Chapter 42

Effective Polish space

In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presen-tation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standardexamples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

42.1 Definition

An effective Polish space is a complete separable metric space X with metric d such that there is a countable denseset C = (c0, c1,...) that makes the following two relations on N4 computable (Moschovakis 2009:96-7):

P (i, j, k,m) ≡ d(ci, cj) ≤m

k + 1

Q(i, j, k,m) ≡ d(ci, cj) <m

k + 1

42.2 References• Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society. ISBN0-8218-4813-5

51

Chapter 43

Elementary definition

In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic,and in particular without reference to set theory or using extensions such as plural quantification.Elementary definitions are of particular interest because they admit a complete proof apparatus while still beingexpressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms suchas ZFC).Saying that a definition is elementary is a weaker condition than saying it is algebraic.

43.1 Related• Elementary sentence

• Elementary theory

43.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

52

Chapter 44

Elementary diagram

For diagrams of electrical circuits, see Circuit diagram.

In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences withparameters from the structure that are true in the structure. It is also called the complete diagram.

44.1 Definition

Let M be a structure in a first-order language L. An extended language L(M) is obtained by adding to L a constantsymbol ca for every element a of M. The structure M can be viewed as an L(M) structure in which the symbols in Lare interpreted as before, and each new constant ca is interpreted as the element a. The elementary diagram of M isthe set of all L(M) sentences that are true in M (Marker 2002:44).

44.2 References• Chang, Chen Chung; Keisler, H. Jerome (1989), Model Theory, Elsevier, ISBN 978-0-7204-0692-4

• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6

• Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York:Springer-Verlag, ISBN 978-0-387-98760-6

53

Chapter 45

Elementary sentence

In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, withoutreference to set theory or using any axioms which have consistency strength equal to set theory.Saying that a sentence is elementary is a weaker condition than saying it is algebraic.

45.1 Related• Elementary theory

• Elementary definition

45.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

54

Chapter 46

Elementary theory

In mathematical logic, an elementary theory is one that involves axioms using only finitary first-order logic, withoutreference to set theory or using any axioms which have consistency strength equal to set theory.Saying that a theory is elementary is a weaker condition than saying it is algebraic.

46.1 Related• Elementary sentence

• Elementary definition

• Elementary theory of the reals

46.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

55

Chapter 47

End extension

In model theory and set theory, which are disciplines within mathematics, a model B = ⟨B,F ⟩ of some axiomsystem of set theory T in the language of set theory is an end extension of A = ⟨A,E⟩ , in symbols A ⊆end B , if

• A is a substructure ofB , and

• b ∈ A whenever a ∈ A and bFa hold, i.e., no new elements are added byB to the elements of A .

The following is an equivalent definition of end extension: A is a substructure of B , and {b ∈ A : bEa} = {b ∈B : bFa} for all a ∈ A .For example, ⟨B,∈⟩ is an end extension of ⟨A,∈⟩ if A and B are transitive sets, and A ⊆ B .

56

Chapter 48

Equisatisfiability

In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa;in other words, either both formulae are satisfiable or both are not. Two equisatisfiable formulae may have differentmodels, provided they both have some or both have none. As a result, equisatisfiability is different from logicalequivalence, as two equivalent formulae always have the same models.Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to becorrect if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept areSkolemization and some translations into conjunctive normal form.

48.1 Examples

A translation from propositional logic into propositional logic in which every binary disjunction a ∨ b is replaced by((a ∨ n) ∧ (¬n ∨ b)) , where n is a new variable (one for each replaced disjunction) is a transformation in whichsatisfiability is preserved: the original and resulting formulae are equisatisfiable. Note that these two formulae are notequivalent: the first formula has the model in which b is true while a and n are false, and this is not a model of thesecond formula, in which n has to be true in this case.

57

Chapter 49

Erasure (logic)

In mathematical logic, a logical system has the erasure property if and only if no subset of the propositions can beadded to another subset of the propositions to refute a consequence.For instance, if proposition A means “the store is open from 8:00 to 22:00” and proposition B means “except Tues-days”, the system AB does not have erasure.

49.1 See also• Monotonic logic in “mathematical logic”

• Peirce’s Logic at the “Stanford Encyclopedia of Philosophy”

58

Chapter 50

Extension (predicate logic)

The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfythe predicate. Such a set of tuples is a relation.For example the statement "d2 is the weekday following d1" can be seen as a truth function associating to each tuple(d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuplesassociated with the value true, i.e.{(Monday, Sunday), (Tuesday, Monday), (Wednesday, Tuesday), (Thursday, Wednesday), (Friday, Thursday), (Sat-urday, Friday), (Sunday, Saturday)}By examining this extension we can conclude that “Tuesday is the weekday following Saturday” (for example) is false.Using set-builder notation, the extension of the n-ary predicate Φ can be written as

{(x1, ..., xn) | Φ(x1, ..., xn)} .

50.1 Relationship with characteristic function

If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectively –making the characteristic function a predicate – , then for all relations R and predicatesΦ the following two statementsare equivalent:

• Φ is the characteristic function of R;

• R is the extension of Φ .

50.2 See also• Extensionality

• Intension

59

Chapter 51

Extensionality

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have thesame external properties. It stands in contrast to the concept of intensionality, which is concerned with whether theinternal definitions of objects are the same.

51.1 Example

Consider the two functions f and g mapping from and to natural numbers, defined as follows:

• To find f(n), first add 5 to n, then multiply by 2.

• To find g(n), first multiply n by 2, then add 10.

These functions are extensionally equal; given the same input, both functions always produce the same value. But thedefinitions of the functions are not equal, and in that intensional sense the functions are not the same.Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionallyidentical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town.Then, the two argument predicates “has one person named”, “is the oldest person in” are intensionally distinct, butextensionally equal for “Joe” in that “town” now.

51.2 In mathematics

The extensional definition of function equality, discussed above, is commonly used in mathematics. Sometimesadditional information is attached to a function, such as an explicit codomain, in which case two functions must notonly agree on all values, but must also have the same codomain, in order to be equal.A similar extensional definition is usually employed for relations: two relations are said to be equal if they have thesame extensions.In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements.In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions—withtheir extension as stated above, so that it is impossible for two relations or functions with the same extension to bedistinguished.Other mathematical objects are also constructed in such a way that the intuitive notion of “equality” agrees with set-level extensional equality; thus, equal ordered pairs have equal elements, and elements of a set which are related byan equivalence relation belong to the same equivalence class.Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonlyused to maintain a difference between intensional equality and a more general equivalence relation (which generallyhas poor constructibility or decidability properties).

60

51.3. SEE ALSO 61

51.3 See also• Duck typing

• Structural typing

• Univalence axiom

51.4 References

Chapter 52

Finite character

In mathematics, a family F of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

52.1 Properties

A family F of sets of finite character enjoys the following properties:

1. For each A ∈ F , every (finite or infinite) subset of A belongs to F .

2. Tukey’s lemma: In F , partially ordered by inclusion, the union of every chain of elements of F also belong toF , therefore, by Zorn’s lemma, F contains at least one maximal element.

52.2 Example

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finitecharacter (because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent). There-fore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family isa vector basis, every vector space has a (possibly infinite) vector basis.This article incorporates material from finite character on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

62

Chapter 53

Fluent (artificial intelligence)

For other meanings of the same term, see fluent (disambiguation)

In artificial intelligence, a fluent is a condition that can change over time. In logical approaches to reasoning aboutactions, fluents can be represented in first-order logic by predicates having an argument that depends on time. Forexample, the condition “the box is on the table”, if it can change over time, cannot be represented by On(box, table); a third argument is necessary to the predicate On to specify the time: On(box, table, t)means that the box is on thetable at time t . This representation of fluents is used in the situation calculus using the sequence of the past actionsin place of the current time.A fluent can also be represented by a function, dropping the time argument. For example, that the box is on the tablecan be represented by on(box, table) , where on is a function and not a predicate. In first order logic, convertingpredicates to functions is called reification; for this reason, fluents represented by functions are said to be reified.When using reified fluents, a separate predicate is necessary to tell when a fluent is actually true or not. For example,HoldsAt(on(box, table), t) means that the box is actually on the table at time t , where the predicate HoldsAtis the one that tells when fluents are true. This representation of fluents is used in the event calculus, in the fluentcalculus, and in the features and fluents logics.Some fluents can be represented as functions in a different way. For example, the position of a box can be representedby a function on(box, t)whose value is the object the box is standing on at time t . Conditions that can be representedin this way are called functional fluents. Statements about the values of such functions can be given in first order logicwith equality using literals such as on(box, t) = table . Some fluents are represented this way in the situation calculus.

53.1 See also• Event calculus

• Fluent calculus

• Frame problem

• Situation calculus

63

Chapter 54

Friedberg numbering

In computability theory, a Friedberg numbering is a numbering (enumeration) of the set of all partial recursivefunctions that has no repetitions: each partial recursive function appears exactly once in the enumeration (Vereščaginand Shen 2003:30).The existence of such numberings was established by Richard M. Friedberg in 1958 (Cutland 1980:78).

54.1 References• Nigel Cutland (1980), Computability: An Introduction to Recursive Function Theory, Cambridge UniversityPress. ISBN 9780521294652.

• Richard M. Friedberg (1958), Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set.III. Enumeration Without Duplication, Journal of Symbolic Logic 23:3, pp. 309–316.

• Nikolaj K. Vereščagin and A. Shen (2003), Computable Functions, American Mathematical Soc.

54.2 External links• Institute of Mathematics

64

Chapter 55

Gabbay’s separation theorem

In mathematical logic and computer science, Gabbay’s separation theorem, named after Dov Gabbay, states thatany arbitrary temporal logic formula can be rewritten in a logically equivalent “past → future” form. I.e. the futurebecomes what must be satisfied.[1] This form can be used as execution rules; a MetateM program is a set of suchrules.[2]

55.1 References[1] Fisher, Michael David; Gabbay, Dov M.; Vila, Lluis (2005), Handbook of Temporal Reasoning in Artificial Intelligence,

Foundations of Artificial Intelligence 1, Elsevier, p. 150, ISBN 9780080533360.

[2] Kowalski, Robert A.; Sadri, Fariba (1996), “Towards a Unified Agent Architecture That Combines Rationality with Re-activity”, Logic in Databases: International Workshop LID '96, San Miniato, Italy, July 1ÔÇô2, 1996, Proceedings, LectureNotes in Computer Science 1154, Springer-Verlag, pp. 137–149, doi:10.1007/BFb0031739, ISBN 3-540-61814-7.

65

Chapter 56

Ground axiom

In set theory, the ground axiom was introduced by Hamkins (2005) and Reitz (2007). It states that the universe isnot a nontrivial set forcing extension of an inner model.

56.1 References• Hamkins, Joel David (2005), “The Ground Axiom”, Oberwolfach Report 55: 3160–3162

• Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh (2008), “The ground axiom is consistent with V ≠HOD”, Proceedings of the American Mathematical Society 136 (8): 2943–2949, doi:10.1090/S0002-9939-08-09285-X, ISSN 0002-9939, MR 2399062

• Reitz, Jonas (2007), “The ground axiom”, Journal of Symbolic Logic 72 (4): 1299–1317, doi:10.2178/jsl/1203350787,ISSN 0022-4812, MR 2371206

• Jonas Reitz (2008). The Ground Axiom (Ph.D.). CUNY Graduate Center.

66

Chapter 57

Herbrand interpretation

In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbolsare assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol isinterpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset ofthe relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows thesymbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then thereis a Herbrand interpretation that satisfies them. Moreover, Herbrand’s theorem states that if S is unsatisfiable thenthere is a finite unsatisfiable set of ground instances from the Herbrand universe defined by S. Since this set is finite,its unsatisfiability can be verified in finite time. However there may be an infinite number of such sets to check.It is named after Jacques Herbrand.

57.1 See also• Herbrand structure

• Interpretation (logic)

• Interpretation (model theory)

67

Chapter 58

Herbrand structure

In mathematics, for a language L , define the Herbrand universe to be the set of ground terms of L .A structureM forL is aHerbrand structure if the domain ofM is the Herbrand universe ofL and the interpretationof M is a Herbrand interpretation. This fixes the domain of M , and so each Herbrand structure can be identifiedwith its interpretation.A Herbrand model of a theory T is a Herbrand structure that is a model of T.

58.1 See also• Herbrand base

• Herbrand’s theorem

This article incorporates material from Herbrand structure on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

68

Chapter 59

Heyting arithmetic

In mathematical logic,Heyting arithmetic (sometimes abbreviated HA) is an axiomatization of arithmetic in accor-dance with the philosophy of intuitionism (Troelstra 1973:18). It is named after Arend Heyting, who first proposedit.

59.1 Introduction

Heyting arithmetic adopts the axioms of Peano arithmetic (PA), but uses intuitionistic logic as its rules of inference.In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used toprove many specific cases. For instance, one can prove that ∀ x, y ∈ N : x = y ∨ x ≠ y is a theorem (any two naturalnumbers are either equal to each other, or not equal to each other). In fact, since "=" is the only predicate symbolin Heyting arithmetic, it then follows that, for any quantifier-free formula p, ∀ x, y, z, … ∈ N : p ∨ ¬p is a theorem(where x, y, z… are the free variables in p).

59.2 History

Kurt Gödel studied the relationship between Heyting arithmetic and Peano arithmetic. He used the Gödel–Gentzennegative translation to prove in 1933 that if HA is consistent, then PA is also consistent.

59.3 Related concepts

Heyting arithmetic should not be confused with Heyting algebras, which are the intuitionistic analogue of Booleanalgebras.

59.4 See also

• Harrop formula

• BHK interpretation

59.5 References

• Ulrich Kohlenbach (2008), Applied proof theory, Springer.

• Anne S. Troelstra, ed. (1973),Metamathematical investigation of intuitionistic arithmetic and analysis, Springer,1973.

69

70 CHAPTER 59. HEYTING ARITHMETIC

59.6 External links• Stanford Encyclopedia of Philosophy: "Intuitionistic Number Theory" by Joan Moschovakis.

• Fragments of Heyting Arithmetic by Wolfgang Burr

Chapter 60

High (computability)

In computability theory, a Turing degree [X] is high if it is computable in 0′, and the Turing jump [X′] is 0′′, whichis the greatest possible degree in terms of Turing reducibility for the jump of a set which is computable in 0′ (Soare1987:71).Similarly, a degree is high n if its n'th jump is the (n+1)'st jump of 0. Even more generally, a degree d is generalizedhigh n if its n'th jump is the n'th jump of the join of d with 0′.

60.1 See also

Low (computability)

60.2 References

Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin,1987. ISBN 3-540-15299-7

71

Chapter 61

Hilbert–Bernays provability conditions

In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays,are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224).These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They are also closelyrelated to axioms of provability logic.

61.1 The conditions

Let T be a formal theory of arithmetic with a formalized provability predicate Prov(n), which is expressed as aformula of T with one free number variable. For each formula φ in the theory, let #(φ) be the Gödel number of φ.The Hilbert–Bernays provability conditions are:

1. If T proves a sentence φ then T proves Prov(#(φ)).

2. For every sentence φ, T proves Prov(#(φ)) → Prov(#(Prov(#(φ))))

3. T proves that Prov(#(φ → ψ)) and Prov(#(φ)) imply Prov (#(ψ))

61.2 References• Smith, Peter (2007). An introduction to Gödel’s incompleteness theorems. Cambridge University Press. ISBN978-0-521-67453-9

72

Chapter 62

Honest leftmost branch

In set theory, an honest leftmost branch of a tree T on ω × γ is a branch (maximal chain) ƒ ∈ [T] such that for eachbranch g ∈ [T], one has ∀ n ∈ ω : ƒ(n) ≤ g(n). Here, [T] denotes the set of branches of maximal length of T, ω is theordinal (represented by the natural numbers N) and γ is some other ordinal.

62.1 See also• scale (computing)

• Suslin set

62.2 References• Akihiro Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer, Berlin, 1997.

• Yiannis N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980.

73

Chapter 63

Indiscernibles

In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation definedby a formula. Usually only first-order formulas are considered.

63.1 Examples

If a, b, and c are distinct and {a, b, c} is a set of indiscernibles, then, for example, for each binary formula φ, wemust have

[φ(a, b)∧φ(b, a)∧φ(a, c)∧φ(c, a)∧φ(b, c)∧φ(c, b)]∨[¬φ(a, b)∧¬φ(b, a)∧¬φ(a, c)∧¬φ(c, a)∧¬φ(b, c)∧¬φ(c, b)] .

Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz.

63.2 Generalizations

In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indis-cernibles often refers implicitly to this weaker notion. In our example of binary formulas, to say that the triple (a, b,c) of distinct elements is a sequence of indiscernibles implies

([φ(a, b)∧φ(a, c)∧φ(b, c)]∨[¬φ(a, b)∧¬φ(a, c)∧¬φ(b, c)])∧([φ(b, a)∧φ(c, a)∧φ(c, b)]∨[¬φ(b, a)∧¬φ(c, a)∧¬φ(c, b)]) .

