Systems of set theoryFrom Wikipedia, the free encyclopedia

Contents

1 Ackermann set theory 11.1 The language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Relation to ZermeloFraenkel set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Ackermann set theory and Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Alternative set theory 32.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Constructive set theory 43.1 Intuitionistic ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1.1 Predicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Myhills constructive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Aczels constructive ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Interpretability in type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Interpretability in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Fuzzy set 74.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Fuzzy interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 Fuzzy relation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.6 Axiomatic denition of credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.7 Credibility inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.9 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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4.10 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 General set theory 145.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.5 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Internal set theory 176.1 Intuitive justication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.1.1 Principles of the standard predicate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2 Formal axioms for IST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.2.1 I: Idealisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2.2 S: Standardisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2.3 T: Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.3 Formal justication for the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 Related theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 KripkePlatek set theory 217.1 The axioms of KP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Proof that Cartesian products exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Admissible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 KripkePlatek set theory with urelements 238.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.2.1 Additional assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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9 MorseKelley set theory 259.1 MK axioms and ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9.2.1 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9.3 The axioms in Kelleys General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10 Naive set theory 3010.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.1.1 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.1.2 Cantors theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.1.3 Axiomatic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.1.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.1.5 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.2 Sets, membership and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2.1 Note on consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2.2 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2.4 Empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.3 Specifying sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.4 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.5 Universal sets and absolute complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.6 Unions, intersections, and relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.7 Ordered pairs and Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.8 Some important sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.9 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11 Near sets 3911.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.4 Efremovi proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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11.8 Descriptive -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.10Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.11Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.12Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 5111.13Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.17Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12 New Foundations 5812.1 The type theory TST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.2 Quinean set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

12.2.1 Axioms and stratication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2.2 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2.3 Admissibility of useful large sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

12.3 Finite axiomatizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.4 Cartesian closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.5 The consistency problem and related partial results . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.6 How NF(U) avoids the set-theoretic paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.7 The system ML (Mathematical Logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.8 Models of NFU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12.8.1 Self-suciency of mathematical foundations in NFU . . . . . . . . . . . . . . . . . . . . 6212.8.2 Facts about the automorphism j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.9 Strong axioms of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13 Non-well-founded set theory 6713.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14 On Numbers and Games 70

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14.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

15 Pocket set theory 7215.1 Arguments supporting PST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.2 The theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.3 Remarks on the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.4 Some PST theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.5 Possible extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

16 Positive set theory 7516.1 Interesting properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.2 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

17 Rough set 7717.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

17.1.1 Information system framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7717.1.2 Example: equivalence-class structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7717.1.3 Denition of a rough set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7817.1.4 Denability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7917.1.5 Reduct and core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017.1.6 Attribute dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

17.2 Rule extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.2.1 Decision matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.2.2 LERS rule induction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

17.3 Incomplete data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.6 Extensions and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.6.1 Rough membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.6.2 Other generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8817.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

18 S (set theory) 8918.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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18.2 Primitive notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

19 ScottPotter set theory 9219.1 ZU etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

19.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.1.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.1.3 Further existence premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

19.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.2.1 Scotts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.2.2 Potters theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

19.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

20 Semiset 9720.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

21 TarskiGrothendieck set theory 9821.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9821.2 Implementation in the Mizar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3 Implementation in Metamath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

22 Universal set 10122.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

22.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10122.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

22.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10122.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10222.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

22.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10222.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10222.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

23 Vague set 10423.1 Mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

CONTENTS vii

23.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10423.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

24 Von NeumannBernaysGdel set theory 10524.1 Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10524.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10524.3 Axiomatizating NBG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

24.3.1 With Class Comprehension schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10724.3.2 Replacing Class Comprehension with nite instances thereof . . . . . . . . . . . . . . . . 108

24.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10924.4.1 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11024.4.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

24.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11024.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11024.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11124.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

25 Zermelo set theory 11325.1 The axioms of Zermelo set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11325.2 Connection with standard set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11325.3 The aim of Zermelos paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.4 The axiom of separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.5 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11525.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11525.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

26 ZermeloFraenkel set theory 11626.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

26.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 11726.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11826.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11826.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11926.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12026.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12026.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

26.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12126.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

26.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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26.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12326.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12326.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12426.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 125

26.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12526.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12726.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter 1

Ackermann set theory

Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.

1.1 The languageAckermann set theory is formulated in rst-order logic. The language LA consists of one binary relation 2 and oneconstant V (Ackermann used a predicate M instead). We will write x 2 y for 2 (x; y) . The intended interpretationof x 2 y is that the object x is in the class y . The intended interpretation of V is the class of all sets.

1.2 The axiomsThe axioms of Ackermann set theory, collectively referred to as A, consists of the universal closure of the followingformulas in the language LA1) Axiom of extensionality:

8x8y(8z(z 2 x$ z 2 y)! x = y):

2) Class construction axiom schema: Let F (y; z1; : : : ; zn) be any formula which does not contain the variable x free.

8y(F (y; z1; : : : ; zn)! y 2 V )! 9x8y(y 2 x$ F (y; z1; : : : ; zn))

3) Reection axiom schema: Let F (y; z1; : : : ; zn) be any formula which does not contain the constant symbol V orthe variable x free. If z1; : : : ; zn 2 V then

8y(F (y; z1; : : : ; zn)! y 2 V )! 9x(x 2 V ^ 8y(y 2 x$ F (y; z1; : : : ; zn))):

4) Completeness axioms for V

x 2 y ^ y 2 V ! x 2 V

x y ^ y 2 V ! x 2 V:5) Axiom of regularity for sets:

x 2 V ^ 9y(y 2 x)! 9y(y 2 x ^ :9z(z 2 y ^ z 2 x)):

1

2 CHAPTER 1. ACKERMANN SET THEORY

1.3 Relation to ZermeloFraenkel set theoryLet F be a rst-order formula in the language L2 = f2g (so F does not contain the constant V ). Dene therestriction of F to the universe of sets (denoted FV ) to be the formula which is obtained by recursively replacingall sub-formulas of F of the form 8xG(x; y1 : : : ; yn) with 8x(x 2 V ! G(x; y1 : : : ; yn)) and all sub-formulas ofthe form 9xG(x; y1 : : : ; yn) with 9x(x 2 V ^G(x; y1 : : : ; yn)) .In 1959 Azriel Levy proved that if F is a formula of L2 and A proves FV , then ZF proves FIn 1970 William Reinhardt proved that if F is a formula of L2 and ZF proves F , then A proves FV .

1.4 Ackermann set theory and Category theoryThe most remarkable feature of Ackermann set theory is that, unlike Von NeumannBernaysGdel set theory, aproper class can be an element of another proper class (see Fraenkel, Bar-Hillel, Levy(1973), p. 153).An extension (named ARC) of Ackermann set theory was developed by F.A. Muller(2001), who stated that ARCfounds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole ofmathematics.

