Ordinal Number 10
159
Ordinal number From Wikipedia, the free encyclopedia
description
1. From Wikipedia, the free encyclopedia2. Lexicographical order
Transcript of Ordinal Number 10
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 Bijection 32.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Cardinal number 93.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
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3.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Cardinality 174.1 Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Definition 1: | A | = | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.2 Definition 2: | A | ≤ | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.3 Definition 3: | A | < | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Finite, countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Infinite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.1 Cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Cofiniteness 225.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Cofinite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.2 Double-pointed cofinite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.1 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Derived set (mathematics) 246.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Topology in terms of derived sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Cantor–Bendixson rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Discrete space 267.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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7.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8 First uncountable ordinal 298.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9 Georg Cantor 309.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.1.1 Youth and studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.1.2 Teacher and researcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.1.3 Late years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.2 Mathematical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2.1 Number theory, trigonometric series and ordinals . . . . . . . . . . . . . . . . . . . . . . 339.2.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.3 Philosophy, religion and Cantor’s mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.4 Cantor’s ancestry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.5 Historiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
10 Hereditary property 4810.1 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
10.2.1 Monotone property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.3 In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4 In matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.5 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11 Injective function 5111.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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11.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12 Integer 5912.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
13 Natural number 6513.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
13.1.1 Modern definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
13.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 6813.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3.6 Algebraic properties satisfied by the natural numbers . . . . . . . . . . . . . . . . . . . . . 68
13.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.5 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
13.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
13.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14 Order isomorphism 7614.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.3 Order types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
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14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
15 Order type 7815.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
16 Ordinal arithmetic 8016.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8616.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
17 Ordinal number 8817.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
17.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.2.2 Definition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 9117.2.3 Von Neumann definition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.2.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
17.3 Transfinite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.4 Transfinite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
17.4.1 What is transfinite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.4.2 Transfinite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
17.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9417.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
17.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.6.2 Cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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17.7 Some “large” countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
18 Partially ordered set 9818.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 10018.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10018.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
19 Set theory 10519.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10619.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10919.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
19.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
CONTENTS vii
19.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 11219.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11319.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
20 Surjective function 11420.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
20.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11620.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
20.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
21 Topological space 12021.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
21.1.1 Neighbourhoods definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12021.1.2 Open sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12121.1.3 Closed sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12221.1.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
21.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12221.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12221.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12321.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.6 Classification of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12421.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12521.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12521.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12521.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
22 Total order 12722.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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22.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12822.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
22.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12822.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12822.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12922.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
22.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 13022.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
23 Transitive set 13223.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13323.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13323.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
24 Trigonometric series 13424.1 The zeros of a trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.2 Zygmund’s book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
24.3.1 Reviews of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13524.3.2 Publication history of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 135
25 Von Neumann cardinal assignment 13625.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
26 Well-order 13826.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13826.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
26.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13926.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13926.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
26.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
CONTENTS ix
26.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14026.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 142
26.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14226.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14726.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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 sufficient to develop most of real analysis.
1.1 Formal statement
The 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₊₁ 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 first 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 Use
DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion ofcountable length, if it is necessary to make a choice at each step.
1.3 Equivalent statements
DC 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 axioms
Unlike full AC, DC is insufficient 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. Sér. Sci.Math. Astronom. Phys. 25 (1977), no. 10, 933-−934.
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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
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 first set. There are no unpaired elements. In mathematical
3
4 CHAPTER 2. BIJECTION
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 finite sets, then the existenceof a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of infinite 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 definitions of isomorphism, homeomorphism,diffeomorphism, permutation group, and projective map.
2.1 Definition
For more details on notation, see Function (mathematics) § Notation.
For a pairing between X and Y (where Y need not be different 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”.
2.2 Examples
2.2.1 Batting line-up of a baseball team
Consider 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 “pairing”is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere inthe list. Property (2) is satisfied 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.
2.2.2 Seats and students of a classroom
In 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.
2.3. MORE MATHEMATICAL EXAMPLES AND SOME NON-EXAMPLES 5
The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.
2.3 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 → (-π/2, π/2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of π/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,+∞) , 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,+∞) , then h becomes bijective; its inverseis the positive square root function.
2.4 Inverses
A bijection f with domain X (“functionally” indicated by f: X→Y) also defines 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 satisfies 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 defined 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.
2.5 Composition
The 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 =
(f−1) ◦ (g−1) .Conversely, if the composition g ◦ f of two functions is bijective, we can only say that f is injective and g is surjective.
2.6 Bijections and cardinality
If X and Y are finite 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 definition of “same number ofelements” (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number,a way to distinguish the various sizes of infinite sets.
6 CHAPTER 2. BIJECTION
X1
2
3
YD
B
C
A
ZP
Q
R
A bijection composed of an injection (left) and a surjection (right).
2.7 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 |f−1(B)| = |B|.
• If X and Y are finite 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 finite 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 set—namely, n!.
2.8 Bijections and category theory
Bijections 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.
2.9. GENERALIZATION TO PARTIAL FUNCTIONS 7
2.9 Generalization to partial functions
The 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 undefined for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]
Another way of defining 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:A′→B′, where A′ is a subset of Aand likewise B′⊆B.[3]
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is the Möbius transformation simply defined on the complex plane, rather than its completion to the extendedcomplex plane.[5]
2.10 Contrast withThis list is incomplete; you can help by expanding it.
• Multivalued function
2.11 See also• Injective function
• Surjective function
• Bijection, injection and surjection
• Symmetric group
• Bijective numeration
• Bijective proof
• Cardinality
• Category theory
• Ax–Grothendieck theorem
2.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satisfies 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 Möbius Inverse Monoid”. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.
8 CHAPTER 2. BIJECTION
2.13 References
This 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 Mathematician’s 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. Foundations of Higher Mathematics. PWS-Kent.
• Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.
• Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.
• D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.
• Cupillari. The Nuts and Bolts of Proofs. Wadsworth.
• Bond. Introduction to Abstract Mathematics. Brooks/Cole.
• Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.
• Ash. A Primer of Abstract Mathematics. MAA.
2.14 External links• Hazewinkel, Michiel, ed. (2001), “Bijection”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-
010-4
• Weisstein, Eric W., “Bijection”, MathWorld.
• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.
Chapter 3
Cardinal number
This article is about the mathematical concept. For number words indicating quantity (“three” apples, “four” birds,etc.), see Cardinal number (linguistics).
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used tomeasure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements inthe set. The transfinite cardinal numbers describe the sizes of infinite sets.Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is aone-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees withthe intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due toGeorg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinalityof the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a propersubset of an infinite set to have the same cardinality as the original set, something that cannot happen with propersubsets of finite sets.There is a transfinite sequence of cardinal numbers:
0, 1, 2, 3, . . . , n, . . . ;ℵ0,ℵ1,ℵ2, . . . ,ℵα, . . . .
This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the alephnumbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under theassumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects thataxiom, the situation is more complicated, with additional infinite cardinals that are not alephs.Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics includingcombinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeletonof the category of sets.
3.1 History
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal,but have the same cardinality, namely three (this is established by the existence of a bijection, i.e. a one-to-onecorrespondence, between the two sets; e.g. {1->4, 2->5, 3->6}).Cantor applied his concept of bijection to infinite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus,all sets having a bijection with N he called denumerable (countably infinite) sets and they all have the same cardinalnumber. This cardinal number is calledℵ0 , aleph-null. He called the cardinal numbers of these infinite sets, transfinitecardinal numbers.Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to runcontrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (whichimplies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers isalso denumerable. Each algebraic number z may be encoded as a finite sequence of integers which are the coefficients
9
10 CHAPTER 3. CARDINAL NUMBER
X 1
2
3
4
YD
B
C
A
A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to thecardinal number 4.
in the polynomial equation of which it is the solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai ∈ Z together with apair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies inthe interval (b0, b1).In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbershas cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but inan 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal numberof the set of real numbers is called the cardinality of the continuum and Cantor used the symbol c for it.Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallesttransfinite cardinal number ( ℵ0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal
(ℵ1,ℵ2,ℵ3, · · · ).
His continuum hypothesis is the proposition that c is the same as ℵ1 . This hypothesis has been found to be inde-pendent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standardassumptions.