63.3 Applications

Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and Zero sharp.

63.4 See also• Rough set

63.5 References• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

74

Chapter 64

Institutional model theory

This page is about the concept in mathematical logic. For the concept in sociology, see Institutional logic .Institutional model theory generalizes a large portion of first-order model theory to an arbitrary logical system.

64.1 Overview

The notion of “logical system” here is formalized as an institution. Institutions constitute a model-oriented meta-theory on logical systems similar to how the theory of rings and modules constitute a meta-theory for classical linearalgebra. Another analogy can be made with universal algebra versus groups, rings, modules etc. By abstracting awayfrom the realities of the actual conventional logics, it can be noticed that institution theory comes in fact closer to therealities of non-conventional logics.Institutional model theory analyzes and generalizes classical model-theoretic notions and results, like

• Elementary diagrams

• Elementary embeddings

• Ultraproducts, Los’ theorem

• Saturated models

• axiomatizability

• Varieties, Birkhoff axiomatizability

• Craig interpolation

• Robinson consistency

• Beth definability

• Gödel's completeness theorem

For each concept and theorem, the infrastructure and properties required are analyzed and formulated as conditionson institutions, thus providing a detailed insight on which properties of first-order logic they rely and how much theycan be generalized to other logics.

64.2 References

64.3 Further reading• Razvan Diaconescu: Institution-Independent Model Theory. Birkhäuser, 2008. ISBN 978-3-7643-8707-5.

75

76 CHAPTER 64. INSTITUTIONAL MODEL THEORY

• Razvan Diaconescu: Jewels of Institution-Independent Model Theory. In: K. Futatsugi, J.-P. Jouannaud, J.Meseguer (eds.): Algebra, Meaning and Computation. Essays Dedicated to Joseph A. Goguen on the Occasionof His 65th Birthday. Lecture Notes in Computer Science 4060, p. 65-98, Springer-Verlag, 2006.

• Marius Petria and Rãzvan Diaconescu: Abstract Beth definability in institutions. Journal of Symbolic Logic71(3), p. 1002-1028, 2006.

• Daniel Gǎinǎ and Andrei Popescu: An institution-independent generalisation of Tarski’s elementary chaintheorem, Journal of Logic and Computation 16(6), p. 713-735, 2006.

• Till Mossakowski, Joseph Goguen, Rãzvan Diaconescu, Andrzej Tarlecki: What is a Logic?. In Jean-YvesBeziau, editor, Logica Universalis, pages 113-133. Birkhauser, 2005.

• Andrzej Tarlecki: Quasi-varieties in abstract algebraic institutions. Journal of Computer and System Sciences33(3), p. 333-360, 1986.

64.4 External links• Razvan Diaconescu’s publication list - contains recent work on institutional model theory

Chapter 65

Jensen’s covering theorem

In set theory, Jensen’s covering theorem states that if 0# does not exist then every uncountable set of ordinals iscontained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universeis close to the universe of all sets. The first proof appeared in (Devlin & Jensen 1975). Silver later gave a fine structurefree proof using his machines and finally Magidor (1990) gave an even simpler proof.The converse of Jensen’s covering theorem is also true: if 0# exists then the countable set of all cardinals less thanℵω cannot be covered by a constructible set of cardinality less than ℵω.In his book Proper Forcing, Shelah proved a strong form of Jensen’s covering lemma.

65.1 References• Devlin, Keith I.; Jensen, R. Björn (1975), “Marginalia to a theorem of Silver”, ISILC Logic Conference (Proc.

Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Lecture notes in mathematics 499, Berlin, New York:Springer-Verlag, pp. 115–142, doi:10.1007/BFb0079419, ISBN 978-3-540-07534-9, MR 0480036

• Magidor, Menachem (1990), “Representing sets of ordinals as countable unions of sets in the core model”,Transactions of the American Mathematical Society 317 (1): 91–126, doi:10.2307/2001455, ISSN 0002-9947,MR 939805

• Mitchell,William (2010), “The covering lemma”,Handbook of Set Theory, Springer, pp. 1497–1594, doi:10.1007/978-1-4020-5764-9_19, ISBN 978-1-4020-4843-2

• Shelah, Saharon (1982), Proper forcing, Lecture Notes in Mathematics 940, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0096536, ISBN 978-3-540-11593-9, MR 675955

77

Chapter 66

Joint embedding property

In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for allstructures A and B in K, there is a structure C in K such that both A and B have embeddings into C.It is one of the three properties used to define the age of a structure.A similar but different notion to the joint embedding property is the amalgamation property. To see the difference,first consider the class K (or simply the set) containing three models with linear orders, L1 of size one, L2 of size two,and L3 of size three. This class K has the joint embedding property because all three models can be embedded intoL3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing asingle element e and extends in two different ways to L3, one in which e is the smallest and the other in which e is thelargest. Now any common model with an embedding from these two extensions must be at least of size five so thatthere are two elements on either side of e.Now consider the class of algebraically closed fields. This class has the amalgamation property since any two fieldextensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embeddedinto a common field when the characteristic of the fields differ.

66.1 References• Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.

78

Chapter 67

Judgment (mathematical logic)

For other uses, see Judgment (disambiguation).

In mathematical logic, a judgment can be an assertion about occurrence of a free variable in an expression of theobject language, or about provability of a proposition (either as a tautology or from a given context), but judgmentscan be also other inductively definable assertions in the metatheory. Judgments are used for example in formalizingdeduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequenceof judgments, and their conclusion is a judgment as well. Also the result of a proof expresses a judgment, and theused hypotheses are formed as a sequence of judgments.A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in anyof their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules.Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalizethe Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simpleform. The same cannot be done with the other two deductions systems: as context is changed in some of their rulesof inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want touse them just for proving derivability of tautologies.This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deductiontheorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as arule of inference in natural deduction.In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the twofields, e.g. Curry-Howard correspondence). The abstraction in the notion of judgment in mathematical logic can beexploited also in foundation of type theory as well.

67.1 See also• Simply typed lambda calculus

• Mathematical logic

67.2 External links• “Judgments in formal systems”. Everything2.

• Pfenning, Frank (Spring 2004). “Natural Deduction” (PDF). 15-815 Automated Theorem Proving.

• Martin-Löf, Per (1983). “On the meaning of the logical constants and the justifications of the logical laws”.Siena Lectures.

79

Chapter 68

Kanamori–McAloon theorem

In mathematical logic, the Kanamori–McAloon theorem, due to Kanamori & McAloon (1987), gives an exampleof an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certainfinitistic special case of a theorem in Ramsey theory due to Erdős and Rado is not provable in Peano arithmetic.

68.1 See also• Paris–Harrington theorem

• Goodstein’s theorem

• Kruskal’s tree theorem

68.2 References• Kanamori, Akihiro; McAloon, Kenneth (1987), “On Gödel incompleteness and finite combinatorics”, Annals

of Pure and Applied Logic 33 (1): 23–41, doi:10.1016/0168-0072(87)90074-1, ISSN 0168-0072, MR 870685

80

Chapter 69

Kleene–Rosser paradox

In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic areinconsistent, in particular the version of Curry's combinatory logic introduced in 1930, and Church's original lambdacalculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibitedby Stephen Kleene and J. B. Rosser in 1935.

69.1 The paradox

Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total,definable number-theoretic functions, which enabled them to construct a term that essentially replicates the Richardparadox in formal language.Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox,and used this to construct a much simpler paradox, now known as Curry’s paradox.

69.2 See also• List of paradoxes

69.3 References• Andrea Cantini, "The inconsistency of certain formal logics", in the Paradoxes and Contemporary Logic entryof Stanford Encyclopedia of Philosophy (2007).

• Kleene, S. C. & Rosser, J. B. (1935). “The inconsistency of certain formal logics”. Annals of Mathematics 36(3): 630–636. doi:10.2307/1968646.

81

Chapter 70

Knaster’s condition

In mathematics, a partially ordered set P is said to have Knaster’s condition upwards (sometimes property (K))if any uncountable subset A of P has an upwards-linked uncountable subset. An analogous definition applies toKnaster’s condition downwards.The property is named after Polish mathematician Bronisław Knaster.Knaster’s condition implies a countable chain condition (ccc), and it is sometimes used in conjunction with a weakerform of Martin’s axiom, where the ccc requirement is replaced with Knaster’s condition. Not unlike ccc, Knaster’scondition is also sometimes used as a property of a topological space, in which case it means that the topology (as in,the family of all open sets) with inclusion satisfies the condition.Furthermore, assuming MA( ω1 ), ccc implies Knaster’s condition, making the two equivalent.

70.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

82

Chapter 71

Least fixed point

The function f(x)=x2−4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2-√17/2.

In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) ofa function from a partially ordered set to itself is the fixed point which is less than each other fixed point, accordingto the set’s order. A function need not have a least fixed point, and cannot have more than one.For example, with the usual order on the real numbers, the least fixed point of the real function f(x) = x2 is x = 0(since the only other fixed point is 1 and 0 < 1). In contrast, f(x) = x+1 has no fixed point at all, let alone a least one,and f(x)=x has infinitely many fixed points, but no least one.

83

84 CHAPTER 71. LEAST FIXED POINT

71.1 Applications

Many fixed-point theorems yield algorithms for locating the least fixed point. Least fixed points often have desirableproperties that arbitrary fixed points do not.In mathematical logic and computer science, the least fixed point is related tomaking recursive definitions (see domaintheory and/or denotational semantics for details).Immerman [1][2] and Vardi [3] independently showed the descriptive complexity result that the polynomial-time com-putable properties of linearly ordered structures are definable in FO(LFP), i.e. in first-order logic with a least fixedpoint operator. However, FO(LFP) is too weak to express all polynomial-time properties of unordered structures(for instance that a structure has even size).

71.2 Greatest fixed points

Greatest fixed points can also be determined, but they are less commonly used than least fixed points. However, incomputer science they, analogously to the least fixed point, give rise to corecursion and codata.

71.3 See also• Fixed point

• Kleene fixed-point theorem

• Knaster–Tarski theorem

71.4 Notes[1] N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1–3) (1986) 86–104.

[2] Immerman, Neil (1982). “Relational Queries Computable in Polynomial Time”. STOC '82: Proceedings of the fourteenthannual ACM symposium on Theory of computing. pp. 147–152. doi:10.1145/800070.802187. Revised version in Infor-mation and Control, 68 (1986), 86–104.

[3] Vardi, Moshe Y. (1982). “The Complexity of Relational Query Languages”. STOC '82: Proceedings of the fourteenthannual ACM symposium on Theory of computing. pp. 137–146. doi:10.1145/800070.802186.

71.5 References• Immerman, Neil. Descriptive Complexity, 1999, Springer-Verlag.

• Libkin, Leonid. Elements of Finite Model Theory, 2004, Springer.

Chapter 72

LEGO (proof assistant)

LEGO is a proof assistant developed by Randy Pollack at the University of Edinburgh. It implements several typetheories: the Edinburgh Logical Framework (LF), the Calculus of Constructions (CC), the Generalized Calculus ofConstructions (GCC) and the Unified Theory of Dependent Types (UTT).

72.1 External links• Official website

85

Chapter 73

Lightface analytic game

In descriptive set theory, a lightface analytic game is a game whose payoff set A is a Σ11 subset of Baire space; that

is, there is a tree T on ω × ω which is a computable subset of (ω × ω)<ω , such that A is the projection of the set ofall branches of T.The determinacy of all lightface analytic games is equivalent to the existence of 0#.

86

Chapter 74

Limited principle of omniscience

In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle ofomniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle(Bridges and Richman 1987). The LPO and LLPO axioms are used to gauge the amount of nonconstructivity requiredfor an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the senseof Brouwer.

74.1 Definitions

The limited principle of omniscience states (Bridges and Richman 1987:3):

LPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, the following holds: either ai = 0for all i, or there is a k with ak = 1.

The lesser limited principle of omniscience states:

LLPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, and such that at most one ai isnonzero, the following holds: either a₂i = 0 for all i, or a₂i₊₁ = 0 for all i, where a₂i and a₂i₊₁ are entrieswith even and odd index respectively.

It can be proved constructively that the law of the excluded middle implies LPO, and LPO implies LLPO. However,none of these implications can be reversed in typical systems of constructive mathematics.The term “omniscience” comes from a thought experiment regarding how a mathematician might tell which of thetwo cases in the conclusion of LPO holds for a given sequence (ai). Answering the question “is there a k with ak =1?" negatively, assuming the answer is negative, seems to require surveying the entire sequence. Because this wouldrequire the examination of infinitely many terms, the axiom stating it is possible to make this determination wasdubbed an “omniscience principle” by Bishop (1967).

74.2 References• Bishop, Errett Foundations of Constructive Analysis, 1967. ISBN 4-87187-714-0

• Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, LondonMathematical Society Lec-ture Notes v. 57, 1987. ISBN 0-521-31802-5

74.3 External links• Constructive mathematics", Douglas Bridges, Stanford Encyclopedia of Philosophy

87

Chapter 75

Lindström’s theorem

In mathematical logic, Lindström’s theorem (named after Swedish logician Per Lindström, who published it in1969) states that first-order logic is the strongest logic [1] (satisfying certain conditions, e.g. closure under classicalnegation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.[2]

Lindström’s theorem is perhaps the best known result of what later became known as abstract model theory,[3] thebasic notion of which is an abstract logic;[4] the more general notion of an institution was later introduced, whichadvances from a set-theoretical notion of model to a category theoretical one.[5] Lindström had previously obtaineda similar result in studying first-order logics extended with Lindström quantifiers.[6]

Lindström’s theorem has been extended to various other systems of logic in particular modal logics by Johan vanBenthem and Sebastian Enqvist.

75.1 Notes[1] In the sense of Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors,

Model-theoretic logics, 1985 ISBN 0-387-90936-2 page 43

[2] A companion to philosophical logic by Dale Jacquette 2005 ISBN 1-4051-4575-7 page 329

[3] Chen Chung Chang; H. Jerome Keisler (1990). Model theory. Elsevier. p. 127. ISBN 978-0-444-88054-3.

[4] Jean-Yves Béziau (2005). Logica universalis: towards a general theory of logic. Birkhäuser. p. 20. ISBN 978-3-7643-7259-0.

[5] Dov M. Gabbay, ed. (1994). What is a logical system?. Clarendon Press. p. 380. ISBN 978-0-19-853859-2.

[6] Jouko Väänänen, Lindström’s Theorem

75.2 References• Per Lindström, “OnExtensions of Elementary Logic”, Theoria 35, 1969, 1–11. doi:10.1111/j.1755-2567.1969.tb00356.x• Johan vanBenthem, “ANewModal LindströmTheorem”, LogicaUniversalis 1, 2007, 125–128. doi:10.1007/s11787-006-0006-3

• Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994),Mathematical Logic (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94258-2

• Sebastian Enqvist, “A General Lindström Theorem for Some Normal Modal Logics”, Logica Universalis 7,2013, 233–264. doi:10.1007/s11787-013-0078-9

• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1

• Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and com-plexity, Oxford University Press, 2004, ISBN 0-19-852981-3, section 9.4

88

Chapter 76

Linked set

In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements Ahave a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound.Note that every centered set is linked, which includes, in particular, every directed set.

76.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

89

Chapter 77

LOGCFL

In computational complexity theory, LOGCFL is the complexity class that contains all decision problems that canbe reduced in logarithmic space to a context-free language. This class is situated between NL and AC1, in the sensethat it contains the former and is contained in the latter. Problems that are complete for LOGCFL include manyproblems whose instances can be characterized by acyclic hypergraphs:

• evaluating acyclic Boolean conjunctive queries

• checking the existence of a homomorphism between two acyclic relational structures

• checking the existence of solutions of acyclic constraint satisfaction problems

77.1 See also• List of complexity classes

77.2 External links• Complexity Zoo: LOGCFL

90

Chapter 78

Logic for Computable Functions

See also: Logic of Computable Functions

Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at the universitiesof Edinburgh and Stanford by Robin Milner and others in 1972. LCF introduced the general-purpose programminglanguage ML to allow users to write theorem-proving tactics. Theorems in the system are propositions of a special“theorem” abstract datatype. The ML type system ensures that theorems are derived using only the inference rulesgiven by the operations of the abstract type.Successors include HOL (Higher Order Logic) and Isabelle.

78.1 References• Gordon, Michael J. C. (1996). “From LCF to HOL: a short history”. Retrieved 2007-10-11.

• Milner, Robin (May 1972). Logic for Computable Functions: description of a machine implementation. (PDF).Stanford University.

91

Chapter 79

Logical assertion

In mathematical logic, logical assertion is a statement that asserts that a certain premise is true, and is useful forstatements in proof. It is equivalent to a sequent with an empty antecedent.For example, if p = "x is even”, the implication

(⊢ p) → (x (mod 2) ≡ 0)

is thus true. We can also write this using the logical assertion symbol, as

⊢ ((⊢ p) → (x (mod 2) ≡ 0))

In computer programming and programming language semantics, these are used in the form of assertions; one ex-ample is a loop invariant.