1.5 See also Zermelo set theory

1.6 References Ackermann, Wilhelm Zur Axiomatik der Mengenlehre in Mathematische Annalen, 1956, Vol. 131, pp.

336-345. Levy, Azriel, On Ackermanns set theory Journal of Symbolic Logic Vol. 24, 1959 154-166 Reinhardt, William, Ackermanns set theory equals ZF Annals of Mathematical Logic Vol. 2, 1970 no. 2,

189-249 A.A.Fraenkel, Y. Bar-Hillel, A.Levy, 1973. Foundations of Set Theory, second edition, North-Holand, 1973. F.A. Muller, Sets, Classes, and Categories British Journal for the Philosophy of Science 52 (2001) 539-573.

Chapter 2

Alternative set theory

Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposedalternative to the standard set theory.Some of the alternative set theories are:

the theory of semisets the set theory New Foundations Positive set theory Internal set theory

Specically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s byPetr Vopnka and his students. It builds on some ideas of the theory of semisets, but also introduces more radicalchanges: for example, all sets are formally nite, which means that sets in AST satisfy the law of mathematicalinduction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalentto the ZermeloFraenkel (or ZF) set theory, in which the axiom of innity is replaced by its negation). However,some of these sets contain subclasses that are not sets, which makes them dierent from Cantor (ZF) nite sets andthey are called innite in AST.

2.1 See also Non-well-founded set theory

2.2 References Vopnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979. Proceedings of the 1st Symposium Mathematics in the Alternative Set Theory. JSMF, Bratislava, 1989. Holmes, Randall M. Alternative Axiomatic Set Theories in the Stanford Encyclopedia of Philosophy.

3

Chapter 3

Constructive set theory

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic settheory. That is, it uses the usual rst-order language of classical set theory, and although of course the logic isconstructive, there is no explicit use of constructive types. Rather, there are just sets, thus it can look very much likeclassical mathematics done on the most common foundations, namely the ZermeloFraenkel axioms (ZFC).

3.1 Intuitionistic ZermeloFraenkel

In 1973, John Myhill proposed a system of set theory based on intuitionistic logic[1] taking the most common foun-dation, ZFC, and throwing away the axiom of choice (AC) and the law of the excluded middle (LEM), leavingeverything else as is. However, dierent forms of some of the ZFC axioms which are equivalent in the classicalsetting are inequivalent in the constructive setting, and some forms imply LEM.The system, which has come to be known as IZF, or Intuitionistic ZermeloFraenkel (ZF refers to ZFC without theaxiom of choice), has the usual axioms of extensionality, pairing, union, innity, separation and power set. The axiomof regularity is stated in the form of an axiom schema of set induction. Also, while Myhill used the axiom schemaof replacement in his system, IZF usually stands for the version with collectionWhile the axiom of replacement requires the relation to be a function over the set A (that is, for every x in A thereis associated exactly one y), the axiom of collection does not: it merely requires there be associated at least one y,and it asserts the existence of a set which collects at least one such y for each such x. The axiom of regularity asit is normally stated implies LEM, whereas the form of set induction does not. The formal statements of these twoschemata are:8A ([8x 2 A 9y (x; y)]! 9B 8x 2 A 9y 2 B (x; y))[8y ([8x 2 y (x)]! (y))]! 8y (y)Adding LEM back to IZF results in ZF, as LEM makes collection equivalent to replacement and set induction equiv-alent to regularity. Even without LEM, IZFs proof-theoretical power equals that of ZF.

3.1.1 Predicativity

While IZF is based on constructive rather than classical logic, it is considered impredicative. It allows formationof sets using the axiom of separation with any proposition, including ones which contain quantiers which are notbounded. Thus new sets can be formed in terms of the universe of all sets. Additionally the power set axiom impliesthe existence of a set of truth values. In the presence of LEM, this set exists and has two elements. In the absence ofit, the set of truth values is also considered impredicative.

4

3.2. MYHILLS CONSTRUCTIVE SET THEORY 5

3.2 Myhills constructive set theoryThe subject was begun by John Myhill to provide a formal foundation for Errett Bishop's program of constructivemathematics. As he presented it, Myhills system CST is a constructive rst-order logic with three sorts: naturalnumbers, functions, and sets. The system is:

Constructive rst-order predicate logic with identity, and basic axioms related to the three sorts. The usual Peano axioms for natural numbers. The usual axiom of extensionality for sets, as well as one for functions, and the usual axiom of union. A form of the axiom of innity asserting that the collection of natural numbers (for which he introduces a

constant N) is in fact a set.

Axioms asserting that the domain and range of a function are both sets. Additionally, an axiom of non-choiceasserts the existence of a choice function in cases where the choice is already made. Together these act likethe usual replacement axiom in classical set theory.

The axiom of exponentiation, asserting that for any two sets, there is a third set which contains all (and only)the functions whose domain is the rst set, and whose range is the second set. This is a greatly weakened formof the axiom of power set in classical set theory, to which Myhill, among others, objected on the grounds ofits impredicativity.

The axiom of restricted, or predicative, separation, which is a weakened form of the separation axiom inclassical set theory, requiring that any quantications be bounded to another set.

An axiom of dependent choice, which is much weaker than the usual axiom of choice.

3.3 Aczels constructive ZermeloFraenkelPeter Aczel's constructive Zermelo-Fraenkel,[2] or CZF, is essentially IZF with its impredicative features removed.It strengthens the collection scheme, and then drops the impredicative power set axiom and replaces it with anothercollection scheme. Finally the separation axiom is restricted, as in Myhills CST. This theory has a relatively simpleinterpretation in a version of constructive type theory and has modest proof theoretic strength as well as a fairly directconstructive and predicative justication, while retaining the language of set theory. Adding LEM to this theory alsorecovers full ZF.The collection axioms are:Strong collection schema: This is the constructive replacement for the axiom schema of replacement. It states thatif is a binary relation between sets which is total over a certain domain set (that is, it has at least one image of everyelement in the domain), then there exists a set which contains at least one image under of every element of thedomain, and only images of elements of the domain. Formally, for any formula :8a((8x 2 a 9y (x; y))! 9b (8x 2 a 9y 2 b (x; y)) ^ (8y 2 b 9x 2 a (x; y)))Subset collection schema: This is the constructive version of the power set axiom. Formally, for any formula :8a; b 9u 8z((8x 2 a 9y 2 b (x; y; z))! 9v 2 u (8x 2 a 9y 2 v (x; y; z)) ^ (8y 2 v 9x 2 a (x; y; z)))This is equivalent to a single and somewhat clearer axiom of fullness: between any two sets a and b, there is a set cwhich contains a total subrelation of any total relation between a and b that can be encoded as a set of ordered pairs.Formally:8a; b 9c P (a; b) 8R 2 P (a; b) 9S 2 c S Rwhere the references to P(a,b) are dened by:R 2 P (a; b) () (8u 2 R 9x 2 a 9y 2 b hx; yi = u) ^ (8x 2 a 9y 2 b hx; yi 2 R)c P (a; b) () 8R 2 c R 2 P (a; b)and some set-encoding of the ordered pair is assumed.