3.2. MOTIVATION 11
Aleph null, the smallest infinite cardinal
3.2 Motivation
In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included:0, 1, 2, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactlywhat can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematicsand logic.More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe theposition of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide,since for every number describing a position in a sequence we can construct a set which has exactly the right size,e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has3 elements. However when dealing with infinite sets it is essential to distinguish between the two — the two notionsare in fact different for infinite sets. Considering the position aspect leads to ordinal numbers, while the size aspect isgeneralized by the cardinal numbers described here.The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or “bigness” ofa set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the numberof elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.
12 CHAPTER 3. CARDINAL NUMBER
A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y.An injective mapping identifies each element of the set X with a unique element of the set Y. This is most easilyunderstood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size wewould observe that there is a mapping:
1 → a2 → b3 → c
which is injective, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has noelement mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injectiveand onto mapping. The advantage of this notion is that it can be extended to infinite sets.We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if thereexists a bijection between X and Y. By the Schroeder–Bernstein theorem, this is equivalent to there being both aninjective mapping from X to Y and an injective mapping from Y to X. We then write |X| = |Y |. The cardinal numberof X itself is often defined as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; forthis definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement isthe well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigningnames to objects.The classic example used is that of the infinite hotel paradox, also called Hilbert’s paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guestarrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest inroom 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:
1 → 22 → 33 → 4...n → n + 1...
In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a bijection betweenthe first and the second has been shown. This motivates the definition of an infinite set being any set which has aproper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}.When considering these large objects, we might also want to see if the notion of counting order coincides with thatof cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we cansee that if some object “one greater than infinity” exists, then it must have the same cardinality as the infinite setwe started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideasof counting and considering each number in turn, and we discover that the notions of cardinality and ordinality aredivergent once we move out of the finite numbers.It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.This can be visualized using Cantor’s diagonal argument; classic questions of cardinality (for instance the continuumhypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite car-dinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality issometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the samecardinality are, respectively, equipotent, equipollent, or equinumerous.
3.3 Formal definition
Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijectionbetween X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not
3.4. CARDINAL ARITHMETIC 13
assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor andexplicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This doesnot work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is toolarge to be a set. In fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × Xand so by the axiom of limitation of size, [X] is a proper class. The definition does work however in type theory andin New Foundations and related systems. However, if we restrict from this class to those equinumerous with X thathave the least rank, then it will work (this is a trick due to Dana Scott:[2] it works because the collection of objectswith any given rank is a set).Formally, the order among cardinal numbers is defined as follows: |X| ≤ |Y | means that there exists an injectivefunction from X to Y. The Cantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y | and |Y | ≤ |X| then |X| = |Y |.The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y | or |Y | ≤ |X|.[3][4]
A set X is Dedekind-infinite if there exists a proper subset Y of X with |X| = |Y |, and Dedekind-finite if such a subsetdoesn't exist. The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if |X| = |n| = n forsome natural number n. Any other set is infinite. Assuming the axiom of choice, it can be proved that the Dedekindnotions correspond to the standard ones. It can also be proved that the cardinal ℵ0 (aleph null or aleph-0, where alephis the first letter in the Hebrew alphabet, represented ℵ ) of the set of natural numbers is the smallest infinite cardinal,i.e. that any infinite set has a subset of cardinality ℵ0. The next larger cardinal is denoted by ℵ1 and so on. For everyordinal α there is a cardinal number ℵα, and this list exhausts all infinite cardinal numbers.
3.4 Cardinal arithmetic
We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers.It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers.Furthermore, these operations share many properties with ordinary arithmetic.
3.4.1 Successor cardinal
For more details on this topic, see Successor cardinal.
If the axiom of choice holds, every cardinal κ has a successor κ+ > κ, and there are no cardinals between κ and itssuccessor. (Without the axiom of choice, using Hartogs’ theorem, it can be shown that, for any cardinal number κ,there is a minimal cardinal κ+, so that κ+ ≰ κ. ) For finite cardinals, the successor is simply κ + 1. For infinitecardinals, the successor cardinal differs from the successor ordinal.
3.4.2 Cardinal addition
If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then theycan be replaced by disjoint sets of the same cardinality, e.g., replace X by X×{0} and Y by Y×{1}.
|X|+ |Y | = |X ∪ Y |.
Zero is an additive identity κ + 0 = 0 + κ = κ.Addition is associative (κ + μ) + ν = κ + (μ + ν).Addition is commutative κ + μ = μ + κ.Addition is non-decreasing in both arguments:
(κ ≤ µ) → ((κ+ ν ≤ µ+ ν) and (ν + κ ≤ ν + µ)).
Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then
κ+ µ = max{κ, µ} .
14 CHAPTER 3. CARDINAL NUMBER
Subtraction
Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ+ κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.
3.4.3 Cardinal multiplication
The product of cardinals comes from the cartesian product.
|X| · |Y | = |X × Y |
κ·0 = 0·κ = 0.κ·μ = 0 → (κ = 0 or μ = 0).One is a multiplicative identity κ·1 = 1·κ = κ.Multiplication is associative (κ·μ)·ν = κ·(μ·ν).Multiplication is commutative κ·μ = μ·κ.Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ.Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite andboth are non-zero, then
κ · µ = max{κ, µ}.
Division
Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κsuch that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π.
3.4.4 Cardinal exponentiation
Exponentiation is given by
|X||Y | =∣∣XY
∣∣where XY is the set of all functions from Y to X.
κ0 = 1 (in particular 00 = 1), see empty function.If 1 ≤ μ, then 0μ = 0.1μ = 1.κ1 = κ.κμ + ν = κμ·κν.κμ · ν = (κμ)ν.(κ·μ)ν = κν·μν.
Exponentiation is non-decreasing in both arguments:
(1 ≤ ν and κ ≤ μ) → (νκ ≤ νμ) and(κ ≤ μ) → (κν ≤ μν).
3.5. THE CONTINUUM HYPOTHESIS 15
Note that 2|X| is the cardinality of the power set of the set X and Cantor’s diagonal argument shows that 2|X| > |X| forany set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal2κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)All the remaining propositions in this section assume the axiom of choice:
If κ and μ are both finite and greater than 1, and ν is infinite, then κν = μν.If κ is infinite and μ is finite and non-zero, then κμ = κ.
If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:
Max (κ, 2μ) ≤ κμ ≤ Max (2κ, 2μ).
Using König’s theorem, one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the cofinalityof κ.
Roots
Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal νsatisfying νµ = κ will be κ.
Logarithms
Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may ormay not be a cardinal λ satisfying µλ = κ . However, if such a cardinal exists, it is infinite and less than κ, and anyfinite cardinality ν greater than 1 will also satisfy νλ = κ .The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithmsof infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants oftopological spaces, though they lack some of the properties that logarithms of positive real numbers possess.[5][6][7]
3.5 The continuum hypothesis
The continuum hypothesis (CH) states that there are no cardinals strictly between ℵ0 and 2ℵ0 . The latter cardinalnumber is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case2ℵ0 = ℵ1. The generalized continuum hypothesis (GCH) states that for every infinite set X, there are no cardinalsstrictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, theZermelo-Fraenkel axioms together with the axiom of choice (ZFC).
3.6 See also
3.7 Notes
3.8 References
Notes
[1] Dauben 1990, pg. 54
[2] Deiser, Oliver (May 2010). “On the Development of the Notion of a Cardinal Number”. History and Philosophy of Logic31 (2): 123–143. doi:10.1080/01445340903545904.
[3] Enderton, Herbert. “Elements of Set Theory”, Academic Press Inc., 1977. ISBN 0-12-238440-7
16 CHAPTER 3. CARDINAL NUMBER
[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "Über das Problem derWohlordnung”, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438–443, ISSN 0025-5831
[5] Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathe-matics 1315, Springer-Verlag.
[6] Eduard Čech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
[7] D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.
Bibliography
• Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton:Princeton University Press, ISBN 0691-02447-2
• Hahn, Hans, Infinity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon andSchuster, 1956.
• Halmos, Paul, 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).
3.9 External links• Hazewinkel, Michiel, ed. (2001), “Cardinal number”, Encyclopedia of Mathematics, Springer, ISBN 978-1-
55608-010-4
• Weisstein, Eric W., “Cardinal Number”, MathWorld.