92

Chapter 80

Logical graph

A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax thatCharles Sanders Peirce developed for logic.In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of agraphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.In the century since Peirce initiated this line of development, a variety of formal systems have branched out fromwhat is abstractly the same formal base of graph-theoretic structures.

80.1 See also• Charles Sanders Peirce bibliography

• Conceptual graph

• Propositional calculus

• Truth table

80.2 External links

Media related to Logical graphs at Wikimedia Commons

• Logical Graph @ Commens Dictionary of Peirce’s Terms

• Existential Graphs, Jay Zeman, ed., U. of Florida. With 4 works by Peirce.

• Frithjof Dau’s page of readings and links on existential graphs includes lists of: books exclusively on existentialgraphs; books containing existential graphs; articles; and some links and downloadables.

• The literature of C.S. Peirce’s Existential Graphs (via Internet Archive), Xin-Wen Liu, Institute of Philosophy,Chinese Academy of Social Sciences, Beijing, PRC. A whole lot there.

93

Chapter 81

Logical machine

Logical machine is a term used by AllanMarquand (1853-1924) in 1883, perhaps in response to the ideas of CharlesSanders Peirce's “Logical Machines” as appearing for example in The American Journal of Psychology, 1. Nov. 1887,p. 165-170 (Google Books Eprint page 165).

81.1 Bibliography• Marquand, Allan

• (1883), “AMachine for Producing Syllogistic Variation” in C. S. Peirce, ed., Studies in Logic, pp. 12–15,along with “Note on an Eight-Term Logical Machine”, p. 16. Google Books Eprint. Book reprinted1983 with introduction by Max Fisch.

• (1886), “A New Logical Machine”, Proceedings of the American Academy of Arts and Sciences 21: 303–07. Google Books Eprint.

• Peirce, C. S.

• (1886 letter), Letter, Peirce to A. Marquand, 1886 December 30, published 1993 in Kloesel, C. et al.,eds., Writings of Charles S. Peirce: A Chronological Edition, Vol. 5. Indiana Univ. Press, pp. 421–3.Google Books Preview.

• (1887), “Logical Machines”, The American Journal of Psychology v. 1, n. 1, Baltimore: N. Murray, pp.165–70. Google Books Eprint. Reprinted in (1976) The New Elements of Mathematics v. III, pt. 1, pp.625–32; (1997) Modern Logic 7:71–77, Project Euclid Eprint; and (2000) Writings of Charles S. Peircev. 6, pp. 65–73.

• Baldwin, Mark James (1902), “Logical Machine”,Dictionary of Philosophy and Psychology, pp. 28–30 GoogleBooks Eprint. Classics in the History of Psychology Eprint.

• Ketner, Kenneth Laine (1984), “The early history of computer design: Charles Sanders Peirce and Marquand’slogical machines”, with the assistance of Arthur Franklin Stewart, Princeton University Library Chronicle, v.45, n. 3, pp. 186–211. PULC 15MB PDF Eprint.

• Dalakov, Georgi (undated), “Charles Peirce and Allan Marquand”, History of Computers and Computing.Eprint.

81.2 See also• Logics for computability

94

Chapter 82

Low (computability)

In computability theory, a Turing degree [X] is low if the Turing jump [X′] is 0′. A set is low if it has low degree.Since every set is computable from its jump, any low set is computable in 0′, but the jump of sets computable in 0′can bound any degree r.e. in 0′ (Schoenfield Jump Inversion). X being low says that its jump X′ has the least possibledegree in terms of Turing reducibility for the jump of a set.A degree is low n if its n'th jump is the n'th jump of 0. A set X is generalized low if it satisfies X′ ≡T X + 0′, that is:if its jump has the lowest degree possible. And a degree d is generalized low n if its n'th jump is the (n-1)'st jumpof the join of d with 0′. More generally, properties of sets which describe their being computationally weak (whenused as a Turing oracle) are referred to under the umbrella term lowness properties.By the Low basis theorem of Jockusch and Soare, any nonempty Π0

1 class in 2ω contains a set of low degree. Thisimplies that, although low sets are computationally weak, they can still accomplish such feats as computing a com-pletion of Peano Arithmetic. In practice, this allows a restriction on the computational power of objects needed forrecursion theoretic constructions: for example, those used in the analyzing the proof-theoretic strength of Ramsey’stheorem.

82.1 See also• High (computability)

• Low Basis Theorem

82.2 References• Soare, Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and com-

putably generated sets. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. ISBN 3-540-15299-7.Zbl 0667.03030.

• Nies, André (2009). Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford UniversityPress. ISBN 978-0-19-923076-1. Zbl 1169.03034.

95

Chapter 83

Low basis theorem

The low basis theorem in computability theory states that every nonemptyΠ01 class in 2ω (see arithmetical hierarchy)

contains a set of low degree (Soare 1987:109). It was first proved by Carl Jockusch and Robert I. Soare in 1972 (Nies2009:57). Cenzer (1993:53) describes it as “perhaps the most cited result in the theory of Π0

1 classes”. The proofuses the method of forcing with Π0

1 classes (Cooper 2004:330).

83.1 References• Cenzer, Douglas (1999). " Π0

1 classes in computability theory”. In Griffor, Edward R. Handbook of com-putability theory. Stud. Logic Found. Math. 140. North-Holland. pp. 37–85. ISBN 0-444-89882-4. MR1720779. Zbl 0939.03047. Theorem 3.6, p. 54.

• Cooper, S. Barry (2004). Computability Theory. Chapman and Hall/CRC. ISBN 1-58488-237-9..

• Jockusch, Carl G., Jr.; Soare, Robert I. (1972). "Π(0, 1) Classes and Degrees of Theories”. Transactionsof the American Mathematical Society 173: 33–56. doi:10.1090/s0002-9947-1972-0316227-0. ISSN 0002-9947. JSTOR 1996261. Zbl 0262.02041. The original publication, including additional clarifying prose.

• Nies, André (2009). Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford UniversityPress. ISBN 978-0-19-923076-1. Zbl 1169.03034. Theorem 1.8.37.

• Soare, Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and com-putably generated sets. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. ISBN 3-540-15299-7.Zbl 0667.03030.

96

Chapter 84

Lusin’s separation theorem

This article is about the separation theorem. For the theorem on continuous functions, see Lusin’s theorem.

In descriptive set theory and mathematical logic, Lusin’s separation theorem states that if A and B are disjointanalytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.[1] It is namedafter Nikolai Luzin, who proved it in 1927.[2]

The theorem can be generalized to show that for each sequence (An) of disjoint analytic sets there is a sequence (Bn)of disjoint Borel sets such that An ⊆ Bn for each n. [1]

An immediate consequence is Suslin’s theorem, which states that if a set and its complement are both analytic, thenthe set is Borel.

84.1 Notes[1] (Kechris 1995, p. 87).

[2] (Lusin 1927).

84.2 References• Kechris, Alexander (1995), Classical descriptive set theory, Graduate texts inmathematics 156, Berlin–Heidelberg–New York: Springer–Verlag, pp. xviii+402, doi:10.1007/978-1-4612-4190-4, ISBN 0-387-94374-9, MR1321597, Zbl 0819.04002 (ISBN 3-540-94374-9 for the European edition)

• Lusin, Nicolas (1927), “Sur les ensembles analytiques”, Fundamenta Mathematicae (in French) 10: 1–95, JFM53.0171.05.

97

Chapter 85

Material nonimplication

Venn diagram of A ↛ B∧⇔⇔ ¬

Material nonimplication or abjunction (latin ab = “from”, junctio =–"joining”) is the negation of material impli-cation. That is to say that for any two propositions P and Q, the material nonimplication from P to Q is true if andonly if P does not imply Q.It may be written using logical notation as:

p⊅qLpqp↛q

98

85.1. DEFINITION 99

85.1 Definition

85.1.1 Truth table

85.2 Properties

falsehood-preserving: The interpretation under which all variables are assigned a truth value of “false” produces atruth value of “false” as a result of material nonimplication.

85.3 Symbol

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is8603 (decimal).

85.4 Natural language

85.4.1 Grammatical

85.4.2 Rhetorical

“It’s not the case that p implies q.”“p but not q.”

85.4.3 Colloquial

“Just because p, doesn't mean q.”

85.5 Boolean algebra

Further information: Boolean algebra

(A'+B)'

85.6 Computer science

Bitwise operation: A&(~B)Logical operation: A&&(!B)

85.7 See also• Implication

85.8 References

Chapter 86

Maximal set

In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerablesubset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers,either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within thelattice of the recursively enumerable sets.Maximal sets have many interesting properties: they are simple, hypersimple, hyperhypersimple and r-maximal; thelatter property says that every recursive set R contains either only finitely many elements of the complement of A oralmost all elements of the complement of A. There are r-maximal sets that are not maximal; some of them do evennot have maximal supersets. Myhill (1956) asked whether maximal sets exist and Friedberg (1958) constructed one.Soare (1974) showed that the maximal sets form an orbit with respect to automorphism of the recursively enumerablesets under inclusion (modulo finite sets). On the one hand, every automorphism maps a maximal set A to anothermaximal set B; on the other hand, for every two maximal sets A, B there is an automorphism of the recursivelyenumerable sets such that A is mapped to B.

86.1 References• Friedberg, Richard M. (1958), “Three theorems on recursive enumeration. I. Decomposition. II. Maximal set.III. Enumeration without duplication”, The Journal of Symbolic Logic (Association for Symbolic Logic) 23 (3):309–316, doi:10.2307/2964290, JSTOR 2964290, MR 0109125

• Myhill, John (1956), “Solution of a problem of Tarski”, The Journal of Symbolic Logic (Association for Sym-bolic Logic) 21 (1): 49–51, doi:10.2307/2268485, JSTOR 2268485, MR 0075894

• H. Rogers, Jr., 1967. The Theory of Recursive Functions and Effective Computability, second edition 1987,MIT Press. ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1.

• Soare, Robert I. (1974), “Automorphisms of the lattice of recursively enumerable sets. I.Maximal sets”,Annalsof Mathematics. Second Series (Annals of Mathematics) 100 (1): 80–120, doi:10.2307/1970842, JSTOR1970842, MR 0360235

100

Chapter 87

Michael D. Morley

Michael Morley in Berkeley

Michael DarwinMorley (born 1930) is an American mathematician, currently professor emeritus at Cornell Univer-sity. His research is in advanced mathematical logic and model theory, and he is best known for Morley’s categoricitytheorem, which he proved in his Ph.D. thesis “Categoricity in Power” in 1962.His formal Ph.D. advisor was Saunders MacLane at the University of Chicago, but he actually finished his thesisunder the guidance of Robert Vaught at the University of California, Berkeley.

87.1 See also

• Morley’s problem

101

102 CHAPTER 87. MICHAEL D. MORLEY

87.2 External links• Math genealogy database

• Morley’s home page

Chapter 88

Milner–Rado paradox

In set theory, a branch of mathematics, the Milner – Rado paradox, found by Eric Charles Milner and RichardRado (1965), states that every ordinal number α less than the successor κ+ of some cardinal number κ can be writtenas the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.

88.1 Proof

The proof is by transfinite induction. Let α be a limit ordinal (the induction is trivial for successor ordinals), and foreach β < α , let {Xβ

n}n be a partition of β satisfying the requirements of the theorem.Fix an increasing sequence {βγ}γ<cf (α) cofinal in α with β0 = 0 .Note cf (α) ≤ κ .Define:

Xα0 = {0}; Xα

n+1 =∪γ

Xβγ+1n \ βγ

Observe that:

∪n>0

Xαn =

∪n

∪γ

Xβγ+1n \ βγ =

∪γ

∪n

Xβγ+1n \ βγ =

∪γ

βγ+1 \ βγ = α \ β0

and so∪

nXαn = α .

Let ot (A) be the order type of A . As for the order types, clearly ot(Xα0 ) = 1 = κ0 .

Noting that the sets βγ+1 \ βγ form a consecutive sequence of ordinal intervals, and that each Xβγ+1n \ βγ is a tail

segment of Xβγ+1n we get that:

ot(Xαn+1) =

∑γ

ot(Xβγ+1n \ βγ) ≤

∑γ

κn = κn · cf(α) ≤ κn · κ = κn+1

88.2 References• Milner, E. C.; Rado, R. (1965), “The pigeon-hole principle for ordinal numbers”, Proc. London Math. Soc.

(3) 15: 750–768, doi:10.1112/plms/s3-15.1.750, MR 0190003

• How to prove Milner-Rado Paradox? - Mathematics Stack Exchange

103

Chapter 89

Minimal logic

Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It isa variant of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does),but also the principle of explosion (ex falso quodlibet).Just like intuitionistic logic, minimal logic can be formulated in a language using→, ∧, ∨, ⊥ (implication, conjunction,disjunction and falsum) as the basic connectives, treating ¬A as an abbreviation for (A → ⊥). In this language it isaxiomatized by the positive fragment (i.e., formulas using only →, ∧, ∨) of intuitionistic logic, with no additionalaxioms or rules about ⊥. Thus minimal logic is a subsystem of intuitionistic logic, and it is strictly weaker as it doesnot derive the ex falso quodlibet principle ¬A,A ⊢ B (however, it derives its special case ¬A,A ⊢ ¬B ).Adding the ex falso axiom ¬A → (A → B) to minimal logic results in intuitionistic logic, and adding the doublenegation law ¬¬A→ A to minimal logic results in classical logic.Minimal logic is closely related to simply typed lambda calculus via the Curry-Howard isomorphism, i.e. the typingderivations of simply typed lambda terms are isomorphic to natural deduction proofs in minimal logic.

89.1 References• Johansson, Ingebrigt, 1936, "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus.” Compositio

Mathematica 4, 119–136.

104

Chapter 90

Omega-categorical theory

Inmathematical logic, an omega-categorical theory is a theory that has only one countablemodel up to isomorphism.Omega-categoricity is the special case κ = ℵ0 = ω of κ-categoricity, and omega-categorical theories are also referredto as ω-categorical. The notion is most important for countable first-order theories.

90.1 Equivalent conditions for omega-categoricity

Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, CzesławRyll-Nardzewski and Lars Svenonius, proved several independently.[1] Despite this, the literature still widely refers tothe Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary betweenauthors.[2][3]

Given a countable complete first-order theory T with infinite models, the following are equivalent:

• The theory T is omega-categorical.

• Every countable model of T has an oligomorphic automorphism group.

• Some countable model of T has an oligomorphic automorphism group.[4]

• The theory T has a model which, for every natural number n, realizes only finitely many n-types, that is, theStone space Sn(T) is finite.

• For every natural number n, T has only finitely many n-types.

• For every natural number n, every n-type is isolated.

• For every natural number n, up to equivalence modulo T there are only finitely many formulas with n freevariables, in other words, every nth Lindenbaum-Tarski algebra of T is finite.

• Every model of T is atomic.

• Every countable model of T is atomic.

• The theory T has a countable atomic and saturated model.

• The theory T has a saturated prime model.

90.2 Notes[1] Rami Grossberg, José Iovino and Olivier Lessmann, A primer of simple theories

[2] Hodges, Model Theory, p. 341.

[3] Rothmaler, p. 200.

[4] Cameron (1990) p.30

105

106 CHAPTER 90. OMEGA-CATEGORICAL THEORY

90.3 References• Cameron, Peter J. (1990), Oligomorphic permutation groups, London Mathematical Society Lecture Note Se-ries 152, Cambridge: Cambridge University Press, ISBN 0-521-38836-8, Zbl 0813.20002

• Chang, Chen Chung; Keisler, H. Jerome (1989) [1973], Model Theory, Elsevier, ISBN 978-0-7204-0692-4

• Hodges, Wilfrid (1993), Model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-30442-9

• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6

• Poizat, Bruno (2000), A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Berlin,New York: Springer-Verlag, ISBN 978-0-387-98655-5

• Rothmaler, Philipp (2000), Introduction to Model Theory, New York: Taylor & Francis Group, ISBN 978-90-5699-313-9

Chapter 91

Ordinal logic

In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to asequence of previous logics.[1][2] The concept was introduced in 1938 by Alan Turing in his PhD dissertation atPrinceton in view of Gödel’s incompleteness theorems.[3][1]

While Gödel showed that every system of logic suffers from some form of incompleteness, Turing focused on amethod so that from a given system of logic a more complete system may be constructed. By repeating the processa sequence L1, L2, … of logics is obtained, each more complete than the previous one. A logic L can then beconstructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc.Thus Turing showed how one can associate a logic with any constructive ordinal.[3]

91.1 References[1] Solomon Feferman, Turing in the Land of O(z) in “The universal Turing machine: a half-century survey” by Rolf Herken

1995 ISBN 3-211-82637-8 page 111

[2] Concise Routledge encyclopedia of philosophy 2000 ISBN 0-415-22364-4 page 647

[3] Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp.161–228.