6 CHAPTER 3. CONSTRUCTIVE SET THEORY

The axiom of fullness implies CSTs axiom of exponentiation: given two sets, the collection of all total functionsfrom one to the other is also in fact a set.The remaining axioms of CZF are: the axioms of extensionality, pairing, union, and innity are the same as in ZF;and set induction and predicative separation are the same as above.

3.4 Interpretability in type theoryIn 1977 Aczel showed that CZF can be interpreted in Martin-Lf type theory,[3] (using the now consecrated propositions-as-types approach) providing what is now seen a standard model of CZF in type theory.[4] In 1989 Ingrid Lindstrmshowed that non-well-founded sets obtained by replacing the axiom of foundation in CZF with Aczels anti-foundationaxiom (CZFA) can also be interpreted in Martin-Lf type theory.[5]

3.5 Interpretability in category theoryPresheaf models for constructive set theory were introduced by Nicola Gambino in 2004. They are analogous to thePresheaf models for intuitionistic set theory developed by Dana Scott in the 1980s (which remained unpublished).[6][7]

3.6 References[1] Myhill, Some properties of Intuitionistic Zermelo-Fraenkel set theory, Proceedings of the 1971 Cambridge Summer School

in Mathematical Logic (Lecture Notes in Mathematics 337) (1973) pp 206-231[2] Peter Aczel and Michael Rathjen, Notes on Constructive Set Theory, Reports Institut Mittag-Leer, Mathematical Logic -

2000/2001, No. 40[3] Aczel, Peter: 1978. The type theoretic interpretation of constructive set theory. In: A. MacIntyre et al. (eds.), Logic

Colloquium '77, Amsterdam: North-Holland, 5566.[4] Rathjen, M. (2004), Predicativity, Circularity, and Anti-Foundation, in Link, Godehard, One Hundred Years of Russell

s Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, ISBN 978-3-11-019968-0[5] Lindstrm, Ingrid: 1989. A construction of non-well-founded sets within Martin-Lf type theory. Journal of Symbolic

Logic 54: 5764.[6] Gambino, N. (2005). PRESHEAF MODELS FOR CONSTRUCTIVE SET THEORIES. In Laura Crosilla and Peter

Schuster. From Sets and Types to Topology and Analysis. pp. 6296. doi:10.1093/acprof:oso/9780198566519.003.0004.ISBN 9780198566519.

[7] Scott, D. S. (1985). Category-theoretic models for Intuitionistic Set Theory. Manuscript slides of a talk given at Carnegie-Mellon University

3.7 Further reading Troelstra, Anne; van Dalen, Dirk (1988). Constructivism in Mathematics, Vol. 2. Studies in Logic and the

Foundations of Mathematics. p. 619. ISBN 0-444-70358-6. Aczel, P. and Rathjen, M. (2001). Notes on constructive set theory. Technical Report 40, 2000/2001. Mittag-

Leer Institute, Sweden.

3.8 External links Laura Crosilla, Set Theory: Constructive and Intuitionistic ZF, Stanford Encyclopedia of Philosophy, Feb 20,

2009 Benno van den Berg, Constructive set theory an overview, slides from Heyting dag, Amsterdam, 7 September

2012

Chapter 4

Fuzzy set

In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced byLot A. Zadeh[1] and Dieter Klaua[2] in 1965 as an extension of the classical notion of set. At the same time, Salii(1965) dened a more general kind of structures called L-relations, which he studied in an abstract algebraic context.Fuzzy relations, which are used now in dierent areas, such as linguistics (De Cock, et al., 2000), decision-making(Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1].In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessmentof the membership of elements in a set; this is described with the aid of a membership function valued in the real unitinterval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases ofthe membership functions of fuzzy sets, if the latter only take values 0 or 1.[3] In fuzzy set theory, classical bivalentsets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which informationis incomplete or imprecise, such as bioinformatics.[4]

It has been suggested by Thayer Watkins that Zadehs ethnicity is an example of a fuzzy set because His father wasTurkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijanin the Soviet Union...Lot was born in Baku in 1921 and lived there until his family moved to Tehran in 1931.[5]

4.1 DenitionA fuzzy set is a pair (U;m) where U is a set and m : U ! [0; 1]:For each x 2 U; the valuem(x) is called the grade of membership of x in (U;m): For a nite setU = fx1; : : : ; xng;the fuzzy set (U;m) is often denoted by fm(x1)/x1; : : : ;m(xn)/xng:Let x 2 U: Then x is called not included in the fuzzy set (U;m) if m(x) = 0 , x is called fully included ifm(x) = 1 , and x is called a fuzzy member if 0 < m(x) < 1 .[6] The set fx 2 U j m(x) > 0g is called thesupport of (U;m) and the set fx 2 U j m(x) = 1g is called its kernel or core. The function m is called themembership function of the fuzzy set (U;m):Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a(xed or variable) algebra or structureL of a given kind; usually it is required thatL be at least a poset or lattice. Theseare usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membershipfunctions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizationswere rst considered in 1967 by Joseph Goguen, who was a student of Zadeh.[7]

4.2 Fuzzy logicMain article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations ( : Vo ! W ) of propositional variables ( Vo ) into aset of membership degrees ( W ) can be thought of as membership functions mapping predicates into fuzzy sets (or

7

8 CHAPTER 4. FUZZY SET

more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logiccan be extended to allow for fuzzy premises from which graded conclusions may be drawn.[8]

This extension is sometimes called fuzzy logic in the narrow sense as opposed to fuzzy logic in the wider sense,which originated in the engineering elds of automated control and knowledge engineering, and which encompassesmany topics involving fuzzy sets and approximated reasoning.[9]

Industrial applications of fuzzy sets in the context of fuzzy logic in the wider sense can be found at fuzzy logic.

4.3 Fuzzy numberMain article: Fuzzy number

A fuzzy number is a convex, normalized fuzzy set ~A R whose membership function is at least segmentallycontinuous and has the functional value A(x) = 1 at precisely one element.This can be likened to the funfair game guess your weight, where someone guesses the contestants weight, withcloser guesses being more correct, and where the guesser wins if he or she guesses near enough to the contestantsweight, with the actual weight being completely correct (mapping to 1 by the membership function).

4.4 Fuzzy intervalA fuzzy interval is an uncertain set ~A R with a mean interval whose elements possess the membership functionvalue A(x) = 1 . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentallycontinuous.[10]

4.5 Fuzzy relation equationThe fuzzy relation equation is an equation of the form A R = B, where A and B are fuzzy sets, R is a fuzzy relation,and A R stands for the composition of A with R.