• Cardinality at ProvenMath proofs of the basic theorems on cardinality.
Chapter 4
Cardinality
For other uses, see Cardinality (disambiguation).
In mathematics, the cardinality of a set is a measure of the “number of elements of the set”. For example, the setA = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality– one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1] Thecardinality of a set is also called its size, when no confusion with other notions of size[2] is possible.The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolutevalue and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A,card(A), or # A.
4.1 Comparing sets
While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usuallystarts with defining the notion of comparison of arbitrary (in particular infinite) sets.
4.1.1 Definition 1: | A | = | B |
Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjectivefunction, from A to B. Such sets are said to be equipotent, equipollent, or equinumerous.
For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as theset N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E.
4.1.2 Definition 2: | A | ≤ | B |
A has cardinality less than or equal to the cardinality of B if there exists an injective function from Ainto B.
4.1.3 Definition 3: | A | < | B |
A has cardinality strictly less than the cardinality of B if there is an injective function, but no bijectivefunction, from A to B.
For example, the setN of all natural numbers has cardinality strictly less than the cardinality of the setRof all real numbers , because the inclusion map i : N→R is injective, but it can be shown that there doesnot exist a bijective function fromN toR (see Cantor’s diagonal argument or Cantor’s first uncountabilityproof).
17
18 CHAPTER 4. CARDINALITY
EN
012
n
024
2n
. .
. .
Bijective function from N to E. Although E is a proper subset of N, both sets have the same cardinality.
If | A | ≤ | B | and | B | ≤ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). The axiom of choice isequivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A,B.[3][4]
4.2 Cardinal numbers
Main article: Cardinal number
Above, “cardinality” was defined functionally. That is, the “cardinality” of a set was not defined as a specific objectitself. However, such an object can be defined as follows.The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the classof all sets. The equivalence class of a set A under this relation then consists of all those sets which have the samecardinality as A. There are two ways to define the “cardinality of a set":
1. The cardinality of a set A is defined as its equivalence class under equinumerosity.
2. A representative set is designated for each equivalence class. The most common choice is the initial ordinal inthat class. This is usually taken as the definition of cardinal number in axiomatic set theory.
Assuming AC, the cardinalities of the infinite sets are denoted
ℵ0 < ℵ1 < ℵ2 < . . . .
For each ordinal α , ℵα+1 is the least cardinal number greater than ℵα .
4.3. FINITE, COUNTABLE AND UNCOUNTABLE SETS 19
The cardinality of the natural numbers is denoted aleph-null (ℵ0 ), while the cardinality of the real numbers is denotedby " c " (a lowercase fraktur script “c”), and is also referred to as the cardinality of the continuum. Cantor showed,using the diagonal argument, that c > ℵ0 . We can show that c = 2ℵ0 , this also being the cardinality of the set ofall subsets of the natural numbers. The continuum hypothesis says that ℵ1 = 2ℵ0 , i.e. 2ℵ0 is the smallest cardinalnumber bigger than ℵ0 , i.e. there is no set whose cardinality is strictly between that of the integers and that of thereal numbers. The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that is, it isimpossible to prove the continuum hypothesis or its negation from ZFC (provided ZFC is consistent). See below formore details on the cardinality of the continuum.[5][6][7]
4.3 Finite, countable and uncountable sets
If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:
• Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set.
• Any set X that has the same cardinality as the set of the natural numbers, or | X | = | N | = ℵ0 , is said to be acountably infinite set.
• Any set X with cardinality greater than that of the natural numbers, or | X | > | N |, for example | R | = c > | N|, is said to be uncountable.
4.4 Infinite sets
Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century GeorgCantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) thatthe whole cannot be the same size as the part. One example of this is Hilbert’s paradox of the Grand Hotel. Indeed,Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (thatis, having the same size in Cantor’s sense); this notion of infinity is called Dedekind infinite. Cantor introduced thecardinal numbers, and showed that (according to his bijection-based definition of size) some infinite sets are greaterthan others. The smallest infinite cardinality is that of the natural numbers ( ℵ0 ).
4.4.1 Cardinality of the continuum
Main article: Cardinality of the continuum
One of Cantor’s most important results was that the cardinality of the continuum ( c ) is greater than that of the naturalnumbers ( ℵ0 ); that is, there are more real numbers R than whole numbers N. Namely, Cantor showed that
c = 2ℵ0 > ℵ0
(see Cantor’s diagonal argument or Cantor’s first uncountability proof).
The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardi-nality of the natural numbers, that is,
c = ℵ1 = ℶ1
(see Beth one).
However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory,if ZFC is consistent.Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to thenumber of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed,in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper
20 CHAPTER 4. CARDINALITY
subsets and proper supersets of an infinite set S that have the same size as S, although S contains elements that do notbelong to its subsets, and the supersets of S contain elements that are not included in it.The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-onecorrespondence between the interval (−½π, ½π) and R (see also Hilbert’s paradox of the Grand Hotel).The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when GiuseppePeano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, orcube, or hypercube, or finite-dimensional space. These curves are not a direct proof that a line has the same numberof points as a finite-dimensional space, but they can be used to obtain such a proof.Cantor also showed that sets with cardinality strictly greater than c exist (see his generalized diagonal argument andtheorem). They include, for instance:
• the set of all subsets of R, i.e., the power set of R, written P(R) or 2R
• the set RR of all functions from R to R
Both have cardinality
2c = ℶ2 > c
(see Beth two).
The cardinal equalities c2 = c, cℵ0 = c, and cc = 2c can be demonstrated using cardinal arithmetic:
c2 =(2ℵ0
)2= 22×ℵ0 = 2ℵ0 = c,
cℵ0 =(2ℵ0
)ℵ0= 2ℵ0×ℵ0 = 2ℵ0 = c,
cc =(2ℵ0
)c= 2c×ℵ0 = 2c.
4.5 Examples and properties
• If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | because { (a, apples), (b, oranges), (c,peaches)} is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.
• If | X | < | Y |, then there exists Z such that | X | = | Z | and Z ⊆ Y.
• If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known asCantor–Bernstein–Schroeder theorem.
• Sets with cardinality of the continuum
4.6 Union and intersection
If A and B are disjoint sets, then
|A ∪B| = |A|+ |B| .
From this, one can show that in general the cardinalities of unions and intersections are related by[8]
|C ∪D|+ |C ∩D| = |C|+ |D| .
4.7. SEE ALSO 21
4.7 See also• Aleph number
• Beth number
• Countable set
• Ordinality
4.8 References[1] Weisstein, Eric W., “Cardinal Number”, MathWorld.
[2] Such as length and area in geometry. – A line of finite length is a set of points that has infinite cardinality.
[3] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "Über das Problemder Wohlordnung”, Mathematische Annalen (Leipzig: B. G. Teubner) 76 (4): 438–443, doi:10.1007/bf01458215, ISSN0025-5831
[4] Felix Hausdorff (2002), Egbert Brieskorn, Srishti D. Chatterji et al., eds., Grundzüge derMengenlehre (1. ed.), Berlin/Heidelberg:Springer, p. 587, ISBN 3-540-42224-2 - Original edition (1914)
[5] Cohen, Paul J. (December 15, 1963). “The Independence of the Continuum Hypothesis”. Proceedings of the NationalAcademy of Sciences of the United States of America 50 (6): 1143–1148. doi:10.1073/pnas.50.6.1143. JSTOR 71858.PMC 221287. PMID 16578557.
[6] Cohen, Paul J. (January 15, 1964). “The Independence of the Continuum Hypothesis, II”. Proceedings of the NationalAcademy of Sciences of the United States of America 51 (1): 105–110. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC300611. PMID 16591132.
[7] Penrose, R (2005), The Road to Reality: A Complete guide to the Laws of the Universe, Vintage Books, ISBN 0-09-944068-7
[8] Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983, ISBN 0-85312-612-7 (student edition),ISBN 0-85312-563-5 (library edition)
Chapter 5
Cofiniteness
Not to be confused with cofinality.
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, Acontains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the setis cocountable.These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as inthe product topology or direct sum.