107

Chapter 92

Paraconsistent mathematics

Paraconsistent mathematics (sometimes called inconsistent mathematics) represents an attempt to develop theclassical infrastructure of mathematics (e.g. analysis) based on a foundation of paraconsistent logic instead of classicallogic. A number of reformulations of analysis can be developed, for example functions which both do and do nothave a given value simultaneously.Chris Mortensen claims (see references):

One could hardly ignore the examples of analysis and its special case, the calculus. There prove to bemany places where there are distinctive inconsistent insights; see Mortensen (1995) for example. (1)Robinson’s non-standard analysis was based on infinitesimals, quantities smaller than any real number,as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advan-tages for calculation in being able to discard higher-order infinitesimals. The theory of differentiationturned out to have these advantages, while the theory of integration did not. (2)

92.1 References• Inconsistent Mathematics, by Chris Mortensen, Dordrecht, Kluwer Academic Publishers, 1995 Kluwer Math-

ematics and Its Applications Series, Vol 312 ISBN 0-7923-3186-9

92.2 External links• Entry in the Internet Encyclopedia of Philosophy

• Entry in the Stanford Encyclopedia of Philosophy

• Lectures by Manuel Bremer of the University of Düsseldorf

108

Chapter 93

Polyadic algebra

Polyadic algebras (more recently called Halmos algebras[1]) are algebraic structures introduced by Paul Halmos.They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositionallogic (see Lindenbaum-Tarski algebra).There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras[1] (when equality ispart of the logic) and Lawvere's functorial semantics (categorical approach).[2]

93.1 References[1] Michiel Hazewinkel (2000). Handbook of algebra 2. Elsevier. pp. 87–89. ISBN 978-0-444-50396-1.

[2] Jon Barwise (1989). Handbook of mathematical logic. Elsevier. p. 293. ISBN 978-0-444-86388-1.

93.2 Further reading• Paul Halmos, Algebraic Logic, Chelsea Publishing, New York (1962)

109

Chapter 94

Predicate logic

For the specific term, see First-order logic.

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. This formal system is distinguished from other systems in that itsformulae contain variables which can be quantified. Two common quantifiers are the existential ∃ (“there exists”)and universal ∀ (“for all”) quantifiers. The variables could be elements in the universe under discussion, or perhapsrelations or functions over that universe. For instance, an existential quantifier over a function symbol would beinterpreted as modifier “there is a function”. The foundations of predicate logic were developed independently byGottlob Frege and Charles Sanders Peirce.[1]

In informal usage, the term “predicate logic” occasionally refers to first-order logic. Some authors consider thepredicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from aninformal, more intuitive development.[2]

Predicate logics also include logics mixing modal operators and quantifiers. See Modal logic, Saul Kripke, BarcanMarcus formulae, A. N. Prior, and Nicholas Rescher.

94.1 See also

• First-order logic

• Propositional logic

• Existential graph

94.2 Footnotes[1] Eric M. Hammer: Semantics for Existential Graphs, Journal of Philosophical Logic, Volume 27, Issue 5 (October 1998),

page 489: “Development of first-order logic independently of Frege, anticipating prenex and Skolem normal forms”

[2] Among these authors is Stolyar, p. 166. Hamilton considers both to be calculi but divides them into an informal calculusand a formal calculus.

94.3 References

• A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK ISBN 0-521-21838-1

• Abram Aronovic Stolyar 1970, Introduction to Elementary Mathematical Logic, Dover Publications, Inc. NY.ISBN 0-486-645614

110

94.3. REFERENCES 111

• George F Luger, Artificial Intelligence, Pearson Education, ISBN 978-81-317-2327-2

• Hazewinkel, Michiel, ed. (2001), “Predicate calculus”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 95

Principle of distributivity

The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logicalconjunction and logical disjunction are distributive over each other so that for any propositions A, B and C theequivalences

A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)

and

A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)

hold.The principle of distributivity is valid in classical logic, but invalid in quantum logic. The article Is logic empirical?discusses the case that quantum logic is the correct, empirical logic, on the grounds that the principle of distributivityis inconsistent with a reasonable interpretation of quantum phenomena.

112

Chapter 96

Proof compression

In proof theory, an area of mathematical logic, proof compression is the problem of algorithmically compressingformal proofs. The developed algorithms can be used to improve the proofs generated by automated theorem provingtools such as sat-solvers, SMT-solvers, first-order theorem provers and proof assistants.

96.1 Problem Representation

In propositional logic a resolution proof of a clause κ from a set of clauses C is a directed acyclic graph (DAG): theinput nodes are axiom inferences (without premises) whose conclusions are elements of C, the resolvent nodes areresolution inferences, and the proof has a node with conclusion κ .[1]

The DAG contains an edge from a node η1 to a node η2 if and only if a premise of η1 is the conclusion of η2 . Inthis case, η1 is a child of η2 , and η2 is a parent of η1 . A node with no children is a root.A proof compression algorithm will try to create a new DAG with less nodes that represents a valid proof of κ or, insome cases, a valid proof of a subset of κ .

96.1.1 A simple example

Let’s take a resolution proof for the clause {a, b, c} from the set of clauses

{η1 : {a, b, p} , η2 : {c,¬p}} η1 : a, b, p η2 : c,¬pη3 : a, b, c

p

Here we can see:

• η1 and η2 are input nodes.

• The node η3 has a pivot p ,

• left resolved literal p• right resolved literal ¬p

• η3 conclusion is the clause {a, b, c}

• η3 premises are the conclusion of nodes η1 and η2 (its parents)

• The DAG would be

η1 η2↖↗η3

113

114 CHAPTER 96. PROOF COMPRESSION

• η1 and η2 are parents of η3• η3 is a child of η1 and η2• η3 is a root of the proof

A (resolution) refutation of C is a resolution proof of ⊥ from C. It is a common that given a node η , to refer to theclause η or η ’s clause meaning the conclusion clause of η , and (sub)proof η meaning the (sub)proof having η as itsonly root.In some works it can be found an algebraic representation of a resolution inference. The resolvent of κ1 and κ2 withpivot p can be denoted as κ1 ⊙p κ2 . When the pivot is uniquely defined or irrelevant, we omit it and write simplyκ1 ⊙ κ2 . In this way, the set of clauses can be seen as an algebra with a commutative operator; and terms in thecorresponding term algebra denote resolution proofs in a notation style that is more compact and more convenientfor describing resolution proofs than the usual graph notation.In our last example the notation of the DAG would be {a, b, p} ⊙p {c,¬p} or simply {a, b, p} ⊙ {c,¬p} .

We can identifyη1︷ ︸︸ ︷

{a, b, p}⊙

η2︷ ︸︸ ︷{c,¬p}︸ ︷︷ ︸

η3

96.2 Compression algorithms

Algorithms for compression of sequent calculus proofs include Cut-introduction and Cut-elimination.Algorithms for compression of propositional resolution proofs includeRecycleUnits,[2] RecyclePivots,[3] RecyclePivotsWithIntersection,[4]LowerUnits,[5] LowerUnivalents,[6] Split,[7] Reduce&Reconstruct,[8] and Subsumption.

96.3 Notes[1] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial

Regularization. 23rd International Conference on Automated Deduction, 2011.

[2] Bar-Ilan, O.; Fuhrmann, O.; Hoory, S. ; Shacham, O. ; Strichman, O. Linear-time Reductions of Resolution Proofs. Hard-ware and Software: Verification and Testing, p. 114–128, Springer, 2011.

[3] Bar-Ilan, O.; Fuhrmann, O.; Hoory, S. ; Shacham, O. ; Strichman, O. Linear-time Reductions of Resolution Proofs. Hard-ware and Software: Verification and Testing, p. 114–128, Springer, 2011.

[4] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via PartialRegularization. 23rd International Conference on Automated Deduction, 2011.

[5] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via PartialRegularization. 23rd International Conference on Automated Deduction, 2011.

[6] https://github.com/Paradoxika/Skeptik/tree/develop/doc/papers/LUniv

[7] Cotton, Scott. “Two Techniques for Minimizing Resolution Proofs”. 13th International Conference on Theory and Appli-cations of Satisfiability Testing, 2010.

[8] Simone, S.F. ; Brutomesso, R. ; Sharygina, N. “An Efficient and Flexible Approach to Resolution Proof Reduction”. 6thHaifa Verification Conference, 2010.

Chapter 97

Proof mining

In proof theory, a branch of mathematical logic, proof mining (or unwinding) is a research program that analyzesformalized proofs, especially in analysis, to obtain explicit bounds or rates of convergence from proofs that, whenexpressed in natural language, appear to be nonconstructive.[1] This research has led to improved results in analysisobtained from the analysis of classical proofs.

97.1 References[1] Ulrich Kohlenbach (2008). Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Verlag,

Berlin. pp. 1–536.

• Ulrich Kohlenbach and Paulo Oliva, “Proof Mining: A systematic way of analysing proofs in mathematics”,Proc. Steklov Inst. Math, 242:136–164, 2003

• Paulo Oliva, “Proof Mining in Subsystems of Analysis”, BRICS PhD thesis citeseer

115

Chapter 98

Pseudo-order

In constructive mathematics, a pseudo-order is a constructive generalisation of a linear order to the continuouscase. The usual trichotomy law does not hold in the constructive continuum because of its indecomposability, so thiscondition is weakened.A pseudo-order is a binary relation satisfying the following conditions:

1. It is not possible for two elements to each be less than the other. That is, ∀x, y : ¬ (x < y ∧ y < x) .2. For all x, y, and z, if x < y then either x < z or z < y. That is, ∀x, y, z : x < y → (x < z ∨ z < y) .3. Every two elements for which neither one is less than the other must be equal. That is, ∀x, y : ¬ (x < y ∨ y <x) → x = y

This first condition is simply antisymmetry. It follows from the first two conditions that a pseudo-order is transitive.The second condition is often called co-transitivity or comparison and is the constructive substitute for trichotomy. Ingeneral, given two elements of a pseudo-ordered set, it is not always the case that either one is less than the other orelse they are equal, but given any nontrivial interval, any element is either above the lower bound, or below the upperbound.The third condition is often taken as the definition of equality. The natural apartness relation on a pseudo-ordered setis given by

x#y ↔ x < y ∨ y < x

and equality is defined by the negation of apartness.The negation of the pseudo-order is a partial order which is close to a total order: if x ≤ y is defined as the negationof y < x, then we have

¬ (¬ (x ≤ y) ∧ ¬ (y ≤ x)).

Using classical logic one would then conclude that x ≤ y or y ≤ x, so it would be a total order. However, this inferenceis not valid in the constructive case.The prototypical pseudo-order is that of the real numbers: one real number is less than another if there exists (onecan construct) a rational number greater than the former and less than the latter. In other words, x < y if there existsa rational number z such that x < z < y.

98.1 References• Arend Heyting (1966) Intuitionism: An introduction. Second revised edition North-Holland Publishing Co.,Amsterdam.

http://books.google.com/books/about/Intuitionism.html?id=4rhLAAAAMAAJ

116

Chapter 99

Reduced product

For the reduced product in algebraic topology, see James reduced product.

Inmodel theory, a branch ofmathematical logic, and in algebra, the reduced product is a construction that generalizesboth direct product and ultraproduct.Let {Si | i ∈ I} be a family of structures of the same signature σ indexed by a set I, and let U be a filter on I. Thedomain of the reduced product is the quotient of the Cartesian product

∏i∈I

Si

by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if

{i ∈ I : ai = bi} ∈ U

If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the originalCartesian product. If U is an ultrafilter, the reduced product is an ultraproduct.Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are inter-preted by

R((a1i )/∼, . . . , (ani )/∼) ⇐⇒ {i ∈ I | RSi(a1i , . . . , ani )} ∈ U.

For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as(a + b)i = ai + bi and multiplication by a scalar c as (ca)i = c ai.

99.1 References• Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundationsof Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3., Chapter 6.

117

Chapter 100

Redundant proof

In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. Thatis, a proof ψ of κ is considered redundant if there exists another proof ψ′ of κ′ such that κ′ ⊆ κ (i.e. κ′ subsumes κ) and |ψ′| < |ψ| where |φ| is the number of nodes in φ .[1]

100.1 Local redundancy

A proof containing a subproof of the shapes (here omitted pivots indicate that the resolvents must be uniquely defined)

(η ⊙ η1)⊙ (η ⊙ η2) or η ⊙ (η1 ⊙ (η ⊙ η2))

is locally redundant.Indeed, both of these subproofs can be equivalently replaced by the shorter subproof η ⊙ (η1 ⊙ η2) . In the caseof local redundancy, the pairs of redundant inferences having the same pivot occur close to each other in the proof.However, redundant inferences can also occur far apart in the proof.The following definition generalizes local redundancy by considering inferences with the same pivot that occur withindifferent contexts. We write ψ [η] to denote a proof-context ψ [−] with a single placeholder replaced by the subproofη .

100.2 Global redundancy

A proof

ψ[ψ1[η ⊙p η1]⊙ ψ2[η ⊙p η2]] or ψ[ψ1[η ⊙p (η1 ⊙ ψ2[η ⊙p η2])]]

is potentially (globally) redundant. Furthermore, it is (globally) redundant if it can be rewritten to one of the followingshorter proofs:

ψ[η ⊙p (ψ1[η1]⊙ ψ2[η2])] or η ⊙p ψ[ψ1[η1]⊙ ψ2[η2]] or ψ[ψ1[η1]⊙ ψ2[η2]].

100.2.1 Example

The proof

118

100.3. NOTES 119

η : p, q η1 : ¬p, rq, r

pη3 : ¬q

rq

η η2 : ¬p, s,¬rq, s,¬r

pη3

s,¬rq

ψ : sr

is locally redundant as it is an instance of the first pattern in the definition ((η ⊙p η1)⊙ η3)⊙ ((η ⊙p η2)⊙ η3).

• The pattern is ψ[ψ1[η ⊙p η1]⊙ ψ2[η ⊙p η2]]

• ψ1[−] = ψ2[−] = _⊙ η3 and ψ[−] = _

But it is not globally redundant because the replacement terms according to the definition contain ψ1[η1]⊙ψ2[η2] inall the cases and ψ1[η1]⊙ ψ2[η2] = (η1 ⊙ η3)⊙ (η2 ⊙ η3) does not correspond to a proof. In particular, neither η1nor η2 can be resolved with η3 , as they do not contain the literal q .The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to thewell-known notion of regularity. [This link to “regularity” is (obviously) a link to a disambiguation page.] Informally,a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once asa pivot in this path.

100.3 Notes[1] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial

Regularization. 23rd International Conference on Automated Deduction, 2011.

Chapter 101

Richardson’s theorem

In mathematics, Richardson’s theorem establishes a limit on the extent to which an algorithm can decide whethercertainmathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is undecidablewhether a particular expression E satisfies the equation E = 0, and similarly undecidable whether the functions de-fined by expressions E and F are everywhere equal (in fact E = F if and only if E - F = 0). It was proved in 1968 bycomputer scientist Daniel Richardson of the University of Bath.Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the numberπ, the number log 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin,exp, and abs functions.For some classes of expressions (generated by other primitives than in Richardson’s theorem) there exist algorithmsthat can determine whether an expression is zero.[1]

101.1 Statement of the theorem

Richardson’s theorem can be stated as follows:[2] Let E be a set of expressions in the variable x which contains x and,as constant expressions, all rational numbers, and is such that if A(x) and B(x) are in E, then A(x) + B(x), A(x) - B(x),A(x)B(x), and A(B(x)) are also in E. Then:

• if x, log 2, π, ex, sin x ∈ E, then the problem of determining, for an expression A(x) in E, whether A(x) < 0 forsome x is unsolvable;

• if also |x| ∈ E then the problem of determining whether A(x) = 0 for all x is also unsolvable;

• if furthermore there is a function B(x) ∈ E without an antiderivative in E then the integration problem isunsolvable. (Example: eax2 has an antiderivative in the elementary functions if and only if a = 0.)

101.2 Extensions

After Hilbert’s Tenth Problem was solved in 1970, B. F. Caviness observed that the use of ex and log 2 could beremoved.[3] P. S. Wang[4] later noted that under the same assumptions under which the question of whether there wasx with A(x) < 0 was insoluble, the question of whether there was x with A(x) = 0 was also insoluble.Miklós Laczkovich[5] removed also the need for π and reduced the use of composition. In particular, given anexpression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn), both the question of whether A(x) >0 for some x and whether A(x) = 0 for some x are unsolvable.By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is notpossible to remove the sine function entirely.

120

101.3. SEE ALSO 121

101.3 See also• Constant problem

101.4 References[1] The identity problem for elementary functions and constants by Richardson and Fitch (pdf file)

[2] “Some Undecidable Problems Involving Elementary Functions of a Real Variable”, Daniel Richardson, J. Symbolic Logic33, #4 (1968), pp. 514-520, JSTOR 2271358.

[3] On Canonical Forms and Simplification, B. F. Caviness, JACM, 17, #2 (April 1970), pp. 385-396.

