Ordinal numberFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of dependent choice 11.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Equivalent statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Relation with other axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Axiom of regularity 32.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 No innite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Simpler set-theoretic denition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 42.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 4

2.2 The axiom of dependent choice and no innite descending sequence of sets implies regularity . . . . 42.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Regularity and Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Axiom of union 83.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Bijection 104.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Burali-Forti paradox 165.1 Stated in terms of von Neumann ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Stated more generally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Resolution of the paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Cardinal number 186.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Cardinality 267.1 Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.1.1 Denition 1: | A | = | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.1.2 Denition 2: | A | | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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7.1.3 Denition 3: | A | < | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Finite, countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 Innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.4.1 Cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.5 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.6 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Coniteness 318.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 Conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2.2 Double-pointed conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.3.1 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.3.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9 Derived set (mathematics) 339.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2 Topology in terms of derived sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.3 CantorBendixson rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10 Discrete space 3510.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Equivalence class 3811.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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11.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12 Equivalence relation 4312.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

12.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13 Finite set 5213.1 Denition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.3 Necessary and sucient conditions for niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.4 Foundational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.5 Set-theoretic denitions of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.5.1 Other concepts of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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13.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5613.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

14 First uncountable ordinal 5714.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15 Georg Cantor 5815.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

15.1.1 Youth and studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.1.2 Teacher and researcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.1.3 Late years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

15.2 Mathematical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.2.1 Number theory, trigonometric series and ordinals . . . . . . . . . . . . . . . . . . . . . . 6115.2.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

15.3 Philosophy, religion and Cantors mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6315.4 Cantors ancestry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6415.5 Historiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

16 Hereditary property 7616.1 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

16.2.1 Monotone property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3 In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.4 In matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.5 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

17 If and only if 7917.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7917.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

17.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7917.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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17.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

18 Inmum and supremum 8218.1 Inma of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.2 Inma in partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.3 Supremum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

18.3.1 Supremum of a set of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.3.2 Suprema within partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8518.3.3 Comparison with other order theoretical notions . . . . . . . . . . . . . . . . . . . . . . . 8618.3.4 Least-upper-bound property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

18.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

19 Injective function 8819.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8919.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

20 Integer 9620.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10020.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10020.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10020.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

21 John von Neumann 10221.1 Early life and education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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21.2 Career and abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.2.1 Beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.2.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.2.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10421.2.4 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10421.2.5 Ergodic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10521.2.6 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10521.2.7 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10521.2.8 Mathematical formulation of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . 10521.2.9 Quantum logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10621.2.10 Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10721.2.11 Mathematical economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10721.2.12 Linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.2.13 Mathematical statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.2.14 Nuclear weapons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.2.15 The Atomic Energy Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.2.16 The ICBM Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.2.17 Mutual assured destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.2.18 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11121.2.19 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11221.2.20 Politics and social aairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11221.2.21 On the eve of World War II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11321.2.22 Greece versus Rome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11421.2.23 Weather systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11421.2.24 Cognitive abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11421.2.25 Mastery of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

21.3 Personal life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11521.4 Later life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11621.5 Honors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

21.5.1 Infopark and Neumann Jnos Street . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11721.6 Selected works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11721.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11821.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

21.8.1 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11821.8.2 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

21.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12321.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

22 Natural number 12622.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

22.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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22.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12822.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

22.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12822.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 12922.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 129

22.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

22.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

22.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13222.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13222.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13322.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

23 New Foundations 13723.1 The type theory TST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13723.2 Quinean set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

23.2.1 Axioms and stratication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13823.2.2 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13823.2.3 Admissibility of useful large sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

23.3 Finite axiomatizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13923.4 Cartesian closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13923.5 The consistency problem and related partial results . . . . . . . . . . . . . . . . . . . . . . . . . . 13923.6 How NF(U) avoids the set-theoretic paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13923.7 The system ML (Mathematical Logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14023.8 Models of NFU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

23.8.1 Self-suciency of mathematical foundations in NFU . . . . . . . . . . . . . . . . . . . . 14123.8.2 Facts about the automorphism j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

23.9 Strong axioms of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14223.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14423.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14423.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14423.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

24 Order isomorphism 14624.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14624.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14624.3 Order types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14624.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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24.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14724.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