4.6 Axiomatic denition of credibility[11] Let A be a non-empty set and P(A) be the power set of A . The set function Cr is known as credibility measureif it satises following condition

Axiom 1: CrfAg = 1

Axiom 2: If B is subset of C, then, CrfBg CrfCg

Axiom 3: CrfBg+ CrfBcg = 1

Axiom 4: Crf[Aig = supi(Cr(Ai)) , for any event Ai with supi CrfAig < 0:5

Cr{B} indicates how frequently event B will occur.

4.7 Credibility inversion theorem[12] Let A be a fuzzy variable with membership function u. Then for any set B of real numbers, we have

CrfA 2 Bg = 12

supt2B

u(t) + 1 supt2Bc

u(t)

4.8. EXPECTED VALUE 9

4.8 Expected Value[13] Let A be a fuzzy variable. Then the expected value is

E[A] =

Z 10

CrfA tg dtZ 01

CrfA tg dt:

4.9 Entropy[14] Let A be a fuzzy variable with a continuous membership function. Then its entropy is

H[A] =

Z 11

S(CrfA tg) dt:

Where

S(y) = y lny (1 y) ln(1 y)

4.10 GeneralizationsThere are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were intro-duced in 1965, a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity,and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory,while others try to mathematically model imprecision and uncertainty in a dierent way (Burgin and Chunihin, 1997;Kerre, 2001; Deschrijver and Kerre, 2003).The diversity of such constructions and corresponding theories includes:

interval sets (Moore, 1966), L-fuzzy sets (Goguen, 1967), ou sets (Gentilhomme, 1968), Boolean-valued fuzzy sets (Brown, 1971), type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975), set-valued sets (Chapin, 1974; 1975), interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975), functions as generalizations of fuzzy sets and multisets (Lake, 1976), level fuzzy sets (Radecki, 1977) underdetermined sets (Narinyani, 1980), rough sets (Pawlak, 1982), intuitionistic fuzzy sets (Atanassov, 1983), fuzzy multisets (Yager, 1986), intuitionistic L-fuzzy sets (Atanassov, 1986), rough multisets (Grzymala-Busse, 1987),

10 CHAPTER 4. FUZZY SET

fuzzy rough sets (Nakamura, 1988), real-valued fuzzy sets (Blizard, 1989), vague sets (Wen-Lung Gau and Buehrer, 1993), Q-sets (Gylys, 1994) shadowed sets (Pedrycz, 1998), -level sets (Yao, 1997), genuine sets (Demirci, 1999), soft sets (Molodtsov, 1999), intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003) blurry sets (Smith, 2004) L-fuzzy rough sets (Radzikowska and Kerre, 2004), generalized rough fuzzy sets (Feng, 2010) rough intuitionistic fuzzy sets (Thomas and Nair, 2011), soft rough fuzzy sets (Meng, Zhang and Qin, 2011) soft fuzzy rough sets (Meng, Zhang and Qin, 2011) soft multisets (Alkhazaleh, Salleh and Hassan, 2011) fuzzy soft multisets (Alkhazaleh and Salleh, 2012)

4.11 See also Alternative set theory Defuzzication Fuzzy concept Fuzzy mathematics Fuzzy measure theory Fuzzy set operations Fuzzy subalgebra Linear partial information Neuro-fuzzy Rough fuzzy hybridization Rough set Srensen similarity index Type-2 Fuzzy Sets and Systems Uncertainty Interval nite element Multiset

4.12. REFERENCES 11

4.12 References[1] L. A. Zadeh (1965) Fuzzy sets. Information and Control 8 (3) 338353.

[2] Klaua, D. (1965) ber einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859876.A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). An early approach toward graded iden-tity and graded membership in set theory. Fuzzy Sets and Systems 161 (18): 23692379. doi:10.1016/j.fss.2009.12.005.

[3] D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.

[4] Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, FM-test: AFuzzy-Set-Theory-Based Approach to Dierential Gene Expression Data Analysis, BMC Bioinformatics, 7 (Suppl 4):S7. 2006.

[5] Fuzzy Logic: The Logic of Fuzzy Sets

[6] AAAI

[7] Goguen, Joseph A., 196, "L-fuzzy sets. Journal of Mathematical Analysis and Applications 18: 145174

[8] Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies PressLtd., ISBN 978-0-86380-262-1

[9] The concept of a linguistic variable and its application to approximate reasoning, Information Sciences 8: 199249,301357; 9: 4380.

[10] Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1: 328

[11] Liu, Baoding. Uncertain theory: an introduction to its axiomatic foundations. Berlin: Springer-Verlag (2004).

[12] Liu, Baoding, and Yian-Kui Liu. Expected value of fuzzy variable and fuzzy expected value models. Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[13] Liu, Baoding, and Yian-Kui Liu. Expected value of fuzzy variable and fuzzy expected value models. Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[14] Xuecheng, Liu. Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy sets and systems52.3 (1992): 305-318.

4.13 Further reading Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID

350600, 20 p.

Alkhazaleh, S., Salleh, A.R. and Hassan, N. Soft Multisets Theory, Applied Mathematical Sciences, v. 5, No.72, 2011, pp. 35613573

Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKRs Session, Soa (deposited in Central Sci.-TechnicalLibrary of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)

Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 8796 Bezdek, J.C. (1978) Fuzzy partitions and relations and axiomatic basis for clustering, Fuzzy Sets and Systems,

v.1, pp. 111127

Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 7797 Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 3239 Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619634 Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255267 Chris Cornelis, Martine De Cock and Etienne E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads of

imperfect knowledge, Expert Systems, v. 20, issue 5, pp. 260270, 2003

12 CHAPTER 4. FUZZY SET

Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy settheory: construction, classication, application, International Journal of Approximate Reasoning, v. 35, pp.5595

Martine De Cock, Ulrich Bodenhofer, and Etienne E. Kerre, Modelling Linguistic Expressions Using FuzzyRelations, (2000) Proceedings 6th International Conference on Soft Computing. Iizuka 2000, Iizuka, Japan(14 October 2000) CDROM. p. 353-360

Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377384 Deschrijver, G. and Kerre, E.E. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets

and Systems, v. 133, no. 2, pp. 227235, 2003 Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series7. Springer. ISBN 978-0-7923-7732-0.

Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9,pp 899911

Gentilhomme, Y. (1968) Les ensembles ous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5,pp. 4763

Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145174 Gottwald, S. (2006). Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based

and Axiomatic Approaches. Studia Logica 82 (2): 211244. doi:10.1007/s11225-006-7197-8.. Gottwald,S. (2006). Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category TheoreticApproaches. Studia Logica 84: 2350. doi:10.1007/s11225-006-9001-1. preprint..

Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math.Logik. Grundladen Math. 22, pp. 149160.

Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd InternationalSymposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325332

Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 931. Ulrich Hhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measure

theory. The Handbooks of Fuzzy Sets Series 3. Springer. ISBN 978-0-7923-8388-8. Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115132 Kerre, E.E. A rst view on the alternatives of fuzzy set theory, Computational Intelligence in Theory and

Practice (B. Reusch, K-H . Temme, eds) Physica-Verlag, Heidelberg (ISBN 3-7908-1357-5), 2001, pp. 5572

George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN978-0-13-101171-7.

Kuzmin,V.B. Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations, Nauka, Moscow, 1982(in Russian)

Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323326 Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematics

with Applications, v. 62, issue 12, 2011, pp. 46354645 Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225235,

2001 Molodtsov, O. (1999) Soft set theory rst results, Computers & Mathematics with Applications, v. 37, No.

4/5, pp. 1931 Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966 Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 18

4.14. EXTERNAL LINKS 13

Narinyani, A.S. Underdetermined Sets A new datatype for knowledge representation, Preprint 232, ProjectVOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980

Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, andCybernetics, Part B, 28, 103-109, 1998.

Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics, Volume 7, Issue 3-4, 1977 Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Articial Intelligence and Soft

Computing - ICAISC 2004, 7th International Conference, Zakopane, Poland, June 711, 2004, Proceedings;01/2004

Salii, V.N. (1965) Binary L-relations, Izv. Vysh. Uchebn. Zaved., Matematika, v. 44, No.1, pp. 133145 (inRussian)

Sambuc, R. Fonctions -oues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. ThesisUniv. Marseille, France, 1975.

Seising, Rudolf: The Fuzzication of Systems. The Genesis of Fuzzy Set Theory and Its Initial ApplicationsDevelopments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]:Springer 2007.

Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165235 Thomas, K.V. and L. S. Nair, Rough intuitionistic fuzzy sets in a lattice, 'International Mathematical Forum',

Vol. 6, 2011, no. 27, 1327 - 1335

Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 2337 Yao, Y.Y., Combination of rough and fuzzy sets based on -level sets, in: Rough Sets and Data Mining:

Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp.301321, 1997.

Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1-4, 1998,pp. 227 242

Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoningI, Inform.Sci., v. 8, pp. 199249

Hans-Jrgen Zimmermann (2001). Fuzzy set theoryand its applications (4th ed.). Kluwer. ISBN 978-0-7923-7435-0.

Gianpiero Cattaneo and Davide Ciucci, Heyting Wajsberg Algebras as an Abstract Environment LinkingFuzzy and Rough Sets in J.J. Alpigini et al. (Eds.): RSCTC 2002, LNAI 2475, pp. 7784, 2002. doi:10.1007/3-540-45813-1_10

4.14 External links Uncertainty model Fuzziness Fuzzy Systems Journal ScholarPedia The Algorithm of Fuzzy Analysis Fuzzy Image Processing

Chapter 5

General set theory

General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST issucient for all mathematics not requiring innite sets, and is the weakest known set theory whose theorems includethe Peano axioms.

5.1 OntologyThe ontology of GST is identical to that of ZFC, and hence is thoroughly canonical. GST features a single primitiveontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe ofdiscourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; thatset a is a member of set b is written ab (usually read "a is an element of b").

5.2 AxiomsThe symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. The naturallanguage versions of the axioms are intended to aid the intuition. The background logic is rst order logic withidentity.1) Axiom of Extensionality: The sets x and y are the same set if they have the same members.

8x8y[8z[z 2 x$ z 2 y]! x = y]:The converse of this axiom follows from the substitution property of equality.2) Axiom Schema of Specication (or Separation or Restricted Comprehension): If z is a set and is any propertywhich may be satised by all, some, or no elements of z, then there exists a subset y of z containing just those elementsx in z which satisfy the property . The restriction to z is necessary to avoid Russells paradox and its variants. Moreformally, let (x)be any formula in the language of GST in which x is free and y is not. Then all instances of thefollowing schema are axioms:

8z9y8x[x 2 y $ (x 2 z ^ (x))]:3) Axiom of Adjunction: If x and y are sets, then there exists a set w, the adjunction of x and y, whose members arejust y and the members of x.[1]

8x8y9w8z[z 2 w $ (z 2 x _ z = y)]:Adjunction refers to an elementary operation on two sets, and has no bearing on the use of that term elsewhere inmathematics, including in category theory.

14

5.3. DISCUSSION 15

5.3 DiscussionGST is the fragment of Z obtained by omitting the axioms Union, Power Set, Innity, and Choice, then takingAdjunction, a theorem of Z, as an axiom. The result is a rst order theory.Setting (x) in Separation to xx, and assuming that the domain is nonempty, assures the existence of the emptyset. Adjunction implies that if x is a set, then so is S(x) = x [ fxg . Given Adjunction, the usual construc-tion of the successor ordinals from the empty set can proceed, one in which the natural numbers are dened as?; S(?); S(S(?)); : : : ; (see Peanos axioms). More generally, given any model M of ZFC, the collection ofhereditarily nite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence of even acountable innite set, that is, of a set whose cardinality is 0. Even if GST did aord a countably innite set, GSTcould not prove the existence of a set whose cardinality is @1 , because GST lacks the axiom of power set. HenceGST cannot ground analysis and geometry, and is too weak to serve as a foundation for mathematics.Boolos was interested in GST only as a fragment of Z that is just powerful enough to interpret Peano arithmetic. Henever lingered over GST, only mentioning it briey in several papers discussing the systems of Frege's Grundlagenand Grundgesetze, and how they could be modied to eliminate Russells paradox. The system A'[0] in Tarskiand Givant (1987: 223) is essentially GST with an axiom schema of induction replacing Specication, and with theexistence of a null set explicitly assumed.GST is called STZ in Burgess (2005), p. 223.[2] Burgesss theory ST[3] is GST with Null Set replacing the axiomschema of specication. That the letters ST also appear in GST is a coincidence.

5.4 MetamathematicsThe most remarkable fact about ST (and hence GST), is that these tiny fragments of set theory give rise to suchrich metamathematics. While ST is a small fragment of the well-known canonical set theories ZFC and NBG, STinterprets Robinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example, ST isessentially undecidable because Q is, and every consistent theory whose theorems include the ST axioms is alsoessentially undecidable.[4] This includes GST and every axiomatic set theory worth thinking about, assuming theseare consistent. In fact, the undecidability of ST implies the undecidability of rst-order logic with a single binarypredicate letter.[5]

Q is also incomplete in the sense of Gdels incompleteness theorem. Any axiomatizable theory, such as ST andGST, whose theorems include the Q axioms is likewise incomplete. Moreover, the consistency of GST cannot beproved within GST itself, unless GST is in fact inconsistent.GST is:

Mutually interpretable with Peano arithmetic (thus it has the same proof-theoretic strength as PA); Immune to the three great antinomies of nave set theory: Russells, Burali-Fortis, and Cantors; Not nitely axiomatizable. Montague (1961) showed that ZFC is not nitely axiomatizable, and his argument

carries over to GST. Hence any axiomatization of GST must either include at least one axiom schema such asSeparation;

Interpretable in relation algebra because no part of any GST axiom lies in the scope of more than threequantiers. This is the necessary and sucient condition given in Tarski and Givant (1987).