[4] P. S. Wang, The undecidability of the existence of zeros of real elementary functions, Journal of the Association forComputing Machinery 21:4 (1974), pp. 586–589.

[5] Miklós Laczkovich, The removal of π from some undecidable problems involving elementary functions, Proc. Amer. Math.Soc. 131:7 (2003), pp. 2235–2240.

101.5 Further reading• Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996). A = B. A. K. Peters. p. 212. ISBN 1-56881-063-6.

• Richardson, Daniel (1968). “Some undecidable problems involving elementary functions of a real variable”.Journal of Symbolic Logic 33 (4) (Association for Symbolic Logic). pp. 514–520. doi:10.2307/2271358.JSTOR 10.2307/2271358.

101.6 External links• Weisstein, Eric W., “Richardson’s theorem”, MathWorld.

Chapter 102

Robinson’s joint consistency theorem

Robinson’s joint consistency theorem is an important theorem of mathematical logic. It is related to Craig inter-polation and Beth definability.The classical formulation of Robinson’s joint consistency theorem is as follows:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and the intersection T1 ∩ T2 is complete (in thecommon language of T1 and T2 ), then the union T1 ∪ T2 is consistent. Note that a theory is complete if it decidesevery formula, i.e. either T ⊢ φ or T ⊢ ¬φ .Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and if there is no formula φ in the common languageof T1 and T2 such that T1 ⊢ φ and T2 ⊢ ¬φ , then the union T1 ∪ T2 is consistent.

102.1 References• Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge Uni-versity Press. p. 264. ISBN 0-521-00758-5.

122

Chapter 103

Scattered order

This article is about order theory. For the Australian post-punk band, see Scattered Order.

In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with morethan one element (Harzheim 2005:193ff.)A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orderswhich contains the singleton orders and is closed under well-ordered and reverse well-ordered sums.Laver's theorem (generalizing Fraïssé's conjecture) states that the embedding relation on the class of countable unionsof scattered orders is a well-quasi-order (Harzheim 2005:265).The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by thelexicographic order on Q× Z .

103.1 References• Egbert Harzheim (2005). Ordered Sets. Springer. ISBN 0-387-24219-8.

• Laver, Richard (1971). “On Fraïssé's order type conjecture”. Annals ofMathematics 93 (1): 89–111. doi:10.2307/1970754.JSTOR 1970754.

123

Chapter 104

Semicomputable function

In computability theory, a semicomputable function is a partial function f : Q → R that can be approximatedeither from above or from below by a computable function.More precisely a partial function f : Q → R is upper semicomputable, meaning it can be approximated fromabove, if there exists a computable function ϕ(x, k) : Q× N → Q , where x is the desired parameter for f(x) andk is the level of approximation, such that:

• limk→∞ ϕ(x, k) = f(x)

• ∀k ∈ N : ϕ(x, k + 1) ≤ ϕ(x, k)

Completely analogous a partial function f : Q → R is lower semicomputable iff −f(x) is upper semicomputableor equivalently if there exists a computable function ϕ(x, k) such that

• limk→∞ ϕ(x, k) = f(x)

• ∀k ∈ N : ϕ(x, k + 1) ≥ ϕ(x, k)

If a partial function is both upper and lower semicomputable it is called computable.

104.1 See also• computability theory

104.2 References• Ming Li and Paul Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, pp 37–38, Springer,1997.

124

Chapter 105

Separating set

This article is about separating sets for functions. For use in graph theory, see connectivity (graph theory).

In mathematics a set of functions S from a set D to a set C is called a separating set for D or said to separate thepoints of D if for any two distinct elements x and y of D, there exists a function f in S so that f(x) ≠ f(y).[1]

Separating sets can be used to formulate a version of the Stone-Weierstrass theorem for real-valued functions on acompact Hausdorff space X, with the topology of uniform convergence. It states that any subalgebra of this space offunctions is dense if and only if it separates points. This is the version of the theorem originally proved by MarshallH. Stone.[1]

105.1 Examples• The singleton set consisting of the identity function on R separates the points of R.

• If X is a T1 normal topological space, then Urysohn’s lemma states that the set C(X) of continuous functionson X with real (or complex) values separates points on X.

105.2 References[1] Carothers, N. L. (2000), Real Analysis, Cambridge University Press, pp. 201–204, ISBN 9781139643160.

125

Chapter 106

Set constraint

?•?

while (x>1) {

?

if (x%2==0)

����

x=x/2; else

@@@@R

x=x*3+1;

@@@@@R

���

��} •

-

?

A = Int

B = A ∪ I

C = B ∩ {n ∈ Int | n > 1}

E = C ∩ {2 · n | n ∈ Int}F = C ∩ {2 · n + 1 | n ∈ Int}

G = {n/2 | n ∈ E}H = {3 · n + 1 | n ∈ F}

D = B ∩ {n ∈ Int | n 6 1}

I = G ∪H

Set contraints obtained from abstract interpretation of the Collatz algorithm.

In mathematics and theoretical computer science, a set constraint is an equation or an inequation between setsof terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of setconstraints. Different approaches admit different operators (like "∪", "∩", "\", and function application)[note 1] on setsand different (in)equation relations (like "=", "⊆", and "⊈") between set expressions.Systems of set constraints are useful to describe (in particular infinite) sets of ground terms.[note 2] They arise inprogram analysis, abstract interpretation, and type inference.

106.1 Relation to regular tree grammars

Each regular tree grammar can be systematically transformed into a system of set inclusions such that its minimalsolution corresponds to the tree language of the grammar.For example, the grammar (terminal and nonterminal symbols indicated by lower and upper case initials, respectively)with the rules

126

106.2. LITERATURE 127

is transformed to the set inclusion system (constants and variables indicated by lower and upper case initials, respec-tively):

This system has a minimal solution, viz. ("L(N)" denoting the tree language corresponding to the nonterminal N inthe above tree grammar):

The maximal solution of the system is trivial; it assigns the set of all terms to every variable.

106.2 Literature• Aiken, A. (1995). Set Constraints: Results, Applications and Future Directions (Technical report). Univ. Berke-ley.

• Aiken, A., Kozen, D., Vardi, M., Wimmers, E.L. (May 1993). The Complexity of Set Constraints (Technicalreport). Computer Science Department, Cornell University. 93–1352.

• Aiken, A., Kozen, D., Vardi, M., Wimmers, E.L. (1994). “The Complexity of Set Constraints”. ComputerScience Logic'93. LNCS 832. Springer. pp. 1–17.

• Aiken, A., Wimmers, E.L. (1992). “Solving Systems of Set Constraints (Extended Abstract)". Seventh AnnualIEEE Symposium on Logic in Computer Science. pp. 329–340.

• Bachmair, Leo, Ganzinger, Harald, Waldmann, Uwe (1992). Set Constraints are the Monadic Class (Technicalreport). Max-Planck-Institut für Informatik. p. 13. MPI-I-92-240.

• Bachmair, Leo, Ganzinger, Harald, Waldmann, Uwe (1993). “Set Constraints are the Monadic Class”. EightAnnual IEEE Symposium on Logic in Computer Science. pp. 75–83.

• Charatonik, W. (Sep 1994). “Set Constraints in Some Equational Theories”. Proc. 1st Int. Conf. on Constraintsin Computational Logics (CCL). LNCS 845. Springer. pp. 304–319.

• Charatonik, Witold; Podelski, Andreas (2002). “Set Constraints with Intersection” (PDF). Information andComputation 179: 213–229. doi:10.1006/inco.2001.2952. Retrieved 11 May 2014.

• Charatonik, W., Podelski, A. (1998). Tobias Nipkow, ed. Co-definite Set Constraints. LNCS 1379. Springer-Verlag. pp. 211–225.

• Charatonik, W., Talbot, J.-M. (2002). Tison, S., ed. Atomic Set Constraints with Projection. LNCS 2378.Springer. pp. 311–325.

• Gilleron, R., Tison, S., Tommasi, M. (1993). “Solving Systems of Set Constraints using Tree Automata”. 10thAnnual Symposium on Theoretical Aspects of Computer Science. LNCS 665. Springer. pp. 505–514.

• Heintze, N., Jaffar, J. (1990). “A Decision Procedure for a Class of Set Constraints (Extended Abstract)".Fifth Annual IEEE Symposium on Logic in Computer Science. pp. 42–51.

• Heintze, N., Jaffar, J. (Feb 1991). A Decision Procedure for a Class of Set Constraints (Technical report).School of Computer Science, Carnegie Mellon University.

• Kozen, D. (1993). “Logical Aspects of Set Constraints”. Computer Science Logic'93 (PDF). LNCS 832. pp.175–188.

• Kozen, D. (1994). “Set Constraints and Logic Programming”. CCL. LNCS 845.

• Dexter Kozen (1998). “Set Constraints and Logic Programming” (PDF). Information and Computation 142:2–25. doi:10.1006/inco.1997.2694.

• Uribe, T.E. (1992). “Sorted Unification Using Set Constraints”. Proc. CADE–11. LNCS 607. pp. 163–177.

128 CHAPTER 106. SET CONSTRAINT

106.2.1 Literature on negative constraints

• Aiken, A., Kozen, D., Wimmers, E.L. (Jun 1993). Decidability of Systems of Set Constraints with NegativeConstraints (Technical report). Computer Science Department, Cornell University. 93–1362.

• Charatonik, W., Pacholski, L. (Jul 1994). “Negative Set Constraints with Equality”. Ninth Annual IEEESymposium on Logic in Computer Science. pp. 128–136.

• R. Gilleron, S. Tison, M. Tommasi (1993). “Solving Systems of Set Constraints with Negated Subset Rela-tionships”. Proceedings of the 34th Symp. on Foundations of Computer Science. pp. 372–380.

• Gilleron, R., Tison, S., Tommasi, M. (1993). Solving Systems of Set Constraints with Negated Subset Relation-ships (Technical report). Laboratoire d'Informatique Fondamentale de Lille. IT 247.

• Stefansson, K. (Aug 1993). Systems of Set Constraints with Negative Constraints are NEXPTIME-Complete(Technical report). Computer Science Department, Cornell University. 93–1380.

• Stefansson, K. (1994). “Systems of Set Constraints with Negative Constraints are NEXPTIME-Complete”.Ninth Annual IEEE Symposium on Logic in Computer Science. pp. 137–141.

106.3 Notes[1] If f is an n-ary function symbol admitted in a term, then "f(E1,...,En)" is a set expression denoting the set { f(t1,...,tn) :

t1∈E1 and ... and tn∈En }, where E1,...,En are set expressions in turn.

[2] This is similar to describing e.g. a rational number as a solution to an equation a⋅x + b = 0, with integer coefficients a, b.

Chapter 107

Set function

In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input isa set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.

107.1 Examples

Examples of set functions include:

• The function that assigns to each set its cardinality, i.e. the number of members of the set, is a set function.

• The function

d(A) = limn→∞

|A ∩ {1, . . . , n}|n

,

assigning densities to sufficiently well-behaved subsets A ⊆ {1, 2, 3, ...}, is a set function.

• The Lebesgue measure is a set function that assigns a non-negative real number to each set of real numbers.(Kolmogorov and Fomin 1975)

• A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the emptyset is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1.

• A possibility measure assigns a number between zero and one to each set in the powerset of some given set.See Possibility theory.

• A Random set is a set-valued random variable. See Random compact set.

107.2 References• A.N. Kolmogorov and S.V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0

107.3 Further reading• Sobolev, V.I. (2001), “Set function”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• *Regular set function at Encyclopedia of Mathematics

129

Chapter 108

Soft set

Soft set theory is a generalization of fuzzy set theory, that was proposed byMolodtsov in 1999 to deal with uncertaintyin a non-parametric manner.[1] A soft set is a parametrised family of sets - intuitively, this is “soft” because theboundary of the set depends on the parameters. Formally, a soft set, over a universal set X and set of parameters E isa pair (f, A) where A is a subset of E, and f is a function from A to the power set of X. For each e in A, the set f(e)is called the value set of e in (f, A).One of themost important steps for the new theory of soft sets was to definemappings on soft sets, which was achievedin 2009 by the mathematician Athar Kharal, with the results published in 2011.[2] Soft sets have also been appliedto the problem of medical diagnosis for use in medical expert systems. Fuzzy soft sets have also been introduced.Mappings on fuzzy soft sets were defined and studied by Kharal and Ahmad.[3]

108.1 Notes[1] Molodtsov, D. A. (1999). “Soft set theory—First results”. Computers & Mathematics With Applications 37 (4): 19–31.

doi:10.1016/S0898-1221(99)00056-5.

[2] Kharal, Athar; B. Ahmad (September 2011). “Mappings on Soft Classes”. New Mathematics and Natural Computation 7(3). doi:10.1142/S1793005711002025.

[3] Kharal, Athar; B. Ahmad (2009). “Mappings on Fuzzy Soft Classes”. Advances in Fuzzy Systems 2009. doi:10.1155/2009/407890.

108.2 References• Molodtsov D. A. A theory of soft sets. Moscow: Editorial URSS, 2004.

• Matsievsky S. V. Sets, multisets, fuzzy and soft sets without universe. Vestnik IKSUR, 2007, N. 10, pp. 44–52.

• Ahmad, B., Kharal, A.OnFuzzy Soft Sets. Advances in Fuzzy SystemsVolume 2009 (2009), Article ID 586507,6 pages doi:10.1155/2009/586507.

130

Chapter 109

Strength (mathematical logic)

The relative strength of two systems of formal logic can be defined via model theory. Specifically, a logic α is saidto be as strong as a logic β if every elementary class in β is an elementary class in α .[1]

109.1 See also• Abstract logic

• Lindström’s theorem

109.2 References[1] Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors,Model-theoretic

logics, 1985 ISBN 0-387-90936-2 page 43

131

Chapter 110

Subcountability

In constructive mathematics, a collection is subcountable if there exists a partial surjection from the natural numbersonto it. The name derives from the intuitive sense that such a collection is “no bigger” than the counting numbers.The concept is trivial in classical set theory, where a set is subcountable if and only if it is finite or countably infinite.Constructively it is consistent to assert the subcountability of some uncountable collections such as the real numbers.Indeed there are models of the constructive set theory CZF in which all sets are subcountable[1] and models of IZFin which all sets with apartness relations are subcountable.[2]

110.1 References[1] Rathjen, M. "Choice principles in constructive and classical set theories", Proceedings of the Logic Colloquium, 2002

[2] McCarty, J. "Subcountability under realizability", Notre Dame Journal of Formal Logic, Vol 27 no 2 April 1986

132

Chapter 111

Successor function

For other uses, see Successor.

In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n) =n+1 for each natural number n. For example, S(1) = 2 and S(2) = 3.The successor function is used in the Peano axioms which define the natural numbers. As such, it is not defined byaddition, but rather is used to define all natural numbers beyond 0, as well as addition. For example, 1 is defined tobe S(0), and addition on natural numbers is defined recursively by:

This yields e.g. 5 + 2 = 5 + S(1) = S(5) + 1 = 6 + 1 = 6 + S(0) = S(6) + 0 = 7 + 0 = 7When natural numbers are constructed based on set theory, a common approach is to define the number 0 to be theempty set {}, and the successor S(x) to be x ∪ { x }. The axiom of infinity then guarantees the existence of a set ℕthat contains 0 and is closed with respect to S; members of ℕ are called natural numbers.[1]

The successor function is the level-0 foundation of the infinite hierarchy of hyperoperations (used to build addition,multiplication, exponentiation, tetration, etc.).It is also one of the primitive functions used in the characterization of computability by recursive functions.

111.1 See also• successor ordinal

• successor cardinal

111.2 References

Paul R. Halmos (1968). Naive Set Theory. Nostrand.

[1] Halmos, Chapter 11

133

Chapter 112

Sudan function

In the theory of computation, the Sudan function is an example of a function that is recursive, but not primitiverecursive. This is also true of the better-known Ackermann function. The Sudan function was the first function havingthis property to be published.It was discovered (and published[1]) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student ofDavid Hilbert.

112.1 Definition

F0(x, y) = x+ y,

Fn+1(x, 0) = x, n ≥ 0

Fn+1(x, y + 1) = Fn(Fn+1(x, y), Fn+1(x, y) + y + 1), n ≥ 0.

112.2 Value Tables

In general, F1(x, y) is equal to F1(0, y) + 2y x.