25 Order type 14825.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14825.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14925.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14925.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14925.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14925.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

26 Ordinal arithmetic 15026.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15026.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15126.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15226.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15526.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15526.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15626.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15726.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15726.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

27 Ordinal number 15827.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15927.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

27.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16127.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 16127.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16127.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

27.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16227.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

27.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16227.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16327.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16327.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16327.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

27.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16427.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

27.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16527.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

27.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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27.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16627.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16627.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16627.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16627.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16627.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

28 Partially ordered set 16828.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16928.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16928.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16928.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 17028.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17028.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17128.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17128.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17128.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17228.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17228.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17328.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17328.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17328.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17328.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17428.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17428.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

29 Principia Mathematica 17529.1 Scope of foundations laid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17629.2 Theoretical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

29.2.1 Contemporary construction of a formal theory . . . . . . . . . . . . . . . . . . . . . . . . 17629.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17729.2.3 Primitive ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17829.2.4 Primitive propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

29.3 Ramied types and the axiom of reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17929.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

29.4.1 An introduction to the notation of Section A Mathematical Logic (formulas 15.71) . 18029.4.2 An introduction to the notation of Section B Theory of Apparent Variables (formulas 8

14.34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18229.4.3 Introduction to the notation of the theory of classes and relations . . . . . . . . . . . . . . 184

29.5 Consistency and criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18529.5.1 Gdel 1930, 1931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18629.5.2 Wittgenstein 1919, 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

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29.5.3 Gdel 1944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18729.6 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

29.6.1 Part I Mathematical logic. Volume I 1 to 43 . . . . . . . . . . . . . . . . . . . . . . . 18729.6.2 Part II Prolegomena to cardinal arithmetic. Volume I 50 to 97 . . . . . . . . . . . . . . 18729.6.3 Part III Cardinal arithmetic. Volume II 100 to 126 . . . . . . . . . . . . . . . . . . . . 18729.6.4 Part IV Relation-arithmetic. Volume II 150 to 186 . . . . . . . . . . . . . . . . . . . . 18729.6.5 Part V Series. Volume II 200 to 234 and volume III 250 to 276 . . . . . . . . . . . 18829.6.6 Part VI Quantity. Volume III 300 to 375 . . . . . . . . . . . . . . . . . . . . . . . . . 188

29.7 Comparison with set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18829.8 Dierences between editions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18929.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18929.10Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18929.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19029.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

30 Set theory 19330.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19430.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19530.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19630.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19630.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19730.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

30.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19830.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19830.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19830.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19830.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19930.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19930.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19930.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19930.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

30.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 20030.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20030.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20030.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20130.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

31 Surjective function 20231.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20331.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20331.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

31.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

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31.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20531.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20631.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 20631.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20631.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

31.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20631.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20731.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

32 Topological space 20832.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

32.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20832.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20932.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21032.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

32.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21032.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21032.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21132.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21232.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21232.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21232.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21232.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21232.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21332.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21332.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21332.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

33 Total order 21533.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21533.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21633.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

33.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21633.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21633.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21733.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21733.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21733.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21733.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

33.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 21833.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21833.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

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33.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21833.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

34 Transnite induction 22034.1 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22034.2 Relationship to the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22234.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22234.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22234.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22234.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

35 Transitive set 22435.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22435.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22435.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22435.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22435.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22535.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22535.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

36 Trigonometric series 22636.1 The zeros of a trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22636.2 Zygmunds book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22636.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

36.3.1 Reviews of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22736.3.2 Publication history of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 227

37 Type theory 22837.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22837.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22837.3 Dierence from set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22937.4 Optional features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

37.4.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22937.4.2 Dependent types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22937.4.3 Equality types (or identity types) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23037.4.4 Inductive types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23037.4.5 Universe types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23037.4.6 Computational component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

37.5 Systems of type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23137.5.1 Major . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23137.5.2 Minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23137.5.3 Active . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

37.6 Practical impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

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37.6.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23137.6.2 Mathematical foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23137.6.3 Proof assistants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23237.6.4 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23237.6.5 Social sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

37.7 Relation to category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23237.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23337.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23337.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23337.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

38 Von Neumann cardinal assignment 23538.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23538.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23538.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

39 Well-order 23739.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23739.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

39.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23839.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23839.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

39.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23939.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23939.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24039.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

40 ZermeloFraenkel set theory 24140.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24140.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

40.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24240.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 24240.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24240.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24340.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24340.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24440.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24540.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24540.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

40.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24640.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

40.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

CONTENTS xv

40.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24740.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24840.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24840.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24940.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 250

40.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25040.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25840.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Chapter 1

Axiom of dependent choice

In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that isstill sucient to develop most of real analysis.