5.1 Boolean algebras
The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operationsof union, intersection, and complementation. This Boolean algebra is the finite-cofinite algebra on X. A Booleanalgebra A has a unique non-principal ultrafilter (i.e. a maximal filter not generated by a single element of the algebra)if and only if there is an infinite set X such that A is isomorphic to the finite-cofinite algebra on X. In this case, thenon-principal ultrafilter is the set of all cofinite sets.
5.2 Cofinite topology
The cofinite topology (sometimes called the finite complement topology) is a topology which can be defined onevery set X. It has precisely the empty set and all cofinite subsets of X as open sets. As a consequence, in the cofinitetopology, the only closed subsets are finite sets, or the whole of X. Symbolically, one writes the topology as
T = {A ⊆ X | A = ∅ or X \A is finite}
This topology occurs naturally in the context of the Zariski topology. Since polynomials over a field K are zero onfinite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The sameis true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane.
5.2.1 Properties
• Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.
• Compactness: Since every open set contains all but finitely many points of X, the space X is compact andsequentially compact.
• Separation: The cofinite topology is the coarsest topology satisfying the T1 axiom; i.e. it is the smallest topologyfor which every singleton set is closed. In fact, an arbitrary topology on X satisfies the T1 axiom if and only if
22
5.3. OTHER EXAMPLES 23
it contains the cofinite topology. If X is finite then the cofinite topology is simply the discrete topology. If X isnot finite, then this topology is not T2, regular or normal, since no two nonempty open sets are disjoint (i.e. itis hyperconnected).
5.2.2 Double-pointed cofinite topology
The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topologicalproduct of the cofinite topology with the indiscrete topology. It is not T0 or T1, since the points of the doublet aretopologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable.An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology thatgroups them together. Let X be the set of integers, and let OA be a subset of the integers whose complement is the setA. Define a subbase of open sets Gx for any integer x to be Gx = O{x, x₊₁} if x is an even number, and Gx = O{x−₁,x} if x is odd. Then the basis sets of X are generated by finite intersections, that is, for finite A, the open sets of thetopology are
UA :=∩x∈A
Gx
The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologicallyindistinguishable. The space is, however, a compact space, since it is covered by a finite union of the UA.
5.3 Other examples
5.3.1 Product topology
The product topology on a product of topological spaces∏
Xi has basis∏
Ui where Ui ⊂ Xi is open, and cofinitelymany Ui = Xi .The analog (without requiring that cofinitely many are the whole space) is the box topology.
5.3.2 Direct sum
The elements of the direct sum of modules⊕
Mi are sequences αi ∈ Mi where cofinitely many αi = 0 .The analog (without requiring that cofinitely many are zero) is the direct product.
5.4 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of
1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 18)
Chapter 6
Derived set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the setof all limit points of S. It is usually denoted by S′ .The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derivedsets on the real line.
6.1 Properties
A subset S of a topological space is closed precisely when S′ ⊆ S , when S contains all its limit points. Two subsetsS and T are separated precisely when they are disjoint and each is disjoint from the other’s derived set (though thederived sets don't need to be disjoint from each other).The set S is defined to be a perfect set if S = S′ . Equivalently, a perfect set is a closed set with no isolated points.Perfect sets are particularly important in applications of the Baire category theorem.Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derivedset of the image of any subset is the image of the derived set of that subset.The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and aperfect set. Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδsubset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.
6.2 Topology in terms of derived sets
Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as theprimitive notion in topology. A set of points X can be equipped with an operator * mapping subsets of X to subsetsof X, such that for any set S and any point a:
1. ∅∗ = ∅
2. S∗∗ ⊆ S∗
3. a ∈ S∗ =⇒ a ∈ (S \ {a})∗
4. (S ∪ T )∗ ⊆ S∗ ∪ T ∗
5. S ⊆ T =⇒ S∗ ⊆ T ∗
Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have thefollowing equivalent axioms:
1. ∅∗ = ∅
24
6.3. CANTOR–BENDIXSON RANK 25
2. S∗∗ ⊆ S∗
• 3'. S∗ = (S \ {a})∗
• 4'. (S ∪ T )∗ = S∗ ∪ T ∗
Calling a set S closed if S∗ ⊆ S will define a topology on the space in which * is the derived set operator, that is,S∗ = S′ . If we also require that the derived set of a set consisting of a single element be empty, the resulting spacewill be a T1 space. In fact, 2 and 3' can fail in a space that is not T1.
6.3 Cantor–Bendixson rank
For ordinal numbers α, the α-th Cantor–Bendixson derivative of a topological space is defined by transfinite in-duction as follows:
• X0 = X
• Xα+1 = (Xα)′
• Xλ =∩α<λ
Xα for limit ordinals λ.
The transfinite sequence of Cantor–Bendixson derivatives of X must eventually be constant. The smallest ordinal αsuch that Xα+1 = Xα is called the Cantor–Bendixson rank of X.
6.4 See also• Perfect space
6.5 External links• PlanetMath’s article on the Cantor–Bendixson derivative
6.6 References• Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts in Mathematics 156 ed.). Springer. ISBN
978-0-387-94374-9.
• Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952). General Topology. University of TorontoPress.
Chapter 7
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in whichthe points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discretetopology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, eachsingleton is an open set in the discrete topology.
7.1 Definitions
Given a set X:
• the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is adiscrete topological space if it is equipped with its discrete topology;
• the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × Xbe an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
• the discrete metric ρ on X is defined by
ρ(x, y) =
{1 if x ̸= y,0 if x = y
for any x, y ∈ X . In this case (X, ρ) is called a discrete metric space or a space of isolated points.
• a set S is discrete in a metric space (X, d) , for S ⊆ X , if for every x ∈ S , there exists some δ > 0(depending on x ) such that d(x, y) > δ for all y ∈ S \ {x} ; such a set consists of isolated points. A set S isuniformly discrete in the metric space (X, d) , for S ⊆ X , if there exists ε > 0 such that for any two distinctx, y ∈ S , d(x, y) > ε.
A metric space (E, d) is said to be uniformly discrete if there exists a “packing radius” r > 0 such that, for anyx, y ∈ E , one has either x = y or d(x, y) > r .[1] The topology underlying a metric space can be discrete, withoutthe metric being uniformly discrete: for example the usual metric on the set {1, 1/2, 1/4, 1/8, ...} of real numbers.
7.2 Properties
The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on adiscrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with oneanother. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; anexample is the metric space X := {1/n : n = 1,2,3,...} (with metric inherited from the real line and given by d(x,y) =|x − y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniformspace. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformlydiscrete or metrically discrete.Additionally:
26
7.2. PROPERTIES 27
• The topological dimension of a discrete space is equal to 0.
• A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn'tcontain any accumulation points.
• The singletons form a basis for the discrete topology.
• A uniform space X is discrete if and only if the diagonal {(x,x) : x is in X} is an entourage.
• Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space isHausdorff, that is, separated.
• A discrete space is compact if and only if it is finite.
• Every discrete uniform or metric space is complete.
• Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it isfinite.
• Every discrete metric space is bounded.
• Every discrete space is first-countable; it is moreover second-countable if and only if it is countable.
• Every discrete space with at least two points is totally disconnected.
• Every non-empty discrete space is second category.
• Any two discrete spaces with the same cardinality are homeomorphic.
• Every discrete space is metrizable (by the discrete metric).
• A finite space is metrizable only if it is discrete.
• If X is a topological space and Y is a set carrying the discrete topology, then X is evenly covered by X × Y (theprojection map is the desired covering)
• The subspace topology on the integers as a subspace of the real line is the discrete topology.
• A discrete space is separable if and only if it is countable.
Any function from a discrete topological space to another topological space is continuous, and any function from adiscrete uniform space to another uniform space is uniformly continuous. That is, the discrete space X is free on theset X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformlycontinuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usuallyfree on sets.With metric spaces, things are more complicated, because there are several categories of metric spaces, depending onwhat is chosen for the morphisms. Certainly the discrete metric space is free when the morphisms are all uniformlycontinuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniformor topological structure. Categories more relevant to the metric structure can be found by limiting the morphismsto Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more thanone element). However, the discrete metric space is free in the category of bounded metric spaces and Lipschitzcontinuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any functionfrom a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discretemetric space to another metric space bounded by 1 is short.Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only ifit is locally constant in the sense that every point in Y has a neighborhood on which f is constant.