112.3 References• Cristian Calude, Solomon Marcus, Ionel Tevy, The first example of a recursive function which is not primitive

recursive, Historia Mathematica 6 (1979), no. 4, 380–384 doi:10.1016/0315-0860(79)90024-7

[1] Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171

134

Chapter 113

Supernatural number

In mathematics, the supernatural numbers (sometimes called generalized natural numbers or Steinitz numbers)are a generalization of the natural numbers. They were used by Ernst Steinitz[1] in 1910 as a part of his work on fieldtheory.A supernatural number ω is a formal product:

ω =∏p

pnp ,

where p runs over all prime numbers, and each np is zero, a natural number or infinity. Sometimes vp(ω) is usedinstead of np . If no np = ∞ and there are only a finite number of non-zero np then we recover the positiveintegers. Slightly less intuitively, if all np are∞ , we get zero. Supernatural numbers extend beyond natural numbersby allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide ω “infinitelyoften,” by taking that prime’s corresponding exponent to be the symbol∞ .There is no natural way to add supernatural numbers, but they can bemultiplied, with

∏p p

np ·∏

p pmp =

∏p p

np+mp

. Similarly, the notion of divisibility extends to the supernaturals with ω1 | ω2 if vp(ω1) ≤ vp(ω2) for all p . Thenotion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers,by defining

lcm({ωi}) =∏p

psup(vp(ωi))

gcd({ωi}) =∏p

pinf(vp(ωi))

With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernaturalnumber. We can also extend the usual p -adic order functions to supernatural numbers by defining vp(ω) = np foreach pSupernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case manyof the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of afinite field.[2] They are also used implicitly in many number-theoretical proofs, such as the density of the square-freeintegers and bounds for odd perfect numbers.

113.1 See also

• profinite integer

135

136 CHAPTER 113. SUPERNATURAL NUMBER

113.2 References[1] Steinitz, Ernst (1910). “Algebraische Theorie der Körper”. Journal für die reine und angewandte Mathematik: 167–309.

ISSN 0075-4102. JFM 41.0445.03.

[2] Brawley & Schnibben (1989) pp.25-26

• Brawley, Joel V.; Schnibben, George E. (1989). Infinite algebraic extensions of finite fields. ContemporaryMathematics 95. Providence, RI: American Mathematical Society. pp. 23–26. ISBN 0-8218-5101-2. Zbl0674.12009.

• Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124.Providence, RI: American Mathematical Society. p. 125. ISBN 0-8218-4041-X. Zbl 1103.12002.

• Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd ed.). Springer-Verlag. p. 520. ISBN 978-3-540-77269-9. Zbl 1145.12001.

113.3 External links• Planet Math: Supernatural number

Chapter 114

Superposition calculus

The superposition calculus is a calculus for reasoning in equational first-order logic. It has been developed in theearly 1990s and combines concepts from first-order resolution with ordering-based equality handling as developedin the context of (unfailing) Knuth–Bendix completion. It can be seen as a generalization of either resolution (toequational logic) or unfailing completion (to full clausal logic). As most first-order calculi, superposition tries to showthe unsatisfiability of a set of first-order clauses, i.e. it performs proofs by refutation. Superposition is refutation-complete — given unlimited resources and a fair derivation strategy, from any unsatisfiable clause set a contradictionwill eventually be derived.As of 2007, most of the (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the Eequational theorem prover), although only a few implement the pure calculus.

114.1 Implementations• E

• SPASS

• Vampire

• Waldmeister

• Ayane at Google Code

114.2 References• Rewrite-Based Equational TheoremProvingwith Selection and Simplification, Leo Bachmair andHaraldGanzinger,Journal of Logic and Computation 3(4), 1994.

• Paramodulation-Based Theorem Proving, Robert Nieuwenhuis and Alberto Rubio, Handbook of AutomatedReasoning I(7), Elsevier Science and MIT Press, 2001.

137

Chapter 115

Switching circuit theory

Switching circuit theory is the mathematical study of the properties of networks of idealized switches.Such networks may be strictly combinational logic, in which their output state is only a function of the present stateof their inputs; or may also contain sequential elements, where the present state depends on the present state and paststates; in that sense, sequential circuits are said to include “memory” of past states. An important class of sequentialcircuits are state machines. Switching circuit theory is applicable to the design of telephone systems, computers, andsimilar systems.In the paper A Symbolic Analysis of Relay and Switching Circuits of 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. The principles of Boolean algebra are appliedto switches, providing mathematical tools for analysis and synthesis of any switching system.Ideal switches are considered as having only two exclusive states, for example, open or closed. In some analysis, thestate of a switch can be considered to have no influence on the output of the system and is designated as a “don't care”state. In complex networks it is necessary to also account for the finite switching time of physical switches; wheretwo or more different paths in a network may affect the output, these delays may result in a “logic hazard” or "racecondition" where the output state changes due to the different propagation times through the network.

115.1 See also• Karnaugh map

• Boolean circuit

• C-element

• Circuit minimization

• Circuit complexity

• Circuit switching

• Logic design

• Logic in computer science

• Logic gate

• Nonblocking minimal spanning switch

• Quine–McCluskey algorithm

• Relay - the kind of logic device Shannon was concerned with in 1938

• Programmable logic controller - computer software mimics relay circuits for industrial applications

• Switching lemma

• Unate function

138

115.2. REFERENCES 139

115.2 References• Keister, William; Ritchie, Alistair E.; Washburn, Seth H. (1963) [1951]. The Design of Switching Circuits.The Bell Telephone Laboratories Series. Princeton, NJ: D. Van Nostrand Company.

• Caldwell, Samuel H. (1965) [1958]. Switching Circuits and Logical Design. New York: John Wiley & Sons.

• Shannon, C. E. (1938). “A Symbolic Analysis of Relay and Switching Circuits”. Trans. AIEE 57 (12): 713–723. doi:10.1109/T-AIEE.1938.5057767.

Chapter 116

Systems of Logic Based on Ordinals

Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing.[1][2]

The thesis is an exploration of formal mathematical systems after Gödel’s theorem. Gödel showed for that any formalsystem S powerful enough to represent arithmetic, there is a theorem G which is true but the system is unable toprove. G could be added as an additional axiom to the system in place of a proof. However this would create anew system S' with its own unprovable true theorem G', and so on. Turing’s thesis considers iterating the process toinfinity, creating a system with an infinite set of axioms.The thesis was completed at Princeton under Alonzo Church and was a classic work in mathematics which introducedthe concept of ordinal logic.[3]

Martin Davis states that although Turing’s use of a computing oracle is not a major focus of the dissertation, it hasproven to be highly influential in theoretical computer science, e.g. in the polynomial time hierarchy.[4]

116.1 References[1] Turing, Alan (1938). Systems of Logic Based on Ordinals (PhD thesis). Princeton University. doi:10.1112/plms/s2-

45.1.161.

[2] Turing, A. M. (1939). “Systems of Logic Based on Ordinals”. Proceedings of the London Mathematical Society: 161–228.doi:10.1112/plms/s2-45.1.161.

[3] Solomon Feferman, Turing in the Land of O(z) in “The universal Turing machine: a half-century survey” by Rolf Herken1995 ISBN 3-211-82637-8 page 111

[4] Martin Davis “Computability, Computation and the Real World”, in Imagination and Rigor edited by Settimo Termini2006 ISBN 88-470-0320-2 pages 63-66

116.2 External links• “Turing’s Princeton Dissertation”. Princeton University Press. Retrieved January 10, 2012.

• Solomon Feferman (November 2006), “Turing’s Thesis”, Notices of the AMS 53 (10)

140

Chapter 117

Takeuti’s conjecture

In mathematics, Takeuti’s conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-orderlogic has cut-elimination (Takeuti 1953). It was settled positively:

• By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966);

• Independently by Takahashi by a similar technique (Takahashi 1967);

• It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.

Takeuti’s conjecture is equivalent1 to the consistency of second-order arithmetic and to the strong normalization ofthe Girard/Reynold’s System F.

117.1 See also• Hilbert’s second problem

117.2 Notes• ^ Equivalent in the sense that each of the statements can be derived from each other in the weak system PRAof arithmetic; consistency refers here to the truth of the Gödel sentence for second-order arithmetic. Seeconsistency proof for more discussion.

117.3 References• William W. Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In

Bulletin of the American Mathematical Society, 72:980–983.

• Gaisi Takeuti, 1953. On a generalized logic calculus. In Japanese Journal of Mathematics, 23:39–96. Anerrata to this article was published in the same journal, 24:149–156, 1954.

• Moto-o Takahashi, 1967. A proof of cut-elimination in simple type theory. In Japanese Mathematical Society,10:44–45.

141

Chapter 118

Tarski–Kuratowski algorithm

In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithmwhich provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy.The algorithm is named after Alfred Tarski and Kazimierz Kuratowski.

118.1 Algorithm

The Tarski–Kuratowski algorithm for the arithmetical hierarchy:

1. Convert the formula to prenex normal form.

2. If the formula is quantifier-free, it is in Σ00 and Π0

0 .

3. Otherwise, count the number of alternations of quantifiers; call this k.

4. If the first quantifier is ∃, the formula is in Σ0k+1 .

5. If the first quantifier is ∀, the formula is in Π0k+1 .

118.2 References• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1;ISBN 0-07-053522-1

142

Chapter 119

The Paradoxes of the Infinite

The Paradoxes of the Infinite (German title: Paradoxien des Unendlichen) is a mathematical work by BernardBolzano on the theory of sets. It was published in 1851, three years after Bolzano’s death, by a friend. The workcontained many interesting results in set theory. Bolzano expanded on the theme of Galileo’s paradox, giving moreexamples of correspondences between the elements of an infinite set and proper subsets of infinite sets. In the workhe also coined the term Menge, rendered in English as “set”.

119.1 References• Paradoxes of the Infinite; trans. by D.A.Steele; London: Routledge, 1950

• Bolzano, Bernard (1851), Paradoxien des Unendlichen, C.H. Reclam (German original) Faksimile

• Burton, David (1997), The History of Mathematics: An Introduction (Third ed.), McGraw-Hill, p. 592

143

Chapter 120

Theory of pure equality

In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only theequality relation symbol, and includes no non-logical axioms at all (Monk 1976:240–242). This theory is consistent,as any set with the usual equality relation provides an interpretation.The theory of pure equality was proven to be decidable by Löwenheim in 1915. If an additional axiom is addedsaying either that there are exactly m objects, for a fixed natural number m, or an axiom scheme is added stating thereare infinitely many objects, the resulting theory is complete.

120.1 References• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1

144

Chapter 121

Trichotomy (mathematics)

In mathematics, the Law of Trichotomy states that every real number is either positive, negative, or zero.[1] Moregenerally, trichotomy is the property of an order relation < on a setX that for any x and y, exactly one of the followingholds: x < y , x = y , or x > y .In mathematical notation, this is

∀x ∈ X ∀y ∈ X ((x < y ∧¬(y < x)∧¬(x = y) )∨ (¬(x < y)∧ y < x∧¬(x = y) )∨ (¬(x < y)∧¬(y < x)∧x = y )) .

Assuming that the ordering is irreflexive and transitive, this can be simplified to

∀x ∈ X ∀y ∈ X ((x < y) ∨ (y < x) ∨ (x = y)) .

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore alsofor comparisons between integers and between rational numbers. The law does not hold in general in intuitionisticlogic.In ZF set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderablesets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinalnumbers (because they are all well-orderable in that case).[2]

More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds.If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example,in the case of three element set {a,b,c} the relation R given by aRb, aRc, bRc is a strict total order, while the relationR given by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as morefoundational than the law of total order.A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it istrivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

121.1 See also• Dichotomy

• Law of noncontradiction

• Law of excluded middle

121.2 References[1] http://mathworld.wolfram.com/TrichotomyLaw.html

145

146 CHAPTER 121. TRICHOTOMY (MATHEMATICS)

[2] Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.

Chapter 122

Truth-table reduction

In computability theory, a truth-table reduction is a reduction from one set of natural numbers to another. As a“tool”, it is weaker than Turing reduction, since not every Turing reduction between sets can be performed by a truth-table reduction, but every truth-table reduction can be performed by a Turing reduction. For the same reason it issaid to be a stronger reducibility than Turing reducibility, because it implies Turing reducibility. Aweak truth-tablereduction is a related type of reduction which is so named because it weakens the constraints placed on a truth-tablereduction, and provides a weaker equivalence classification; as such, a “weak truth-table reduction” can actually bemore powerful than a truth-table reduction as a “tool”, and perform a reduction which is not performable by truthtable.A Turing reduction from a set B to a set A computes the membership of a single element in B by asking questionsabout the membership of various elements in A during the computation; it may adaptively determine which questionsit asks based upon answers to previous questions. In contrast, a truth-table reduction or a weak truth-table reductionmust present all of its (finitely many) oracle queries at the same time. In a truth-table reduction, the reduction alsogives a boolean function (a truth table) which, when given the answers to the queries, will produce the final answer ofthe reduction. In a weak truth-table reduction, the reduction uses the oracle answers as a basis for further computationwhich may depend on the given answers but may not ask further questions of the oracle.Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by acomputable function. For this reason, they are sometimes referred to as bounded Turing (bT) reductions rather thanas weak truth-table (wtt) reductions.

122.1 Properties

As every truth-table reduction is a Turing reduction, if A is truth-table reducible to B (A ≤ B), then A is alsoTuring reducible to B (A ≤T B). Considering also one-one reducibility, many-one reducibility and weak truth-tablereducibility,

A ≤1 B ⇒ A ≤m B ⇒ A ≤tt B ⇒ A ≤wtt B ⇒ A ≤T B

or in other words, one-one reducibility implies many-one reducibility, which implies truth-table reducibility, whichin turn implies weak truth-table reducibility, which in turn implies Turing reducibility.

122.2 References• H. Rogers, Jr., 1967. The Theory of Recursive Functions and Effective Computability, second edition 1987,MIT Press. ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1

147

Chapter 123

Zero dagger

In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovayin unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on somebrowsers.) The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions.Specifically, if ZFC is consistent, then ZFC + “0† does not exist” is consistent. ZFC + “0† exists” is not known to beinconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (seelarge cardinal for a discussion). It is usually formulated as follows:

0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the rela-tivized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ ismeasurable.

If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unboundedsubset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for thestructure (L,∈, U) , and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscerniblesin L[U].Solovay showed that the existence of 0† follows from the existence of two measurable cardinals. It is traditionallyconsidered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.

123.1 See also• 0#: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.

123.2 References• Kanamori, Akihiro; Awerbuch-Friedlander, Tamara (1990). “The compleat 0†". Zeitschrift für Mathematische

Logik und Grundlagen der Mathematik 36 (2): 133–141. doi:10.1002/malq.19900360206. ISSN 0044-3050.MR 1068949

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

123.3 External links• Definition by “Zentralblatt math database” (PDF)

148

Chapter 124

Łoś–Tarski preservation theorem

The Łoś–Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formulaspreserved under taking substructures is exactly the set of universal formulas (Hodges 1997).

124.1 Statement

Let T be a theory in a first-order language L and Φ(x) a set of formulas of L . (The set of sequence of variables xneed not be finite.) Then the following are equivalent:

1. If A andB are models of T , A ⊆ B , a is a sequence of elements of A andB |=∧Φ(a) , then A |=

∧Φ(a)

.( Φ is preserved in substructures for models of T )

2. Φ is equivalent modulo T to a set Ψ(x) of ∀1 formulas of L .

A formula is ∀1 if and only if it is of the form ∀x[ψ(x)] where ψ(x) is quantifier-free.Note that this property fails for finite models.

124.2 References• Peter G. Hinman (2005), Fundamentals of Mathematical Logic, A K Peters, ISBN 1568812620.

• Hodges (1997), A Shorter Model Theory, Cambridge University Press, ISBN 0521587131.

149

Chapter 125

Ω-logic

Not to be confused with ω-logic.

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part ofan attempt to generalize the theory of determinacy of pointclasses to cover the structure Hℵ2 . Just as the axiom ofprojective determinacy yields a canonical theory ofHℵ1 , he sought to find axioms that would give a canonical theoryfor the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis isfalse.Woodin’s Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most resultsin the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completenesstheorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive overHℵ2

(inΩ-logic), it must imply that the continuum is not ℵ1 . Woodin also isolated a specific axiom, a variation of Martin’smaximum, which states that any Ω-consistentΠ2 (overHℵ2

) sentence is true; this axiom implies that the continuumis ℵ2 .Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a “large cardinalproperty” to be a Σ2 property P (α) of ordinals which implies that α is a strong inaccessible, and which is invariantunder forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large modelscontaining a large cardinal, this fact will be provable in Ω-logic.The theory involves a definition of Ω-validity: A statement is an Ω-valid consequence of a set theory T if it holds inevery model of T having the form V B

α for some ordinal α and some forcing notion B . This notion is clearly preservedunder forcing, and in the presence of a proper class ofWoodin cardinals it will also be invariant under forcing (in otherwords, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability; here the “proofs”consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory,and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the “proof”,restricted its own reals. For a proof-setA the condition to be checked here is called "A-closed”. A complexity measurecan be given on the proofs by their ranks in the Wadge hierarchy. Woodin showed that this notion of “provability”implies Ω-validity for sentences which are Π2 over V. The Ω-conjecture states that the converse of this result alsoholds. In all currently known core models, it is known to be true; moreover the consistency strength of the largecardinals corresponds to the least proof-rank required to “prove” the existence of the cardinals.