1.1 Formal statementThe axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence(xn) in X such that xnR xn for each n in N. (Here an entire binary relation on X is one such that for each a in X thereis a b in X such that aRb.) Note that even without such an axiom we could form the rst n terms of such a sequence,for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way.If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DCR.

1.2 UseDC is the fragment of AC required to show the existence of a sequence constructed by transnite recursion ofcountable length, if it is necessary to make a choice at each step.

1.3 Equivalent statementsDC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is alsoequivalent[1] to the Baire category theorem for complete metric spaces.

1.4 Relation with other axiomsUnlike full AC, DC is insucient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a setof reals without the property of Baire or without the perfect set property.The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.

1.5 Footnotes

[1] Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sr. Sci.Math. Astronom. Phys. 25 (1977), no. 10, 933-934.

1

2 CHAPTER 1. AXIOM OF DEPENDENT CHOICE

1.6 References Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN

3-540-44085-2.

Chapter 2

Axiom of regularity

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

8x (x 6= ?! 9y 2 x (y \ x = ?))

The axiom implies that no set is an element of itself, and that there is no innite 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 innite 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 downwardinnite 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 f(n; )jn 2! ^ ordinal an is g :Given the other axioms of ZermeloFraenkel 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.

2.1 Elementary implications of regularity

2.1.1 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 denition of disjoint).

2.1.2 No innite 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. Dene 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 denitionof 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

3

4 CHAPTER 2. 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 innite and constant.Notice that this argument only applies to functions f that can be represented as sets as opposed to undenable classes.The hereditarily nite sets, V, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom ofinnity). So if one forms a non-trivial ultrapower of V, then it will also satisfy the axiom of regularity. The resultingmodel will contain elements, called non-standard natural numbers, that satisfy the denition 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 innite descending sequences of elements. For example, suppose n is anon-standard natural number, then (n 1) 2 n and (n 2) 2 (n 1) , and so on. For any actual natural numberk, (n k 1) 2 (n k) . This is an unending descending sequence of elements. But this sequence is not denablein the model and thus not a set. So no contradiction to regularity can be proved.

2.1.3 Simpler set-theoretic denition of the ordered pairThe axiom of regularity enables dening the ordered pair (a,b) as {a,{a,b}}. See ordered pair for specics. Thisdenition eliminates one pair of braces from the canonical Kuratowski denition (a,b) = {{a},{a,b}}.

2.1.4 Every set has an ordinal rankThis was actually the original form of von Neumanns axiomatization.

2.1.5 For every two sets, only one can be an element of the otherLet 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 denition of disjoint then, we must have either Y is notan element of X or vice versa.

2.2 The axiom of dependent choice and no innite descending sequence ofsets 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 dene a binary relation R on S by aRb :, b 2 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 aninnite descending chain, we arrive at a contradiction and so, no such S exists.

2.3 Regularity and the rest of ZF(C) axiomsRegularity 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. 210212).

2.4 Regularity and Russells paradoxNaive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent dueto Russells paradox. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension

2.5. REGULARITY, THE CUMULATIVE HIERARCHY, AND TYPES 5

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 innity) 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, Russells 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 from ZFC. Various non-wellfoundedset theories allow safe circular sets, such as Quine atoms, without becoming inconsistent by means of Russellsparadox.(Rieger 2011, pp. 175,178)

2.5 Regularity, the cumulative hierarchy, and typesIn ZF it can be proven that the classS V (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 satises it can be constructed by taking only sets inS V .Herbert Enderton (1977, p. 206) wrote that The idea of rank is a descendant of Russells concept of type". Com-paring ZF with type theory, Alasdair Urquhart wrote that Zermelos 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 Russells and Zermelos intuitions. Indeed thebest way to regard Zermelos theory is as a simplication and extension of Russells. (We mean Russellssimple theory of types, of course.) The simplication 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 transnitejust 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. (Lvy 2002, p. 73)