28 CHAPTER 7. DISCRETE SPACE
7.3 Uses
A discrete structure is often used as the “default structure” on a set that doesn't carry any other natural topology,uniformity, or metric; discrete structures can often be used as “extreme” examples to test particular suppositions.For example, any group can be considered as a topological group by giving it the discrete topology, implying thattheorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topologicalgroups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example incombination with Pontryagin duality. A 0-dimensional manifold (or differentiable or analytical manifold) is nothingbut a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group.A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space ofirrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countablyinfinite copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphicto the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by using ternarynotation of numbers. (See Cantor space.)In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topo-logical approach to the ultrafilter principle, which is a weak form of choice.
7.4 Indiscrete spaces
Main article: Trivial topology
In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), whichhas the fewest possible open sets (just the empty set and the space itself). Where the discrete topology is initialor free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space iscontinuous, etc.
7.5 Quotation• Stanislaw Ulam characterized Los Angeles as “a discrete space, in which there is an hour’s drive between
points”.[2]
7.6 See also• Cylinder set
• Taxicab geometry
7.7 References[1] Pleasants, Peter A.B. (2000). “Designer quasicrystals: Cut-and-project sets with pre-assigned properties”. In Baake,
Michael. Directions in mathematical quasicrystals. CRM Monograph Series 13. Providence, RI: American MathematicalSociety. pp. 95–141. ISBN 0-8218-2629-8. Zbl 0982.52018.
[2] Stanislaw Ulam's autobiography, Adventures of a Mathematician.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York:Springer-Verlag. ISBN 3-540-90312-7. MR 507446. Zbl 0386.54001.
Chapter 8
First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted byω1 or sometimes byΩ, is the smallest ordinalnumber that, considered as a set, is uncountable. It is the supremum of all countable ordinals. The elements of ω1
are the countable ordinals, of which there are uncountably many.Like any ordinal number (in von Neumann’s approach), ω1 is a well-ordered set, with set membership ("∈") servingas the order relation. ω1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω1.The cardinality of the set ω1 is the first uncountable cardinal number, ℵ1 (aleph-one). The ordinal ω1 is thus theinitial ordinal of ℵ1. Indeed, in most constructions ω1 and ℵ1 are equal as sets. To generalize: if α is an arbitraryordinal we define ωα as the initial ordinal of the cardinal ℵα.The existence of ω1 can be proven without the axiom of choice. (See Hartogs number.)
8.1 Topological properties
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topologicalspace, ω1 is often written as [0,ω1) to emphasize that it is the space consisting of all ordinals smaller than ω1.Every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union(=supremum) of every countable set of countable ordinals is another countable ordinal.The topological space [0,ω1) is sequentially compact but not compact. As a consequence, it is not metrizable. It ishowever countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω1) is first countable butnot separable nor second countable.The space [0, ω1] = ω1 + 1 is compact and not first countable. ω1 is used to define the long line and the Tychonoffplank, two important counterexamples in topology.
8.2 See also• Ordinal arithmetic
• Large countable ordinal
• Continuum hypothesis
8.3 References• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN
3-540-44085-2.
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York,1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
29
Chapter 9
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ];March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventorof set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the realnumbers are “more numerous” than the natural numbers. In fact, Cantor’s method of proof of this theorem impliesthe existence of an "infinity of infinities”. He defined the cardinal and ordinal numbers and their arithmetic. Cantor’swork is of great philosophical interest, a fact of which he was well aware.[2]
Cantor’s theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encoun-tered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later fromHermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devoutLutheran,[4] believed the theory had been communicated to him by God.[5] Some Christian theologians (particularlyneo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God[6] –on one occasion equating the theory of transfinite numbers with pantheism[7] – a proposition that Cantor vigorouslyrejected.The objections to Cantor’s work were occasionally fierce: Poincaré referred to his ideas as a “grave disease” infectingthe discipline of mathematics,[8] and Kronecker's public opposition and personal attacks included describing Cantoras a “scientific charlatan”, a “renegade” and a “corrupter of youth.”[9] Kronecker objected to Cantor’s proofs thatthe algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included ina standard mathematics curriculum. Writing decades after Cantor’s death, Wittgenstein lamented that mathematicsis “ridden through and through with the pernicious idioms of set theory,” which he dismissed as “utter nonsense”that is “laughable” and “wrong”.[10] Cantor’s recurring bouts of depression from 1884 to the end of his life havebeen blamed on the hostile attitude of many of his contemporaries,[11] though some have explained these episodes asprobable manifestations of a bipolar disorder.[12]
The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its SylvesterMedal, the highest honor it can confer for work in mathematics.[13] David Hilbert defended it from its critics byfamously declaring: “No one shall expel us from the Paradise that Cantor has created.”[14][15]
9.1 Life
9.1.1 Youth and studies
Cantor was born in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he waseleven. Georg, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm(1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperialorchestra.[16] Cantor’s father had been a member of the Saint Petersburg stock exchange; when he became ill, thefamily moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of SaintPetersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skillsin mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the University of Zürich. Afterreceiving a substantial inheritance upon his father’s death in 1863, Cantor shifted his studies to the University of
30
9.1. LIFE 31
Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of1866 at the University of Göttingen, then and later a center for mathematical research.
9.1.2 Teacher and researcher
Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly ina Berlin girls’ school, Cantor took up a position at the University of Halle, where he spent his entire career. Hewas awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon hisappointment at Halle.[17]
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was ableto support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoonin the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he hadmet two years earlier while on Swiss holiday.Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank atthe age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular atBerlin, at that time the leading German university. However, his work encountered too much opposition for that to bepossible.[18] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortablewith the prospect of having Cantor as a colleague,[19] perceiving him as a “corrupter of youth” for teaching his ideas toa younger generation of mathematicians.[20] Worse yet, Kronecker, a well-established figure within the mathematicalcommunity and Cantor’s former professor, disagreed fundamentally with the thrust of Cantor’s work. Kronecker, nowseen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor’s set theory becauseit asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose membersdid indeed satisfy those properties. Cantor came to believe that Kronecker’s stance would make it impossible for himever to leave Halle.In 1881, Cantor’s Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor’s suggestionthat it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair afterbeing offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.In 1882, the mathematical correspondence between Cantor and Richard Dedekind came to an end, apparently as aresult of Dedekind’s declining the chair at Halle.[21] Cantor also began another important correspondence, with GöstaMittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler’s journal Acta Mathematica. But in 1885,Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submittedto Acta.[22] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about onehundred years too soon.” Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler,writing to a third party:
Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed toogreat a demand! ... But of course I never want to know anything again about Acta Mathematica.[23]
Cantor suffered his first known bout of depression in 1884.[24] Criticism of his work weighed on his mind: every oneof the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these lettersis revealing of the damage to Cantor’s self-confidence:
... I don't know when I shall return to the continuation of my scientific work. At the moment I cando absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how muchhappier I would be to be scientifically active, if only I had the necessary mental freshness.[25]
This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study ofElizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to Shakespeare(see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.[26]
Cantor recovered soon thereafter, and subsequently made further important contributions, including his famousdiagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84.He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreementsand difficulties dividing them persisted.In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting inHalle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker’s
32 CHAPTER 9. GEORG CANTOR
opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosityKronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to doso because his wife was dying from injuries sustained in a skiing accident at the time.
9.1.3 Late years
After Cantor’s 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.[24] Soon afterthat second hospitalization, Cantor’s youngest son Rudolph died suddenly (while Cantor was delivering a lecture onhis views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion formathematics.[27] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paperpresented by Julius König at the Third International Congress of Mathematicians. The paper attempted to provethat the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters andcolleagues, Cantor perceived himself as having been publicly humiliated.[28] Although Ernst Zermelo demonstratedless than a day later that König’s proof had failed, Cantor remained shaken, and momentarily questioning God.[13]
Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on severaloccasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizationsat intervals of two or three years.[29] He did not abandon mathematics completely, however, lecturing on the para-doxes of set theory (Burali-Forti paradox, Cantor’s paradox, and Russell’s paradox) to a meeting of the DeutscheMathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in1904.In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the foundingof the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newlypublished Principia Mathematica repeatedly cited Cantor’s work, but this did not come about. The following year,St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[30] The publiccelebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatoriumwhere he had spent the final year of his life.