125.1 References

• Bagaria, Joan; Castells, Neus; Larson, Paul (2006), “An Ω-logic primer”, Set theory, Trends Math., Basel,Boston, Berlin: Birkhäuser, pp. 1–28, ISBN 978-3-7643-7691-8, MR 2267144

• Dehornoy, Patrick (2004), “Progrès récents sur l'hypothèse du continu (d'après Woodin)", Astérisque (294):147–172, ISSN 0303-1179, MR 2111643

• Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walterde Gruyter, ISBN 3-11-015708-X, MR 1713438

150

125.2. EXTERNAL LINKS 151

• Woodin, W. Hugh (2001), “The continuum hypothesis. I”, Notices of the American Mathematical Society 48(6): 567–576, ISSN 0002-9920, MR 1834351

• Woodin, W. Hugh (2001b), “The Continuum Hypothesis, Part II”, Notices of the AMS 48 (7): 681–690

• Woodin, W. Hugh (2005), “The continuum hypothesis”, in Cori, Rene; Razborov, Alexander; Todorcevic,Stevo et al., Logic Colloquium 2000, Lect. Notes Log. 19, Urbana, IL: Assoc. Symbol. Logic, pp. 143–197,MR 2143878

125.2 External links• W. H. Woodin Slides for 3 talks

152 CHAPTER 125. Ω-LOGIC

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• Continuous function (set theory) Source: https://en.wikipedia.org/wiki/Continuous_function_(set_theory)?oldid=544970583 Contrib-utors: Tobias Bergemann, SmackBot, Jason22~enwiki, JRSpriggs, CBM, Julian Mendez and Imthegayestguy

• Continuum (set theory) Source: https://en.wikipedia.org/wiki/Continuum_(set_theory)?oldid=654402964 Contributors: Tobias Berge-mann, Clementi, Rgdboer, Nbarth, Daqu, CBM, Konradek, Paolo.dL, Jsondow, Oniongas, Dyaa, TheStrayCat, Solomonfromfinland,KLBot2 and Anonymous: 1

• Countryman line Source: https://en.wikipedia.org/wiki/Countryman_line?oldid=554330467Contributors: Hyacinth, Vivacissamamente,Jeodesic, Oleg Alexandrov, Rdore, CBM, David Eppstein, Boleyn, DOI bot, Kilom691, Citation bot, VladimirReshetnikov and Illia Con-nell

• Cyclic negation Source: https://en.wikipedia.org/wiki/Cyclic_negation?oldid=632663694 Contributors: Rich Farmbrough, Jfraser, Nor-texoid, Kuru, Albmont, David Eppstein, Paradoctor, Iamthedeus, Erik9bot, Disambigutron and Anonymous: 2

• Dense order Source: https://en.wikipedia.org/wiki/Dense_order?oldid=635100176 Contributors: EmilJ, Physicistjedi, MarSch, MichaelSlone, SmackBot, Imz, Melchoir, Turms, JAnDbot, VolkovBot, TXiKiBoT, Palnot, Addbot, ב ,.דניאל Pcap, Erik9bot, ZéroBot, HelpfulPixie Bot, Qetuth, Brirush and Anonymous: 6

• Diagonal intersection Source: https://en.wikipedia.org/wiki/Diagonal_intersection?oldid=542564744 Contributors: Charles Matthews,Melikamp, Porcher, The Giant Puffin, Amakuru, JRSpriggs, David Eppstein, Matthew Yeager, Addbot, 777sms, ZéroBot, ב ,.אנדרייLuizpuodzius and Makecat-bot

• Double recursion Source: https://en.wikipedia.org/wiki/Double_recursion?oldid=587235152 Contributors: Michael Hardy, Sligocki,CBM, Cydebot, DemocraticLuntz and Anonymous: 1

• Double turnstile Source: https://en.wikipedia.org/wiki/Double_turnstile?oldid=630094132Contributors: Paul A, Hyacinth, GPHemsley,Jason Quinn, Apokrif, DePiep, Dbmag9, Arthur Rubin, Javalenok, Leon..., Gregbard, Cydebot, David Eppstein, Plastikspork, Yobot,Tbvdm, LilHelpa, Xqbot, SporkBot, Paulmiko and Anonymous: 3

• Effective Polish space Source: https://en.wikipedia.org/wiki/Effective_Polish_space?oldid=626071484 Contributors: Tobias Berge-mann, CBM, Cydebot, RebelRobot, CBM2, Andrewbt and Helpful Pixie Bot

• Elementary definition Source: https://en.wikipedia.org/wiki/Elementary_definition?oldid=327373287 Contributors: Michael Hardy,AshtonBenson, Durova, Classicalecon, Hans Adler, Pcap and Erik9bot

154 CHAPTER 125. Ω-LOGIC

• Elementary diagram Source: https://en.wikipedia.org/wiki/Elementary_diagram?oldid=545907035Contributors: Michael Hardy, CBM,Addbot and Luckas-bot

• Elementary sentence Source: https://en.wikipedia.org/wiki/Elementary_sentence?oldid=316831275 Contributors: Michael Hardy andAshtonBenson

• Elementary theory Source: https://en.wikipedia.org/wiki/Elementary_theory?oldid=415833358 Contributors: AshtonBenson and Tku-vho

• End extension Source: https://en.wikipedia.org/wiki/End_extension?oldid=266956794 Contributors: Michael Hardy, Malcolma, Turmsand Ksbrown

• Equisatisfiability Source: https://en.wikipedia.org/wiki/Equisatisfiability?oldid=550551298Contributors: ObradovicGoran, Tizio, Greg-bard, David Eppstein, AnomieBOT, EmausBot and Anonymous: 4

• Erasure (logic) Source: https://en.wikipedia.org/wiki/Erasure_(logic)?oldid=481083354 Contributors: Graeme Bartlett, Sstrader, Er-icGPrud, SmackBot, Imaginationac, Gregbard, David Eppstein, AlexNewArtBot, Twinsday and Anonymous: 1

• Extension (predicate logic) Source: https://en.wikipedia.org/wiki/Extension_(predicate_logic)?oldid=502020968 Contributors: PaulAugust, Salix alba, SmackBot, Nbarth, AndrewWarden, Lambiam, CBM, Gregbard, Ktr101, RichardBergmair, Erik9bot, SD5bot andAK456

• Extensionality Source: https://en.wikipedia.org/wiki/Extensionality?oldid=640783551 Contributors: Charles Matthews, Hyacinth, Pop-ulus, Tobias Bergemann, Snobot, Caesura, Oleg Alexandrov, Linas, BD2412, Seliopou, Hairy Dude, SmackBot, Allixpeeke, Mhss, Blue-bot, Byelf2007, Pezant, CBM, Gregbard, David Eppstein, AlleborgoBot, OKBot, Classicalecon, Mild Bill Hiccup, Addbot, TaBOT-zerem, AnomieBOT, Samppi111, The Wiki ghost, D'ohBot, Andrew Cave, D.Lazard, Ihaveacatonmydesk and Anonymous: 10

• Finite character Source: https://en.wikipedia.org/wiki/Finite_character?oldid=544108057Contributors: Michael Hardy, CharlesMatthews,Aleph4, Giftlite, Paul August, Salix alba, YurikBot, Arthur Rubin, Judicael, Dreadstar, David Eppstein, KittyHawker, Addbot, Ptbot-gourou, 777sms and Anonymous: 1

• Fluent (artificial intelligence) Source: https://en.wikipedia.org/wiki/Fluent_(artificial_intelligence)?oldid=648244552Contributors: BD2412,Tizio, MarSch, CBM, Sdorrance, Yobot, Erik9bot, Hobsonlane, PhnomPencil and Qetuth

• Friedberg numbering Source: https://en.wikipedia.org/wiki/Friedberg_numbering?oldid=526274966Contributors: Michael Hardy, CBMand Kumioko

• Gabbay’s separation theorem Source: https://en.wikipedia.org/wiki/Gabbay’{}s_separation_theorem?oldid=675446270 Contributors:Michael Hardy, BlueNovember, SmackBot, Gregbard, David Eppstein, R'n'B, KCinDC, Yobot, DrilBot and Solomon7968

• Ground axiom Source: https://en.wikipedia.org/wiki/Ground_axiom?oldid=610751983Contributors: Rjwilmsi, R.e.b., Headbomb, DavidEppstein, Jesse V., BattyBot, DoctorKubla, Jochen Burghardt and Nynj

• Herbrand interpretation Source: https://en.wikipedia.org/wiki/Herbrand_interpretation?oldid=635276419Contributors: CALR, Linas,Zero sharp, CBM, Gregbard, JaGa, Cyborg1, Hans Adler, Addbot, Proofreader77, Waheedghumman, Omnipaedista, Blas3nik andAnonymous: 3

• Herbrand structure Source: https://en.wikipedia.org/wiki/Herbrand_structure?oldid=635276201 Contributors: Michael Hardy, Linas,Cronholm144, CBM, Ceilican, Addbot, SpBot, 777sms, The ubik and Anonymous: 2

• Heyting arithmetic Source: https://en.wikipedia.org/wiki/Heyting_arithmetic?oldid=635371404 Contributors: Chinju, Markhurd, Hy-acinth, Waltpohl, BD2412, SmackBot, CBM, Sdorrance, Gregbard, Cydebot, Magioladitis, R'n'B, Radagast3, Nnemo, El bot de la dieta,Addbot, Yobot, 9258fahsflkh917fas, LilHelpa, ZéroBot, Brirush and Anonymous: 6

• High (computability) Source: https://en.wikipedia.org/wiki/High_(computability)?oldid=653993997 Contributors: Wknight94, Smack-Bot, CBM, Cydebot, Althai, Quux0r, Birdstends and MarcelB612

• Hilbert–Bernays provability conditions Source: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions?oldid=569588494 Contributors: Tobias Bergemann, EmilJ, CBM, Yobot, Tkuvho and Anonymous: 1

• Honest leftmost branch Source: https://en.wikipedia.org/wiki/Honest_leftmost_branch?oldid=676093878Contributors: Zundark,MichaelHardy, TakuyaMurata, Andreas Kaufmann, MFH, Algebraist, David Eppstein, Cobi, COBot and Solomon7968

• Indiscernibles Source: https://en.wikipedia.org/wiki/Indiscernibles?oldid=635482675 Contributors: Karada, Charles Matthews, BenStandeven, Adamarthurryan, Dfass, Mets501, Zero sharp, JRSpriggs, CBM, Hans Adler, Erik9bot, Deltahedron and Brirush

• Institutional model theory Source: https://en.wikipedia.org/wiki/Institutional_model_theory?oldid=626256433 Contributors: Mdd,Tillmo, Arthur Rubin, SmackBot, Lambiam, CBM, Kyriosity, Tcamps42, Hans Adler, Jochen Burghardt and Anonymous: 1

• Jensen’s covering theorem Source: https://en.wikipedia.org/wiki/Jensen’{}s_covering_theorem?oldid=451291546Contributors: MichaelHardy, R.e.b., CBM, Kope, Citation bot, Xqbot, Omnipaedista, RjwilmsiBot and CitationCleanerBot

• Joint embedding property Source: https://en.wikipedia.org/wiki/Joint_embedding_property?oldid=490338555 Contributors: MichaelHardy, Kuratowski’s Ghost, Bender2k14, Hans Adler, Xx521xx and Anonymous: 1

• Judgment (mathematical logic) Source: https://en.wikipedia.org/wiki/Judgment_(mathematical_logic)?oldid=600394251 Contribu-tors: Michael Hardy, BD2412, Physis, Gregbard, Deadbeef, GoingBatty, JPaestpreornJeolhlna, PlaidPolarity and Anonymous: 2

• Kanamori–McAloon theorem Source: https://en.wikipedia.org/wiki/Kanamori%E2%80%93McAloon_theorem?oldid=569642732Con-tributors: Michael Hardy, Giftlite, Rjwilmsi, R.e.b., Sodin, CBM, Headbomb, David Eppstein, Yobot and Anonymous: 1

• Kleene–Rosser paradox Source: https://en.wikipedia.org/wiki/Kleene%E2%80%93Rosser_paradox?oldid=628611729 Contributors:William Avery, Michael Hardy, Hyacinth, Leonard G., Beland, Rich Farmbrough, Bender235, Mike Schwartz, Spug, Linas, Smack-Bot, CapitalSasha, Lambiam, Zero sharp, CBM, Falcor84, Daniel5Ko, Dorftrottel, Seanmclaughlin, Paradoctor, Neuralwarp, Marc vanLeeuwen, C. A. Russell, Yobot, Pcap, DmitriyZotikov, Jochen Burghardt and Anonymous: 12

• Knaster’s condition Source: https://en.wikipedia.org/wiki/Knaster’{}s_condition?oldid=632652335Contributors: Michael Hardy,Woohookitty,Myasuda, David Eppstein, SchreiberBike, Yaddie, Helpful Pixie Bot, Brad7777 and Mark viking

125.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 155

• Least fixed point Source: https://en.wikipedia.org/wiki/Least_fixed_point?oldid=640516965 Contributors: Michael Hardy, CharlesMatthews, Dcoetzee, Creidieki, Oleg Alexandrov, Jpbowen, Ott2, Mhss, CRGreathouse, Cydebot, Blaisorblade, Heyitspeter, AaronRotenberg, Addbot, SethFogarty, Tijfo098, Wcherowi, Jochen Burghardt and Anonymous: 6

• LEGO (proof assistant) Source: https://en.wikipedia.org/wiki/LEGO_(proof_assistant)?oldid=546222014 Contributors: Ruud Koot,Cydebot, David Eppstein and Addbot

• Lightface analytic game Source: https://en.wikipedia.org/wiki/Lightface_analytic_game?oldid=631352163 Contributors: Sgeo, Trova-tore, TechnoGuyRob, CBM, AnomieBOT, Erik9bot and Mark viking

• Limited principle of omniscience Source: https://en.wikipedia.org/wiki/Limited_principle_of_omniscience?oldid=523663730 Con-tributors: Michael Hardy, CBM and Anonymous: 1

• Lindström’s theorem Source: https://en.wikipedia.org/wiki/Lindstr%C3%B6m’{}s_theorem?oldid=612860185 Contributors: MichaelHardy, Dcoetzee, Giftlite, Chalst, Nortexoid, Gene Nygaard, Sebkha, Tillmo, Alynna Kasmira, SmackBot, RDBury, Turms, Zero sharp,CBM, Gregbard, MarshBot, David Eppstein, VanishedUserABC, Hans Adler, Addbot, Yobot, Pcap, DrilBot, ZéroBot, Tijfo098, HelpfulPixie Bot, BG19bot and Anonymous: 5

• Linked set Source: https://en.wikipedia.org/wiki/Linked_set?oldid=604894240 Contributors: Michael Hardy, David Eppstein, Yaddie,Helpful Pixie Bot, Brad7777 and Anonymous: 1

• LOGCFL Source: https://en.wikipedia.org/wiki/LOGCFL?oldid=607685060 Contributors: Tizio, Sbrools, SmackBot, Gelingvistoj,David Eppstein, JaGa, R'n'B, Jamelan, Addbot, Yobot, Twri, LucienBOT, RobinK and Anonymous: 1

• Logic for Computable Functions Source: https://en.wikipedia.org/wiki/Logic_for_Computable_Functions?oldid=602596773 Contrib-utors: Mav, Michael Hardy, Michaeln, Greenrd, Tobias Bergemann, Urhixidur, Leibniz, Ascánder, Chalst, Cwolfsheep, Daira Hopwood,Ruud Koot, Hairy Dude, Jpbowen, Gregbard, Cydebot, The Wild Falcon, Legobot, Pcap, Masssly and Anonymous: 3

• Logical assertion Source: https://en.wikipedia.org/wiki/Logical_assertion?oldid=648551905 Contributors: Docu, Silverfish, Dysprosia,Hyacinth, Jag123, DiegoMoya, Oleg Alexandrov, SeventyThree, SAE1962, BOT-Superzerocool, Mhss, Nbarth, Mets501, CBM,Addbot,Nate Wessel, ChuispastonBot, Masssly, Dwellee and Anonymous: 2

• Logical graph Source: https://en.wikipedia.org/wiki/Logical_graph?oldid=602607355 Contributors: Michael Hardy, Goethean, Giftlite,Pmanderson, Paul August, El C, Jeffrey O. Gustafson, Linas, DoubleBlue, Mathbot, RussBot, Trovatore, Closedmouth, C.Fred, Trebor,E946, Jon Awbrey, Mostlyharmless, FlyHigh, Wvbailey, JzG, Dbtfz, Antonielly, Slakr, Mets501, OS2Warp, CRGreathouse, CBM, GogoDodo, Hut 8.5, David Eppstein, Brigit Zilwaukee, Yolanda Zilwaukee, On This Continent, CardinalDan, The Tetrast, Seb26, Coffee,TFCforever, Islaammaged126, Hans Adler, Buchanan’s Navy Sec, Overstay, Marsboat, Unco Guid, Viva La Information Revolution!,Dave Chaparral, Poke Salat Annie, Flower Mound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, SoutheastPenna Poppa, Delaware Valley Girl, Addbot, Ichiboku Natabori, MrOllie, Queen of the Dishpan, Twri, Throw it in the Fire, C.W.Murry, Branzillo, Lantern Leatherhead, Gamewizard71, Igotta Lemma, Pangur Ban My Cat, A Thousand Dancing Hamsters, Massslyand Anonymous: 7