2.6 HistoryThe concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimano (1917) cf. Lvy(2002, p. 68) and Hallett (1986, 4.4, esp. p. 186, 188). Mirimano called a set x regular (French: ordinaire) ifevery descending chain x x1 x2 ... is nite. Mirimano however did not consider his notion of regularity (andwell-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 6263); in later papers Mirimanoalso explored what are now called non-well-founded sets (extraordinaire in Mirimanos terminology) (Sangiorgi2011, pp. 1719, 26).According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as superuous (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):

8x (x 6= ; ! 9y 2 x (y \ x = ;))

6 CHAPTER 2. AXIOM OF REGULARITY

2.7 See also Non-well-founded set theory

2.8 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: 215231, doi:10.2307/2025204

reprinted in Boolos, George (1998), Logic, Logic and Logic, Harvard University Press, pp. 1329

Enderton, Herbert B. (1977), Elements of Set Theory, Academic Press Urquhart, Alasdair (2003), The Theory of Types, in Grin, Nicholas, The Cambridge Companion to Bertrand

Russell, 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

Lvy, Azriel (2002) [rst published in 1979], Basic set theory, Dover Publications, ISBN 0-486-42079-5 Hallett, Michael (1996) [rst published 1984], Cantorian set theory and limitation of size, Oxford University

Press, 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, Peter

Clark, Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell., pp. 171187,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

2.8.1 Primary sources Mirimano, D. (1917), Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie

des ensembles, L'Enseignement Mathmatique 19: 3752

von Neumann, J. (1925), Eine axiomatiserung der Mengenlehre, Journal fr die reine und angewandte Math-ematik 154: 219240; translation in van Heijenoort, Jean (1967), From Frege to Gdel: A Source Book inMathematical Logic, 18791931, pp. 393413

von Neumann, J. (1928), "ber die Denition durch transnite Induktion und verwandte Fragen der allge-meinen Mengenlehre, Mathematische Annalen 99: 373391, doi:10.1007/BF01459102

von Neumann, J. (1929), Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journalfur die reine und angewandte Mathematik 160: 227241, doi:10.1515/crll.1929.160.227

Zermelo, Ernst (1930), "ber Grenzzahlen und Mengenbereiche. Neue Untersuchungen ber die Grundlagender Mengenlehre. (PDF), Fundamenta Mathematicae 16: 2947; translation in Ewald, W.B., ed. (1996),From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2, Clarendon Press, pp. 121933

Bernays, P. (1941), A system of axiomatic set theory. Part II, The Journal of Symbolic Logic 6: 117,doi:10.2307/2267281

2.9. EXTERNAL LINKS 7

Bernays, P. (1954), A system of axiomatic set theory. Part VII, The Journal of Symbolic Logic 19: 8196,doi:10.2307/2268864

Riegger, L. (1957), A contribution to Gdels axiomatic set theory (PDF), Czechoslovak Mathematical Jour-nal 7: 323357

Scott, D. (1974), Axiomatizing set theory, Axiomatic set theory. Proceedings of Symposia in Pure MathematicsVolume 13, Part II, pp. 207214

2.9 External links http://www.trinity.edu/cbrown/topics_in_logic/sets/sets.html contains an informative description of the axiom

of regularity under the section on Zermelo-Fraenkel set theory.

Axiom of Foundation at PlanetMath.org.

Chapter 3

Axiom of union

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of unionis one of the axioms of ZermeloFraenkel set theory, stating that, for any set x there is a set y whose elements areprecisely the elements of the elements of x. Together with the axiom of pairing this implies that for any two sets,there is a set that contains exactly the elements of both.

3.1 Formal statementIn the formal language of the ZermeloFraenkel axioms, the axiom reads:

8A 9B 8c (c 2 B () 9D (c 2 D ^D 2 A) )

or in words:

Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is aset D such that c is a member of D and D is a member of A.

or, more simply:

For any set A , there is a setSA which consists of just the elements of the elements of that set.3.2 InterpretationWhat the axiom is really saying is that, given a set A, we can nd a set B whose members are precisely the membersof the members of A. By the axiom of extensionality this set B is unique and it is called the union of A, and denotedSA . Thus the essence of the axiom is:

The union of a set is a set.