9.2 Mathematical work
Cantor’s work between 1874 and 1884 is the origin of set theory.[31] Prior to this work, the concept of a set wasa rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideasof Aristotle.[32] No one had realized that set theory had any nontrivial content. Before Cantor, there were onlyfinite sets (which are easy to understand) and “the infinite” (which was considered a topic for philosophical, ratherthan mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantorestablished that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of afoundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects(for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis andtopology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts ofset theory are now used throughout mathematics.[33]
In one of his earliest papers,[34] Cantor proved that the set of real numbers is “more numerous” than the set of naturalnumbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciatethe importance of one-to-one correspondences (hereinafter denoted “1-to-1 correspondence”) in set theory. He usedthis concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets anduncountable sets (nondenumerable infinite sets).[35]
Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that theCantor set is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals areeverywhere dense, but countable.Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of allpossible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, evenwhen A is an infinite set; this result soon became known as Cantor’s theorem. Cantor developed an entire theory andarithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. Hisnotation for the cardinal numbers was the Hebrew letter ℵ (aleph) with a natural number subscript; for the ordinalshe employed the Greek letter ω (omega). This notation is still in use today.
9.2. MATHEMATICAL WORK 33
The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three openproblems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantor’s work alsoattracted favorable notice beyond Hilbert’s celebrated encomium.[15] The US philosopher Charles Sanders Peircepraised Cantor’s set theory, and, following public lectures delivered by Cantor at the first International Congressof Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration. At thatCongress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded withhis British admirer and translator Philip Jourdain on the history of set theory and on Cantor’s religious ideas. Thiswas later published, as were several of his expository works.
9.2.1 Number theory, trigonometric series and ordinals
Cantor’s first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professorat Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Peter GustavLejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation ofa function by trigonometric series. Cantor solved this difficult problem in 1869. It was while working on this problemthat he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of atrigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedurethat produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. IfSk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. Becausethe sets Sk were closed, they contained their Limit points, and the intersection of the infinite decreasing sequence ofsets S, S1, S2, S3,... formed a limit set, which we would now call Sω, and then he noticed that Sω would also haveto have a set of limit points Sω₊₁, and so on. He had examples that went on forever, and so here was a naturallyoccurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, ...[36]
Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrationalnumbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paperlater that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. Whileextending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxicallyopposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing themas both “an abomination” and “a cholera bacillus of mathematics”.[37] Cantor also published an erroneous “proof” ofthe inconsistency of infinitesimals.[38]
9.2.2 Set theory
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor’s 1874 article,[31]
"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” ("On a Property of the Collection of AllReal Algebraic Numbers").[40] This article was the first to provide a rigorous proof that there was more than one kindof infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of “the samesize” or having the same number of elements).[41] Cantor proved that the collection of real numbers and the collectionof positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs fromdiagonal argument that he gave in 1891.[42] Cantor’s article also contains a new method of constructing transcendentalnumbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.[43]
Cantor established these results using two constructions. His first construction shows how to write the real algebraicnumbers[44] as a sequence a1, a2, a3, .... In other words, the real algebraic numbers are countable. Cantor starts hissecond construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whoseintersection contains a real number not in the sequence. Since every sequence of real numbers can be used to con-struct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are notcountable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendentalnumber. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville’s theo-rem: Every interval contains infinitely many transcendental numbers.[45] Cantor’s next article contains a constructionthat proves the set of transcendental numbers has the same “power” (see below) as the set of real numbers.[46]
Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed anintroduction to his set theory. At the same time, there was growing opposition to Cantor’s ideas, led by Kronecker,who admitted mathematical concepts only if they could be constructed in a finite number of steps from the naturalnumbers, which he took as intuitively given. For Kronecker, Cantor’s hierarchy of infinities was inadmissible, sinceaccepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of
34 CHAPTER 9. GEORG CANTOR
mathematics as a whole.[47] Cantor also introduced the Cantor set during this period.The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre” ("Foundations of a GeneralTheory of Aggregates”), published in 1883,[48] was the most important of the six and was also published as a separatemonograph. It contained Cantor’s reply to his critics and showed how the transfinite numbers were a systematicextension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced asthe order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinalnumbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a specialcase of order types.In 1891, he published a paper containing his elegant “diagonal argument” for the existence of an uncountable set.He applied the same idea to prove Cantor’s theorem: the cardinality of the power set of a set A is strictly larger thanthe cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinalarithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proofof Gödel’s first incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894.In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; thesewere his last significant papers on set theory.[49] The first paper begins by defining set, subset, etc., in ways that wouldbe largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper toinclude a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets andordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalentto a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof,as well as Cantor’s, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the nameCantor–Bernstein–Schroeder theorem.
One-to-one correspondence
Main article: BijectionCantor’s 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use thatphrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points ofa unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positiveinteger n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in ann-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!"(“I see it, but I don't believe it!")[50] The result that he found so astonishing has implications for geometry and thenotion of dimension.In 1878, Cantor submitted another paper to Crelle’s Journal, in which he defined precisely the concept of a 1-to-1correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or “equivalence” of sets:two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor definedcountable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers,and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn hasthe same power as the real numbersR, as does a countably infinite product of copies of R. While he made free use ofcountability as a concept, he did not write the word “countable” until 1883. Cantor also discussed his thinking aboutdimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.This paper displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do soand Weierstrass supported its publication.[51] Nevertheless, Cantor never again submitted anything to Crelle.
Continuum hypothesis
Main article: Continuum hypothesis
Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no setwhose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of thereals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be trueand tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerableanxiety.[11]
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in thefield of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum
9.3. PHILOSOPHY, RELIGION AND CANTOR’S MATHEMATICS 35
hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice(the combination referred to as “ZFC”).[52]
Paradoxes of set theory
Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of theseimplied fundamental problems with Cantor’s set theory program.[53] In an 1897 paper on an unrelated topic, CesareBurali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinalsmust be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intendedto defend the basic tenets of his set theory.[13]
In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it mustbe the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than thecardinal number of A (this fact is now known as Cantor’s theorem). This paradox, together with Burali-Forti paradox,led Cantor to formulate a concept called limitation of size,[54] according to which the collection of all ordinals, or ofall sets, was an “inconsistent multiplicity” that was “too large” to be a set. Such collections later became known asproper classes.One common view among mathematicians is that these paradoxes, together with Russell’s paradox, demonstrate thatit is not possible to take a “naive”, or non-axiomatic, approach to set theory without risking contradiction, and it iscertain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Othersnote, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which canbe seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naiveset theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of theCantorian conception.[55]
9.3 Philosophy, religion and Cantor’s mathematics
The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics,philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although notin the same form as held by his critics, was long a concern of Cantor’s.[56] He directly addressed this intersectionbetween these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where hestressed the connection between his view of the infinite and the philosophical one.[57] To Cantor, his mathematicalviews were intrinsically linked to their philosophical and theological implications – he identified the Absolute Infinitewith God,[58] and he considered his work on transfinite numbers to have been directly communicated to him by God,who had chosen Cantor to reveal them to the world.[5]
Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the natureof actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate,and denied its existence.[59] Mathematicians from three major schools of thought (constructivism and its two off-shoots, intuitionism and finitism) opposed Cantor’s theories in this matter. For constructivists such as Kronecker,this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs suchas Cantor’s diagonal argument are sufficient proof that something exists, holding instead that constructive proofs arerequired. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at thedecision via a different route than constructivism. Firstly, Cantor’s argument rests on logic to prove the existence oftransfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot bereduced to logical propositions, originating instead in the intuitions of the mind.[8] Secondly, the notion of infinity asan expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infi-nite set.[60] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor’swork. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that “mostof the ideas of Cantorian set theory should be banished from mathematics once and for all.”[8] Finally, Wittgenstein'sattacks were finitist: he believed that Cantor’s diagonal argument conflated the intension of a set of cardinal or realnumbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[10]
Some Christian theologians saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature ofGod.[6] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something otherthan God as jeopardizing “God’s exclusive claim to supreme infinity”.[61] Cantor strongly believed that this view was
36 CHAPTER 9. GEORG CANTOR
a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[62]
... the transfinite species are just as much at the disposal of the intentions of the Creator and Hisabsolute boundless will as are the finite numbers.[63]
Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and wasshocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophicalbeliefs.[64]
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his settheory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor hadcorresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[65] as well as theologians suchas Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[7]
Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[62]
Cantor’s philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to positand prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The onlyrestrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction,and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertionthat “the essence of mathematics is its freedom.”[66] These ideas parallel those of Edmund Husserl, whom Cantorhad met in Halle.[67]
Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an “abomination” and “thecholera bacillus of mathematics”.Cantor’s 1883 paper reveals that he was well aware of the opposition his ideas were encountering:
... I realize that in this undertaking I place myself in a certain opposition to views widely heldconcerning the mathematical infinite and to opinions frequently defended on the nature of numbers.[68]
Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freelyintroduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He alsocites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.