• Logicalmachine Source: https://en.wikipedia.org/wiki/Logical_machine?oldid=602607667Contributors: Rich Farmbrough, OlegAlexan-drov, Formativ, Bwpach, CBM, Alaibot, Emeraude, The Tetrast, Addbot, Eumolpo, PigFlu Oink, Masssly and Anonymous: 1

• Low (computability) Source: https://en.wikipedia.org/wiki/Low_(computability)?oldid=657249275 Contributors: Johndburger, Smack-Bot, Toughpigs, Cydebot, Althai, Quux0r, MarcelB612, MaximalIdeal, Deltahedron and Anonymous: 1

• Lowbasis theorem Source: https://en.wikipedia.org/wiki/Low_basis_theorem?oldid=665829033Contributors: Tobias Bergemann, OlegAlexandrov, Rjwilmsi, CBM, Cydebot, Althai, David Eppstein, DavidCBryant, Yobot, AnomieBOT, Deltahedron and Anonymous: 2

• Lusin’s separation theorem Source: https://en.wikipedia.org/wiki/Lusin’{}s_separation_theorem?oldid=616041706Contributors: MichaelHardy, Giftlite, R.e.b., Sodin, CBM, Magioladitis, Yoni, Daniele.tampieri, Helpful Pixie Bot and Anonymous: 1

• Material nonimplication Source: https://en.wikipedia.org/wiki/Material_nonimplication?oldid=662913831Contributors: Kaldari, BD2412,Kbdank71, Chris Capoccia, MacMog, SmackBot, Cybercobra, Bjankuloski06en~enwiki, Gregbard, Cydebot, David Eppstein, MauriceCarbonaro, Anzurio, Francvs, Classicalecon, BANZ111, Alex836, Watchduck, Addbot, Meisam, Luckas-bot, Yobot, FrescoBot, JesseV., EmausBot, Jontturi, Matthew Kastor and Anonymous: 5

• Maximal set Source: https://en.wikipedia.org/wiki/Maximal_set?oldid=638561052Contributors: Michael Hardy, Paul August, C S, Con-scious, SmackBot, Cronholm144, CRGreathouse, Cydebot, Alaibot, Jay1279, Fabrictramp, David Eppstein, Frank Stephan, JackSchmidt,JL-Bot, NClement, Citation bot, Citation bot 1, Trappist the monk, Tagib, Deltahedron, PlaidPolarity and Anonymous: 1

• Michael D. Morley Source: https://en.wikipedia.org/wiki/Michael_D._Morley?oldid=659227815 Contributors: Michael Hardy, Aleph4,Gene Nygaard, Alai, GregAsche, Jaxl, Caerwine, Bluebot, Mr. Lefty, AlsatianRain, Cydebot, Waacstats, Johnpacklambert, Feepflop,Hans Adler, Addbot, Yobot, Laubbaum, Omnipaedista, Joeylinpc, Suslindisambiguator and Anonymous: 3

• Milner–Rado paradox Source: https://en.wikipedia.org/wiki/Milner%E2%80%93Rado_paradox?oldid=618654006Contributors: MichaelHardy, R.e.b., CBM, Gregbard, Headbomb, David Eppstein, Kope, Allispaul, Yobot, Vilimlendvaj and Anonymous: 2

• Minimal logic Source: https://en.wikipedia.org/wiki/Minimal_logic?oldid=624352098 Contributors: Markhurd, EmilJ, Clconway, Cy-debot, A3nm, Addbot, DixonDBot, Hpvpp, Chricho, Jochen Burghardt and Anonymous: 1

• Omega-categorical theory Source: https://en.wikipedia.org/wiki/Omega-categorical_theory?oldid=538098728 Contributors: MichaelHardy, R.e.b., RDBury, CBM, VanishedUserABC, Hans Adler, False vacuum, Ultracoffee, BG19bot, Deltahedron and Anonymous: 1

• Ordinal logic Source: https://en.wikipedia.org/wiki/Ordinal_logic?oldid=490075488 Contributors: Michael Hardy, JRSpriggs, Greg-bard, VanishedUserABC, Tinton5, Helpful Pixie Bot, Brad7777 and Qetuth

• Paraconsistentmathematics Source: https://en.wikipedia.org/wiki/Paraconsistent_mathematics?oldid=602434485Contributors: JohnOwens,Charles Matthews, Henrygb, Eduardoporcher, Mcsee, Porcher, Joelr31, SmackBot, CBM, Gregbard, David Eppstein, Philg88, Embra-ceParadox, Aubreybardo and Anonymous: 3

• Polyadic algebra Source: https://en.wikipedia.org/wiki/Polyadic_algebra?oldid=572263580Contributors: Michael Hardy, Giftlite, Ezhiki,Mdd, Rjwilmsi, Pthag, CBM, Iohannes Animosus, Hugo Herbelin, Charvest, Tijfo098, Helpful Pixie Bot, Jochen Burghardt and Anony-mous: 1

156 CHAPTER 125. Ω-LOGIC

• Predicate logic Source: https://en.wikipedia.org/wiki/Predicate_logic?oldid=668159355 Contributors: Toby Bartels, Michael Hardy,Andres, Hyacinth, Robbot, MathMartin, Giftlite, Leonard G., Mindmatrix, Thekohser, Eubot, Chobot, Sharkface217, Jpbowen, Tomisti,SmackBot, Mhss, Cybercobra, Nakon, Byelf2007, Wvbailey, Bjankuloski06en~enwiki, George100, CBM, Gregbard, Naudefj, Thijs!bot,EdJohnston, JAnDbot, Hypergeek14, Stassa, Vanished user g454XxNpUVWvxzlr, Policron, Dessources, JohnBlackburne, AnonymousDissident, Gerakibot, Soler97, Kumioko, DesolateReality, Xiaq, ClueBot, Taxa, Djk3, TimClicks, Addbot, Jayde239, Yobot, AnomieBOT,Materialscientist, RandomDSdevel, ESSch, Keri, Logichulk, Xnn, EmausBot, WikitanvirBot, Mayur, ClueBot NG, Satellizer, ChesterMarkel, MerlIwBot, Helpful Pixie Bot, Virago250, Brad7777, Jochen Burghardt, BoltonSM3, Tomajohnson and Anonymous: 34

• Principle of distributivity Source: https://en.wikipedia.org/wiki/Principle_of_distributivity?oldid=582122861 Contributors: CharlesMatthews, Tobias Bergemann, Chalst, Oleg Alexandrov, Btyner, Rjwilmsi, Jb-adder, Shirahadasha, CBM, Gregbard, Cydebot, Erudecorp,Gzhanstong, LiederLover1982, AnomieBOT, Erik9bot, Gamewizard71, Eskilp, Mark viking and Anonymous: 6

• Proof compression Source: https://en.wikipedia.org/wiki/Proof_compression?oldid=628438102Contributors: Michael Hardy,Mr. Stradi-varius, Ceilican, LilHelpa, BG19bot, ChrisGualtieri, Mark viking and Ezequiel234

• Proofmining Source: https://en.wikipedia.org/wiki/Proof_mining?oldid=442037556Contributors: Michael Hardy, Jayme, CRGreathouse,CBM, Classicalecon, Pauloboliva, Unzerlegbarkeit, Yobot, Willy xD and Anonymous: 1

• Pseudo-order Source: https://en.wikipedia.org/wiki/Pseudo-order?oldid=537194190 Contributors: Toby Bartels, Cydebot, David Epp-stein, Unzerlegbarkeit, Yobot, DrilBot, EefeG0hi and Anonymous: 1

• Reduced product Source: https://en.wikipedia.org/wiki/Reduced_product?oldid=481254985 Contributors: Michael Hardy, Waltpohl,EmilJ, MarSch, R.e.b., Malcolma, MrShamrock, SmackBot, Popopp and Helpful Pixie Bot

• Redundant proof Source: https://en.wikipedia.org/wiki/Redundant_proof?oldid=607430635 Contributors: Michael Hardy, Bearcat,Boomur, D.Lazard, Ad Orientem, Mark viking and Ezequiel234

• Richardson’s theorem Source: https://en.wikipedia.org/wiki/Richardson’{}s_theorem?oldid=665153030 Contributors: Michael Hardy,Dominus, Giftlite, Jason Quinn, Profzoom, Jason Davies, Mandarax, R.e.b., Jfriedl, Spacepotato, RDBury, CRGreathouse, CBM, Greg-bard, Cydebot, Asmeurer, Laurusnobilis, Haseldon, Gaz v pol, Addbot, AndersBot, AnomieBOT, Citation bot, RedZiz~enwiki, Citationbot 1, Chricho, D.Lazard, Helpful Pixie Bot, Syedhanif86, Kondormari, Cyrapas and Anonymous: 12

• Robinson’s joint consistency theorem Source: https://en.wikipedia.org/wiki/Robinson’{}s_joint_consistency_theorem?oldid=671837343Contributors: Giftlite, Waltpohl, Tillmo, RDBury, CBM, Gregbard, DavidCBryant, Addbot, Amirobot, Stefan.vatev, Trappist the monkand Helpful Pixie Bot

• Scattered order Source: https://en.wikipedia.org/wiki/Scattered_order?oldid=568837076 Contributors: Michael Hardy, Aleph4, EmilJ,Rjwilmsi, CBM, Giggy, David Eppstein, BOTijo and CitationCleanerBot

• Semicomputable function Source: https://en.wikipedia.org/wiki/Semicomputable_function?oldid=465074972 Contributors: MichaelHardy, AtZeuS, Malcolma, SmackBot, CBM, Yobot, Erik9bot and Anonymous: 2

• Separating set Source: https://en.wikipedia.org/wiki/Separating_set?oldid=636544832Contributors: Michael Hardy, JitseNiesen,Math-Martin, Giftlite, Andreas Kaufmann, Salix alba, Silly rabbit, CBM, David Eppstein, Foxj, AnomieBOT, Schmittz and Anonymous: 1

• Set constraint Source: https://en.wikipedia.org/wiki/Set_constraint?oldid=657046623 Contributors: Qwertyus, Rjwilmsi, LogAntiLog,BG19bot, Jochen Burghardt and Monkbot

• Set function Source: https://en.wikipedia.org/wiki/Set_function?oldid=594486647 Contributors: Michael Hardy, Gabbe, Andres, RD-Bury, Kjetil1001, CBM, David Eppstein, Addbot, HRoestBot and Anonymous: 2

• Soft set Source: https://en.wikipedia.org/wiki/Soft_set?oldid=667807295 Contributors: Michael Hardy, Rjwilmsi, Alcides, SmackBot,YellowMonkey, RayAYang, JustAnotherJoe, Freelance Intellectual, David Eppstein, LokiClock, Kharal9, De728631, Addbot, Yobot,Xqbot, RjwilmsiBot, Matsievsky, StarryGrandma, Cerabot~enwiki, Atharkharal, Monkbot and Anonymous: 2

• Strength (mathematical logic) Source: https://en.wikipedia.org/wiki/Strength_(mathematical_logic)?oldid=545998336 Contributors:Gregbard, VanishedUserABC, Addbot, The Interior, Helpful Pixie Bot and Anonymous: 3

• Subcountability Source: https://en.wikipedia.org/wiki/Subcountability?oldid=582355341Contributors: CBM,Cydebot, Avytipat, R'n'B,Whatever511723421343, Yobot, LudovicoVan and Anonymous: 1

• Successor function Source: https://en.wikipedia.org/wiki/Successor_function?oldid=652938516 Contributors: Michael Hardy, Dori,Angela, Charles Matthews, Kaal, Dissident, Pearle, Diego Moya, Oleg Alexandrov, Ruud Koot, Triddle, BD2412, Salix alba, RussBot,Bigmantonyd, CmdrObot, CBM, Nick Number, David Eppstein, Tbvdm, VladikVP, Jochen Burghardt and Anonymous: 6

• Sudan function Source: https://en.wikipedia.org/wiki/Sudan_function?oldid=655501814 Contributors: Charles Matthews, Jerzy, Rameinstein, Factitious, Ben Standeven, Oleg Alexandrov, Hgkamath, Jwy, Ryszard Szopa~enwiki, Thijs!bot, Dfrg.msc, David Eppstein,Epsilon0, Addbot, Lightbot, Yobot, FrescoBot, CarrieVS and Anonymous: 7

• Supernatural number Source: https://en.wikipedia.org/wiki/Supernatural_number?oldid=673942905 Contributors: Michael Hardy,TakuyaMurata, Jason Quinn, Algebraist, Malcolma, CRGreathouse, CBM, Alaibot, JamesBWatson, Jéské Couriano, David Eppstein,Newbyguesses, DFRussia, Addbot, Luckas-bot, Omnipaedista, Aaqqqq, John of Reading, Deltahedron, Spectral sequence, Tall human,GeoffreyT2000, ScrapIronIV and Anonymous: 5

• Superposition calculus Source: https://en.wikipedia.org/wiki/Superposition_calculus?oldid=611708869 Contributors: Michael Hardy,Charles Matthews, Stephan Schulz, Tizio, Ojcit, CRGreathouse, CBM, Benea, Addbot, Luckas-bot, Wolfgang42, RichardMills65, Markviking and Anonymous: 5

• Switching circuit theory Source: https://en.wikipedia.org/wiki/Switching_circuit_theory?oldid=655468193 Contributors: Rpyle731,AJim, Wtshymanski, David Eppstein, Jim.henderson, AnomieBOT, Satellizer, DBigXray, Qetuth and Anonymous: 7

• Systems of Logic Based on Ordinals Source: https://en.wikipedia.org/wiki/Systems_of_Logic_Based_on_Ordinals?oldid=585202308Contributors: Michael Hardy, Aymanshamma, Bejnar, JRSpriggs, Gregbard, Olsonist, Daniel5Ko, Duncan.Hull, VanishedUserABC,Hairhorn, Helpful Pixie Bot, Brad7777, Qetuth, RudolfRed, Charleswfox and Anonymous: 1

• Takeuti’s conjecture Source: https://en.wikipedia.org/wiki/Takeuti’{}s_conjecture?oldid=627015510 Contributors: Michael Hardy,TakuyaMurata, Charles Matthews, Chalst, Mairi, CambridgeBayWeather, ArglebargleIV, Gregbard, David Eppstein, Cobi, Tradereddy,AlptaBot, Anne Bauval, Omnipaedista, FrescoBot, Proof Theorist and Anonymous: 3

125.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 157

• Tarski–Kuratowski algorithm Source: https://en.wikipedia.org/wiki/Tarski%E2%80%93Kuratowski_algorithm?oldid=551279029Con-tributors: Psychonaut, MathMartin, Gene Nygaard, NekoDaemon, Mets501, CRGreathouse, CBM, Cydebot, Yobot and VladimirReshet-nikov

• The Paradoxes of the Infinite Source: https://en.wikipedia.org/wiki/The_Paradoxes_of_the_Infinite?oldid=626151196 Contributors:Michael Hardy, Trovatore, SmackBot, RDBury, Trialsanderrors, Gregbard, Thenub314, KConWiki, David Eppstein, Paradoctor, Addbot,Zero Thrust, EmausBot, Apollineo!, Mogism, Jochen Burghardt and Mark viking

• Theory of pure equality Source: https://en.wikipedia.org/wiki/Theory_of_pure_equality?oldid=596388457 Contributors: BD2412,CBM and Myasuda

• Trichotomy (mathematics) Source: https://en.wikipedia.org/wiki/Trichotomy_(mathematics)?oldid=646849767Contributors: Zundark,Patrick, Michael Hardy, Arthur Frayn, Casu Marzu, Henrygb, UtherSRG, Tobias Bergemann, Macrakis, Paul August, Oleg Alexandrov,VKokielov, Michael Slone, Bota47, JJL, SmackBot, Mhss, Colonies Chris, Jdthood, Cybercobra, Courcelles, JRSpriggs, Meng.benjamin,Gregbard, Difluoroethene, DavidCherney, AntiVandalBot, Indeed123,Minnnnng, YohanN7,MilesAgain, Addbot, Luckas-bot, AnomieBOT,Götz, Xqbot, Constructive editor, ComputScientist, Þjóðólfr, Tkuvho, SporkBot, Paulmiko, Helpful Pixie Bot, Zorglub x and Anonymous:23

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domain Contributors: Own work Original artist: This vector image was created with Inkscape by Elembis, and then manually replaced.• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-

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• File:Pinocchio_paradox.png Source: https://upload.wikimedia.org/wikipedia/commons/2/2b/Pinocchio_paradox.png License: Publicdomain Contributors: File:Pinocchio paradox.jpg Original artist: Carlo Chiostri, User:Mbz1

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