5.5 Footnotes[1] Adjunction is seldom mentioned in the literature. Exceptions are Burgess (2005) passim, and QIII in Tarski and Givant

(1987: 223).

[2] The Null Set axiom in STZ is redundant, because the existence of the null set is derivable from the axiom schema ofSpecication.

[3] Called S' in Tarski et al. (1953: 34).

[4] Burgess (2005), 2.2, p. 91.

16 CHAPTER 5. GENERAL SET THEORY

[5] Tarski et al. (1953), p. 34.

5.6 References George Boolos (1998) Logic, Logic, and Logic. Harvard Univ. Press. Burgess, John, 2005. Fixing Frege. Princeton Univ. Press. Richard Montague (1961) Semantical closure and non-nite axiomatizability in Innistic Methods. Warsaw:

45-69. Alfred Tarski, Andrzej Mostowski, and Raphael Robinson (1953) Undecidable Theories. North Holland. Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMS

Colloquium Publications, v. 41.

5.7 External links Stanford Encyclopedia of Philosophy: Set Theoryby Thomas Jech.

Chapter 6

Internal set theory

Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomaticbasis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elementsto the real numbers, Nelsons approach modies the axiomatic foundations through syntactic enrichment. Thus,the axioms introduce a new term, standard, which can be used to make discriminations not possible under theconventional axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satised for all classicalpredicates, while the new unary predicate standard satises three additional axioms I, S, and T. In particular, suitablenon-standard elements within the set of real numbers can be shown to have properties that correspond to the propertiesof innitesimal and unlimited elements.Nelsons formulation is made more accessible for the lay-mathematician by leaving out many of the complexities ofmeta-mathematical logic that were initially required to justify rigorously the consistency of number systems containinginnitesimal elements.

6.1 Intuitive justicationWhilst IST has a perfectly formal axiomatic scheme, described below, an intuitive justication of the meaning ofthe term 'standard' is desirable. This is not part of the formal theory, but is a pedagogical device that might helpthe student interpret the formalism. The essential distinction, similar to the concept of denable numbers, contraststhe niteness of the domain of concepts that we can specify and discuss with the unbounded innity of the set ofnumbers; compare nitism.

The number of symbols we write with is nite. The number of mathematical symbols on any given page is nite. The number of pages of mathematics a single mathematician can produce in a lifetime is nite. Any workable mathematical denition is necessarily nite. There are only a nite number of distinct objects a mathematician can dene in a lifetime. There will only be a nite number of mathematicians in the course of our (presumably nite) civilization. Hence there is only a nite set of whole numbers our civilization can discuss in its allotted lifespan. What that limit actually is, is unknowable to us, being contingent on many accidental cultural factors. This limitation is not in itself susceptible to mathematical scrutiny, but the fact that there is such a limit, whilst

the set of whole numbers continues forever without bound, is a mathematical truth.

The term standard is therefore intuitively taken to correspond to some necessarily nite portion of accessible wholenumbers. In fact the argument can be applied to any innite set of objects whatsoever - there are only so manyelements that we can specify in nite time using a nite set of symbols and there are always those that lie beyond thelimits of our patience and endurance, no matter how we persevere. We must admit to a profusion of non-standardelements too large or too anonymous to grasp within any innite set.

17

18 CHAPTER 6. INTERNAL SET THEORY

6.1.1 Principles of the standard predicate

The following principles follow from the above intuitive motivation and so should be deducible from the formalaxioms. For the moment we take the domain of discussion as being the familiar set of whole numbers.

Any mathematical expression that does not use the new predicate standard explicitly or implicitly is an internalformula.

Any denition that does so is an external formula.

Any number uniquely specied by an internal formula is standard (by denition).

Non-standard numbers are precisely those that cannot be uniquely specied (due to limitations of time andspace) by an internal formula.

Non-standard numbers are elusive: each one is too enormous to be manageable in decimal notation or anyother representation, explicit or implicit, no matter how ingenious your notation. Whatever you succeed inproducing is by-denition merely another standard number.

Nevertheless, there are (many) non-standard whole numbers in any innite subset of N.

Non-standard numbers are completely ordinary numbers, having decimal representations, prime factorizations,etc. Every classical theorem that applies to the natural numbers applies to the non-standard natural numbers.We have created, not new numbers, but a new method of discriminating between existing numbers.

Moreover, any classical theorem that is true for all standard numbers is necessarily true for all natural numbers.Otherwise the formulation the smallest number that fails to satisfy the theorem would be an internal formulathat uniquely dened a non-standard number.

The predicate non-standard is a logically consistent method for distinguishing large numbers the usualterm will be illimited. Reciprocals of these illimited numbers will necessarily be extremely small real numbers innitesimals. To avoid confusion with other interpretations of these words, in newer articles on IST thosewords are replaced with the constructs i-large and i-small.

There are necessarily only nitely many standard numbers - but caution is required: we cannot gather themtogether and hold that the result is a well-dened mathematical set. This will not be supported by the formalism(the intuitive justication being that the precise bounds of this set vary with time and history). In particularwe will not be able to talk about the largest standard number, or the smallest non-standard number. It will bevalid to talk about some nite set that contains all standard numbers - but this non-classical formulation couldonly apply to a non-standard set.

6.2 Formal axioms for ISTIST is an axiomatic theory in the rst-order logic with equality in a language containing a binary predicate symbol and a unary predicate symbol standard(x). Formulas not involving st (i.e., formulas of the usual language of settheory) are called internal, other formulas are called external. We use the abbreviations

9stx(x) = 9x (standard(x) ^ (x));8stx(x) = 8x (standard(x)! (x)):

IST includes all axioms of the ZermeloFraenkel set theory with the axiom of choice (ZFC). Note that the ZFCschemata of separation and replacement are not extended to the new language, they can only be used with inter-nal formulas. Moreover, IST includes three new axiom schemata conveniently one for each letter in its name:Idealisation, Standardisation, and Transfer.

6.2. FORMAL AXIOMS FOR IST 19

6.2.1 I: Idealisation For any internal formula without free occurrence of z, the universal closure of the following formula is an

axiom:8stz (znite is ! 9y 8x 2 z (x; y; u1; : : : ; un))$ 9y 8stx(x; y; u1; : : : ; un):

In words: For every internal relation R, and for arbitrary values for all other free variables, we have that if foreach standard, nite set F, there exists a g such that R(g, f) holds for all f in F, then there is a particular G suchthat for any standard f we have R(G, f), and conversely, if there exists G such that for any standard f, we haveR(G, f), then for each nite set F, there exists a g such that R(g, f) holds for all f in F.