Note that the union of A and B, commonly written as A [ B , can be written as SfA;Bg . Thus, the ordinaryconstruction of unions is trivial given the axiom of pairing.The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternativeaxiomatization of set theory.Note that there is no corresponding axiom of intersection. If A is a nonempty set containing E, then we can form theintersectionTA using the axiom schema of specication as

{c in E: for all D in A, c is in D},

8

3.3. REFERENCES 9

so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as

{c: for all D in A, c is in D}

is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the universe, butthe notion of a universal set is antithetical to ZermeloFraenkel set theory.)

3.3 References Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag,

New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). 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.

3.4 External links Axiom of Union at PlanetMath.org.

Chapter 4

Bijection

X 1

2

3

4

YD

B

C

A

A bijective function, f: X Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the rst set. There are no unpaired elements. In mathematical

10

4.1. DEFINITION 11

terms, a bijective function f: X Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential to many areas of mathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.

4.1 DenitionFor more details on notation, see Function (mathematics) Notation.

For a pairing between X and Y (where Y need not be dierent from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both one-to-one and onto.

4.2 Examples

4.2.1 Batting line-up of a baseball teamConsider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The pairingis given by which player is in what position in this order. Property (1) is satised since each player is somewhere inthe list. Property (2) is satised since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.

4.2.2 Seats and students of a classroomIn a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

4. No seat had more than one student in it.

12 CHAPTER 4. BIJECTION

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

4.3 More mathematical examples and some non-examples For any set X, the identity function 1X: X X, 1X(x) = x, is bijective. The function f: R R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y 1)/2 such that f(x)

= y. In more generality, any linear function over the reals, f: R R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

The function f: R (-/2, /2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-/2, /2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-/2, /2) was made larger to include an integer multiple of /2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of /2 by this arctan function.

The exponential function, g: R R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ (0;+1) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

The function h: R R+, h(x) = x2 is not bijective: for instance, h(1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+0 [0;+1) , then h becomes bijective; its inverseis the positive square root function.

4.4 InversesA bijection f with domain X (functionally indicated by f: X Y) also denes a relation starting in Y and going to X(by turning the arrows around). The process of turning the arrows around for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X Y is bijective if and only if it satises the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being dened takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

4.5 CompositionThe composition g f of two bijections f: X Y and g: Y Z is a bijection. The inverse of g f is (g f)1 =(f1) (g1) .Conversely, if the composition g f of two functions is bijective, we can only say that f is injective and g is surjective.

4.6 Bijections and cardinalityIf X and Y are nite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the denition of same number ofelements (equinumerosity), and generalising this denition to innite sets leads to the concept of cardinal number,a way to distinguish the various sizes of innite sets.

4.7. PROPERTIES 13

X1

2

3

YD

B

C

A

ZP

Q

R

A bijection composed of an injection (left) and a surjection (right).

4.7 Properties A function f: R R is bijective if and only if its graph meets every horizontal and vertical line exactly once.

If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f1(B)| = |B|.

If X and Y are nite sets with the same cardinality, and f: X Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

For a nite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that setnamely, n!.

4.8 Bijections and category theoryBijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

14 CHAPTER 4. BIJECTION

4.9 Generalization to partial functionsThe notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undened for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. dened everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of dening the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:AB, where A is a subset of Aand likewise BB.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is the Mbius transformation simply dened on the complex plane, rather than its completion to the extendedcomplex plane.[5]

4.10 Contrast withThis list is incomplete; you can help by expanding it.

Multivalued function

4.11 See also Injective function Surjective function Bijection, injection and surjection Symmetric group Bijective numeration Bijective proof Cardinality Category theory AxGrothendieck theorem

4.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satises property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). Groups and semigroups: connections and contrasts. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson, M. V. (1998). The Mbius Inverse Monoid. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.

4.13. REFERENCES 15

4.13 ReferencesThis topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

Wolf (1998). Proof, Logic and Conjecture: A Mathematicians Toolbox. Freeman. Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall. Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole). Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley. O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall. Morash. Bridge to Abstract Mathematics. Random House. Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press. Lay (2001). Analysis with an introduction to proof. Prentice Hall. Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall. Fletcher; Patty. Founda