9.4 Cantor’s ancestry
Cantor’s paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the NapoleonicWars. There is very little direct information on his grandparents.[69] Cantor was sometimes called Jewish in hislifetime,[70] but has also variously been called Russian, German, and Danish as well.Jakob Cantor, Cantor’s grandfather, gave his children Christian saints' names. Further, several of his grandmother’srelatives were in the Czarist civil service, which would not welcome Jews, unless they converted to Christianity.Cantor’s father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his cor-respondence with his son shows both of them as devout Lutherans. Very little is known for sure about GeorgeWoldemar’s origin or education.[71] His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Peters-burg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter fromCantor’s brother Louis to their mother, stating:
Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigungder Hebräer sein, im socialen Leben sind mir Christen lieber ...[71]
(“Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favourof equal rights for Hebrews, in social life I prefer Christians...”) which could be read to imply that she was of Jewishancestry.[72]
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
More often [i.e., than the ancestry of the mother] the question has been discussed of whether GeorgCantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in
9.5. HISTORIOGRAPHY 37
Copenhagen from the year 1937 concerning his father: “It is hereby testified that Georg Woldemar Can-tor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completelywithout doubt was not a Jew ...”[71]
It is also later said in the same document:
Also efforts for a long time by the librarian Josef Fischer, one of the best experts on Jewish geneal-ogy in Denmark, charged with identifying Jewish professors, that Georg Cantor was of Jewish descent,finished without result. [Something seems to be wrong with this sentence, but the meaning seems clearenough.] In Cantor’s published works and also in his Nachlass there are no statements by himself whichrelate to a Jewish origin of his ancestors. There is to be sure in the Nachlass a copy of a letter of hisbrother Ludwig from 18 November 1869 to their mother with some unpleasant antisemitic statements,in which it is said among other things: ...[71]
(the rest of the quote is finished by the very first quote above). In Men of Mathematics, Eric Temple Bell describedCantor as being “of pure Jewish descent on both sides,” although both parents were baptized. In a 1971 articleentitled “Towards a Biography of Georg Cantor,” the British historian of mathematics Ivor Grattan-Guinness mentions(Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also statesthat Cantor’s wife, Vally Guttmann, was Jewish).In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence,Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the SephardicJewish community of Copenhagen. Specifically, Cantor states in describing his father: “Er ist aber in Kopenhagengeboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde...” (“He was born in Copenhagenof Jewish (lit: “Israelite”) parents from the local Portuguese-Jewish community.”)[73]
In addition, Cantor’s maternal great uncle,[74] a Hungarian violinist Josef Böhm, has been described as Jewish,[75]
which may imply that Cantor’s mother was at least partly descended from the Hungarian Jewish community.[76]
In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows:
Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopen-hagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular justGermain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with myfather and mother and brothers and sister, eleven years old in the year 1856, into Germany.[77]
9.5 Historiography
Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927) – largelythe correspondence with Mittag-Leffler – and Fraenkel (1930). Both were at second and third hand; neither hadmuch on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one ofCantor’s modern biographers describes as “perhaps the most widely read modern book on the history of mathematics";and as “one of the worst”.[78] Bell presents Cantor’s relationship with his father as Oedipal, Cantor’s differences withKronecker as a quarrel between two Jews, and Cantor’s madness as Romantic despair over his failure to win acceptancefor his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claimswere true, but they may be found in many books of the intervening period, owing to the absence of any other narrative.There are other legends, independent of Bell – including one that labels Cantor’s father a foundling, shipped to SaintPetersburg by unknown parents.[79] A critique of Bell’s book is contained in Joseph Dauben's biography.[80] WritesDauben:
Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, toattacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics’, whichhad spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italianmathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number wasincomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny theperfection of Cantor’s own creation. Understandably, Cantor launched a thorough campaign to discreditVeronese’s work in every way possible.[81]
38 CHAPTER 9. GEORG CANTOR
9.6 See also• Cantor algebra
• Cantor cube
• Cantor function
• Cantor medal – award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.
• Cantor set
• Cantor space
• Cantor’s back-and-forth method
• Controversy over Cantor’s theory
• Heine–Cantor theorem
• Infinity
• List of German inventors and discoverers
• Pairing function
9.7 Notes[1] Grattan-Guinness 2000, p. 351
[2] The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert andIlgauds 1985 are useful additional sources.
[3] Dauben 2004, p. 1.
[4] Dauben, Joseph Warren (1979). Georg Cantor His Mathematics and Philosophy of the Infinite. princeton university press.pp. introduction. ISBN 9780691024479.
[5] Dauben 2004, pp. 8, 11, 12–13.
[6] Dauben 1977, p. 86; Dauben 1979, pp. 120, 143.
[7] Dauben 1977, p. 102.
[8] Dauben 1979, p. 266.
[9] Dauben 2004, p. 1; Dauben 1977, p. 89 15n.
[10] Rodych 2007.
[11] Dauben 1979, p. 280: "...the tradition made popular by Arthur Moritz Schönflies blamed Kronecker’s persistent criticismand Cantor’s inability to confirm his continuum hypothesis” for Cantor’s recurring bouts of depression.
[12] Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor’s examining physicians atHalle Nervenklinik, referring to Cantor’s mental illness as “cyclic manic-depression”.
[13] Dauben 1979, p. 248.
[14] Hilbert (1926, p. 170): “Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.” (Literally:“Out of the Paradise that Cantor created for us, no one must be able to expel us.”)
[15] Reid, Constance (1996), Hilbert, New York: Springer-Verlag, p. 177, ISBN 0-387-04999-1.
[16] ru: The musical encyclopedia (Музыкальная энциклопедия)
[17] O'Connor, John J, and Robertson, Edmund F (1998). “Georg Ferdinand Ludwig Philipp Cantor”. MacTutor History ofMathematics.
[18] Dauben 1979, p. 163.
9.7. NOTES 39
[19] Dauben 1979, p. 34.
[20] Dauben 1977, p. 89 15n.
[21] Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
[22] Dauben 1979, p. 138.
[23] Dauben 1979, p. 139.
[24] Dauben 1979, p. 282.
[25] Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
[26] Dauben 1979, pp. 281–283.
[27] Dauben 1979, p. 283.
[28] For a discussion of König’s paper see Dauben 1979, pp. 248–250. For Cantor’s reaction, see Dauben 1979, pp. 248, 283.
[29] Dauben 1979, pp. 283–284.
[30] Dauben 1979, p. 284.
[31] Johnson, Phillip E. (1972), “The Genesis and Development of Set Theory”, The Two-Year College Mathematics Journal 3(1): 55, JSTOR 3026799.
[32] This paragraph is a highly abbreviated summary of the impact of Cantor’s lifetime of work. More details and referencescan be found later.
[33] Suppes, Patrick (1972), Axiomatic Set Theory, Dover, p. 1, ISBN 9780486616308, With a few rare exceptions the entitieswhich are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects. ... As aconsequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.
[34] Cantor 1874
[35] A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets.However, this terminology is not universally followed, and sometimes “denumerable” is used as a synonym for “countable”.
[36] Cooke, Roger (1993), “Uniqueness of trigonometric series and descriptive set theory, 1870–1985”, Archive for History ofExact Sciences 45 (4): 281, doi:10.1007/BF01886630.