The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by thesimple statement that elements of standard nite sets are standard. The more important left-to-right implicationexpresses that the collection of all standard sets is contained in a nite (non-standard) set, and moreover, this niteset can be taken to satisfy any given internal property shared by all standard nite sets.This very general axiom scheme upholds the existence of ideal elements in appropriate circumstances. Threeparticular applications demonstrate important consequences.

Applied to the relation

If S is standard and nite, we take for the relation R(g, f): g and f are not equal and g is in S. Since "For every standardnite set F there is an element g in S such that g f for all f in F" is false (no such g exists when F = S), we may useIdealisation to tell us that "There is a G in S such that G f for all standard f" is also false, i.e. all the elements of Sare standard.If S is innite, then we take for the relation R(g, f): g and f are not equal and g is in S. Since "For every standardnite set F there is an element g in S such that g f for all f in F" (the innite set S is not a subset of the nite set F),we may use Idealisation to derive "There is a G in S such that G f for all standard f. In other words, every inniteset contains a non-standard element (many, in fact).The power set of a standard nite set is standard (by Transfer) and nite, so all the subsets of a standard nite set arestandard.If S is non-standard, we take for the relation R(g, f): g and f are not equal and g is in S. Since "For every standardnite set F there is an element g in S such that g f for all f in F" (the non-standard set S is not a subset of the standardand nite set F), we may use Idealisation to derive "There is a G in S such that G f for all standard f." In otherwords, every non-standard set contains a non-standard element.As a consequence of all these results, all the elements of a set S are standard if and only if S is standard and nite.

Applied to the relation >>>>>>>>>>>>>>>>>>:

fO1; O2gfO3; O7; O10gfO4gfO5gfO6gfO8gfO9g

Thus, the two objects within the rst equivalence class, fO1; O2g , cannot be distinguished from each other basedon the available attributes, and the three objects within the second equivalence class, fO3; O7; O10g , cannot bedistinguished from one another based on the available attributes. The remaining ve objects are each discerniblefrom all other objects. The equivalence classes of the P -indiscernibility relation are denoted [x]P .It is apparent that dierent attribute subset selections will in general lead to dierent indiscernibility classes. Forexample, if attribute P = fP1g alone is selected, we obtain the following, much coarser, equivalence-class structure:

8>:fO1; O2gfO3; O5; O7; O9; O10gfO4; O6; O8g

17.1.3 Denition of a rough set

Let X U be a target set that we wish to represent using attribute subset P ; that is, we are told that an arbitraryset of objects X comprises a single class, and we wish to express this class (i.e., this subset) using the equivalenceclasses induced by attribute subset P . In general, X cannot be expressed exactly, because the set may include andexclude objects which are indistinguishable on the basis of attributes P .For example, consider the target set X = fO1; O2; O3; O4g , and let attribute subset P = fP1; P2; P3; P4; P5g , thefull available set of features. It will be noted that the set X cannot be expressed exactly, because in [x]P ; , objectsfO3; O7; O10g are indiscernible. Thus, there is no way to represent any setX which includesO3 but excludes objectsO7 and O10 .However, the target set X can be approximated using only the information contained within P by constructing the P-lower and P -upper approximations of X :

PX = fx j [x]P Xg

PX = fx j [x]P \X 6= ;g

Lower approximation and positive region

The P -lower approximation, or positive region, is the union of all equivalence classes in [x]P which are containedby (i.e., are subsets of) the target set in the example, PX = fO1; O2g [ fO4g , the union of the two equivalenceclasses in [x]P which are contained in the target set. The lower approximation is the complete set of objects in U/Pthat can be positively (i.e., unambiguously) classied as belonging to target set X .

Upper approximation and negative region

The P -upper approximation is the union of all equivalence classes in [x]P which have non-empty intersection withthe target set in the example, PX = fO1; O2g[fO4g[fO3; O7; O10g , the union of the three equivalence classesin [x]P that have non-empty intersection with the target set. The upper approximation is the complete set of objectsthat in U/P that cannot be positively (i.e., unambiguously) classied as belonging to the complement ( X ) of the

17.1. DEFINITIONS 79

target set X . In other words, the upper approximation is the complete set of objects that are possibly members ofthe target set X .The set UPX therefore represents the negative region, containing the set of objects that can be denitely ruled outas members of the target set.

Boundary region

The boundary region, given by set dierence PX PX , consists of those objects that can neither be ruled in norruled out as members of the target set X .In summary, the lower approximation of a target set is a conservative approximation consisting of only those objectswhich can positively be identied as members of the set. (These objects have no indiscernible clones which areexcluded by the target set.) The upper approximation is a liberal approximation which includes all objects that mightbe members of target set. (Some objects in the upper approximation may not be members of the target set.) Fromthe perspective of U/P , the lower approximation contains objects that are members of the target set with certainty(probability = 1), while the upper approximation contains objects that are members of the target set with non-zeroprobability (probability > 0).

The rough set

The tuple hPX;PXi composed of the lower and upper approximation is called a rough set; thus, a rough set iscomposed of two crisp sets, one representing a lower boundary of the target set X , and the other representing anupper boundary of the target set X .The accuracy of the rough-set representation of the set X can be given (Pawlak 1991) by the following:

P (X) =jPXjPX

That is, the accuracy of the rough set representation of X , P (X) , 0 P (X) 1 , is the ratio of the numberof objects which can positively be placed in X to the number of objects that can possibly be placed in X thisprovides a measure of how closely the rough set is approximating the target set. Clearly, when the upper and lowerapproximations are equal (i.e., boundary region empty), then P (X) = 1 , and the approximation is perfect; at theother extreme, whenever the lower approximation is empty, the accuracy is zero (regardless of the size of the upperapproximation).

Objective analysis

Rough set theory is one of many methods that can be employed to analyse uncertain (including vague) systems,although less common than more traditional methods of probability, statistics, entropy and DempsterShafer theory.However a key dierence, and a unique strength, of using classical rough set theory is that it provides an objectiveform of analysis (Pawlak et al. 1995). Unlike other methods, as those given above, classical rough set analysis requiresno additional information, external parameters, models, functions, grades or subjective interpretations to determineset membership instead it only uses the information presented within the given data (Dntsch and Gediga 1995).More recent adaptations of rough set theory, such as dominance-based, decision-theoretic and fuzzy rough sets, haveintroduced more subjectivity to the analysis.

17.1.4 Denability

In general, the upper and lower approximations are not equal; in such cases, we say that target set X is undenableor roughly denable on attribute set P . When the upper and lower approximations are equal (i.e., the boundary isempty), PX = PX , then the target set X is denable on attribute set P . We can distinguish the following specialcases of undenability:

80 CHAPTER 17. ROUGH SET

SetX is internally undenable if PX 6= ; and PX = U . This means that on attribute set P , there are objectswhich we can be certain belong to target setX , but there are no objects which we can denitively exclude fromset X .

Set X is externally undenable if PX = ; and PX 6= U . This means that on attribute set P , there are noobjects which we can be certain belong to target set X , but there are objects which we can denitively excludefrom set X .

SetX is totally undenable if PX = ; and PX = U . This means that on attribute set P , there are no objectswhich we can be certain belong to target set X , and there are no objects which we can denitively excludefrom set X . Thus, on attribute set P , we cannot decide whether any object is, or is not, a member of X .

17.1.5 Reduct and coreAn interesting question is whether there are attributes in the information system (attribute-value table) which are moreimportant to the knowledge represented in the equivalence class structure than other attributes. Often, we wonderwhether there is a subset of attributes which can, by itself, fully characterize the knowledge in the database; such anattribute set is called a reduct.Formally, a reduct is a subset of attributes RED P such that

[x]RED = [x]P , that is, the equivalence classes induced by the reduced attribute set RED are the same as theequivalence class structure induced by the full attribute set P .

the attribute set RED is minimal, in the sense that [x](REDfag) 6= [x]P for any attribute a 2 RED ; in otherwords, no attribute can be removed from set RED without changing the equivalence classes [x]P .

A reduct can be thought of as a sucient set of features sucient, that is, to represent the category structure. In theexample table above, attribute set fP3; P4; P5g is a reduct the information system projected on just these attributespossesses the same equivalence class structure as that expressed by the full attribute set:

8>>>>>>>>>>>>>>>>>>>:

fO1; O2gfO3; O7; O10gfO4gfO5gfO6gfO8gfO9g

Attribute set fP3; P4; P5g is a legitimate reduct because eliminating any of these attributes causes a collapse of theequivalence-class structure, with the result that [x]RED 6= [x]P .The reduct of an information system is not unique: there may be many subsets of attributes which preserve theequivalence-class structure (i.e., the knowledge) expressed in the information system. In the example informationsystem above, another reduct is fP1; P2; P5g , producing the same equivalence-class structure as [x]P .The set of attributes which is common to all reducts is called the core: the core is the set of attributes which is possessedby every legitimate reduct, and therefore consists of attributes which cannot be removed from the information systemwithout causing collapse of the equivalence-class structure. The core may be thought of as the set of necessaryattributes necessary, that is, for the category structure to be represented. In the example, the only such attribute isfP5g ; any one of the other attributes can be removed singly without damaging the equivalence-class structure, andhence these are all dispensable. However, removing fP5g by itself does change the equivalence-class structure, andthus fP5g is the indispensable attribute of this information system, and hence the core.It is possible for the core to be empty, which means that there is no indispensable attribute: any single attribute insuch an information system can be deleted without altering the equivalence-class structure. In such cases, there is noessential or necessary attribute which is required for the class structure to be represented.

17.2. RULE EXTRACTION 81

17.1.6 Attribute dependency

One of the most important aspects of database analysis or data acquisition is the discovery of attribute dependencies;that is, we wish to discover which variables are strongly related to which other variables. Generally, it is these strongrelationships that will warrant further investigation, and that will ultimately be of use in predictive modeling.In rough set theory, the notion of dependency is dened very simply. Let us take two (disjoint) sets of attributes,set P and set Q , and inquire what degree of dependency obtains between them. Each attribute set induces an(indiscernibility) equivalence class structure, the equivalence classes induced byP given by [x]P , and the equivalenceclasses induced by Q given by [x]Q .Let [x]Q = fQ1; Q2; Q3; : : : ; QNg , where Qi is a given equivalence class from the equivalence-class structureinduced by attribute set Q . Then, the dependency of attribute set Q on attribute set P , P (Q) , is given by

P (Q) =

PNi=1 jPQijjUj 1

That is, for each equivalence class Qi in [x]Q , we add up the size of its lower approximation by the attributes in P ,i.e., PQi . This approximation (as above, for arbitrary setX ) is the number of objects which on attribute setP can bepositively identied as belonging to target set Qi . Added across all equivalence classes in [x]Q , the numerator aboverepresents the total number of objects which based on attribute set P can be positively categorized according tothe classication induced by attributes Q . The dependency ratio therefore expresses the proportion (within the entireuniverse) of such classiable objects. The dependency P (Q) can be interpreted as a proportion of such objects inthe information system for which it suces to know the values of attributes in P to determine the values of attributesin Q ".Another, intuitive, way to consider dependency is to take the partition induced by Q as the target class C, and considerP as the attribute set we wish to use in order to re-construct the target class C. If P can completely reconstruct C,then Q depends totally upon P; if P results in a poor and perhaps a random reconstruction of C, then Q does notdepend upon P at all.Thus, this measure of dependency expresses the degree of functional (i.e., deterministic) dependency of attribute setQ on attribute set P ; it is not symmetric. The relationship of this notion of attribute dependency to more traditionalinformation-theoretic (i.e., entropic) notions of attribute dependence has been discussed in a number of sources (e.g.,Pawlak, Wong, & Ziarko 1988; Yao & Yao 2002; Wong, Ziarko, & Ye 1986, Quafafou & Boussouf 2000).

17.2 Rule extractionThe category representations discussed above are all extensional in nature; that is, a category or complex class issimply the sum of all its members. To represent a category is, then, just to be able to list or identify all the objectsbelonging to that category. However, extensional category representations have very limited practical use, becausethey provide no insight for deciding whether novel (never-before-seen) objects are members of the category.What is generally desired is an intentional description of the category, a representation of the category based on a setof rules that describe the scope of the category. The choice of such rules is not unique, and therein lies the issue ofinductive bias. See Version space and Model selection for more about this issue.There are a few rule-extraction methods. We will start from a rule-extraction procedure based on Ziarko & Shan(1995).

17.2.1 Decision matrices

Let us say that we wish to nd the minimal set of consistent rules (logical implications) that characterize our samplesystem. For a set of condition attributes P = fP1; P2; P3; : : : ; Png and a decision attribute Q;Q /2 P , these rulesshould have the form P ai P bj : : : P ck ! Qd , or, spelled out,

(Pi = a) ^ (Pj = b) ^ ^ (Pk = c)! (Q = d)

82 CHAPTER 17. ROUGH SET

where fa; b; c; : : : g are legitimate values from the domains of their respective attributes. This is a form typical ofassociation rules, and the number of items in U which match the condition/antecedent is called the support for therule. The method for extracting such rules given in Ziarko & Shan (1995) is to form a decision matrix correspondingto each individual value d of decision attribute Q . Informally, the decision matrix for value d of decision attributeQ lists all attributevalue pairs that d