[37] Katz, Karin Usadi and Katz, Mikhail G. (2012), “A Burgessian Critique of Nominalistic Tendencies in ContemporaryMathematics and its Historiography”, Foundations of Science 17 (1): 51–89, doi:10.1007/s10699-011-9223-1
[38] Ehrlich, P. (2006), “The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence ofnon-Archimedean systems of magnitudes” (PDF), Arch. Hist. Exact Sci. 60 (1): 1–121, doi:10.1007/s00407-005-0102-4.
[39] This follows closely the first part of Cantor’s 1891 paper.
[40] Cantor 1874. English translation: Ewald 1996, pp. 840–843.
[41] For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumer-ous – see Moore, A.W. (April 1995), “A brief history of infinity” (PDF), Scientific American 272 (4): 112–116 (114),doi:10.1038/scientificamerican0495-112.
[42] For this, and more information on the mathematical importance of Cantor’s work on set theory, see e.g., Suppes 1972.
[43] Liouville, Joseph (13 May 1844). A propos de l'existence des nombres transcendants.
[44] The real algebraic numbers are the real roots of polynomial equations with integer coefficients.
[45] For more details on Cantor’s article, see Cantor’s first uncountability proof and Gray, Robert (1994), “Georg Cantor andTranscendental Numbers” (PDF), American Mathematical Monthly 101: 819–832, doi:10.2307/2975129. Gray (pp. 821–822) describes a computer program that uses Cantor’s constructions to generate a transcendental number.
[46] Cantor’s construction starts with the set of transcendentals T and removes a countable subset {tn} (for example, tn = e / n).Call this set T’. Then T = T’ ∪ {tn} = T’ ∪ {t₂n−₁} ∪ {t₂n}. The set of reals R = T ∪ {an} = T’ ∪ {tn} ∪ {an} where anis the sequence of real algebraic numbers. So both T and R are the union of three disjoint sets: T’ and two countable sets.A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T’, f(t₂n−₁) = tn, and f(t₂n) = an.Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to anyset formed by removing countably many numbers from the set of reals (Cantor 1932, p. 142).
40 CHAPTER 9. GEORG CANTOR
[47] Dauben 1977, p. 89.
[48] Cantor 1883.
[49] Cantor (1895), Cantor (1897). The English translation is Cantor 1955.
[50] Wallace, David Foster (2003), Everything and More: A Compact History of Infinity, New York: W.W. Norton and Com-pany, p. 259, ISBN 0-393-00338-8.
[51] Dauben 1979, pp. 69, 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed itspublication due to Kronecker’s opposition. Weierstrass actively supported it.
[52] Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine theformal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for “natural”or “plausible” axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidencefor or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel’s last papers argues thatthe CH is false, and the continuum has cardinality Aleph-2.
[53] Dauben 1979, pp. 240–270; see especially pp. 241, 259.
[54] Hallett 1986.
[55] Weir, Alan (1998), “Naive Set Theory is Innocent!", Mind 107 (428): 763–798, doi:10.1093/mind/107.428.763 p. 766:"...it may well be seriously mistaken to think of Cantor’s Mengenlehre [set theory] as naive...”
[56] Dauben 1979, p. 295.
[57] Dauben 1979, p. 120.
[58] Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
[59] Dauben 1979, p. 225
[60] Snapper, Ernst (1979), "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism", Mathematics Magazine524: 207–216.
[61] Davenport, Anne A. (1997), “The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century”, Isis 88(2): 263–295, doi:10.1086/383692, JSTOR 236574.
[62] Dauben 1977, p. 85.
[63] Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
[64] Dauben 1979, p. 296.
[65] Dauben 1979, p. 144.
[66] Dauben 1977, pp. 91–93.
[67] On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
[68] Dauben 1979, p. 96.
[69] E.g., Grattan-Guinness’s only evidence on the grandfather’s date of death is that he signed papers at his son’s engagement.
[70] For example, Jewish Encyclopedia, art. “Cantor, Georg"; Jewish Year Book 1896–97, “List of Jewish Celebrities of theNineteenth Century”, p. 119; this list has a star against people with one Jewish parent, but Cantor is not starred.
[71] Purkert and Ilgauds 1985, p. 15.
[72] For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert andIlgauds 1985; the letter is from Aczel 2000, pp. 93–94, from Louis’ trip to Chicago in 1863. It is ambiguous in German,as in English, whether the recipient is included.
[73] Tannery, Paul (1934) Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, p. 306.
[74] Dauben 1979, p. 274.
[75] Mendelsohn, Ezra (ed.) (1993) Modern Jews and their musical agendas, Oxford University Press, p. 9.
[76] Ismerjükoket?: zsidó származású nevezetes magyarok arcképcsarnoka, István Reményi Gyenes Ex Libris, (Budapest 1997),pages 132–133
9.8. REFERENCES 41
[77] Russell, Bertrand. Autobiography, vol. I, p. 229. In English in the original; italics also as in the original.
[78] Grattan-Guinness 1971, p. 350.
[79] Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell’s Jewish stereotypes appear tohave been removed from some postwar editions.)
[80] Dauben 1979
[81] Dauben, J.: The development of the Cantorian set theory, pp.~181-219. See pp.216-217. In Bos, H.; Bunn, R.; Dauben,J.; Grattan-Guinness, I.; Hawkins, T.; Pedersen, K. From the calculus to set theory, 1630-1910. An introductory history.Edited by I. Grattan-Guinness. Gerald Duckworth & Co. Ltd., London, 1980.
9.8 References
• Dauben, Joseph W. (1977), “Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite”,Journal of the History of Ideas 38 (1): 85–108, JSTOR 2708842.
• Dauben, Joseph W. (1979), Georg Cantor: his mathematics and philosophy of the infinite, Boston: HarvardUniversity Press, ISBN 978-0-691-02447-9.
• Dauben, Joseph (2004) [1993], “Georg Cantor and the Battle for Transfinite Set Theory”, Proceedings of the9th ACMS Conference (Westmont College, Santa Barbara, CA), pp. 1–22. Internet version published in Journalof the ACMS 2004.
• Ewald, William B., ed. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations ofMathematics, New York: Oxford University Press, ISBN 978-0-19-853271-2.
• Grattan-Guinness, Ivor (1971), “Towards a Biography of Georg Cantor”, Annals of Science 27: 345–391,doi:10.1080/00033797100203837.
• Grattan-Guinness, Ivor (2000), The Search for Mathematical Roots: 1870–1940, Princeton University Press,ISBN 978-0-691-05858-0.
• Hallett, Michael (1986), Cantorian Set Theory and Limitation of Size, New York: Oxford University Press,ISBN 0-19-853283-0.
• Purkert, Walter; Ilgauds, Hans Joachim (1985), Georg Cantor: 1845–1918, Birkhäuser, ISBN 0-8176-1770-1.
• Suppes, Patrick (1972) [1960], Axiomatic Set Theory, New York: Dover, ISBN 0-486-61630-4. Althoughthe presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor’s results, whichdemonstrates Cantor’s continued importance for the edifice of foundational mathematics.
9.9 Bibliography
Older sources on Cantor’s life should be treated with caution. See Historiography section above.
Primary literature in English
• Cantor, Georg (1955) [1915], Philip Jourdain, ed., Contributions to the Founding of the Theory of TransfiniteNumbers, New York: Dover, ISBN 978-0-486-60045-1.
Primary literature in German
• Cantor, Georg (1874), “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” (PDF),Journal für die Reine und Angewandte Mathematik 77: 258–262, doi:10.1515/crll.1874.77.258.
• Georg Cantor (1879). “Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen 15(1): 1–7. doi:10.1007/bf01444101.
42 CHAPTER 9. GEORG CANTOR
• Georg Cantor (1880). “Ueber unendliche, lineare Punktmannichfaltigkeiten (2)". Mathematische Annalen 17(3): 355–358. doi:10.1007/bf01446232.
• Georg Cantor (1882). “Ueber unendliche, lineare Punktmannichfaltigkeiten (3)". Mathematische Annalen 20(1): 113–121. doi:10.1007/bf01443330.
• Georg Cantor (1883). “Ueber unendliche, lineare Punktmannichfaltigkeiten (4)". Mathematische Annalen 21(1): 51–58. doi:10.1007/bf01442612.
• Georg Cantor (1883). “Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen 21(4): 545–591. doi:10.1007/bf01446819.
• Georg Cantor (1892). “Ueber eine elementare Frage der Mannigfaltigkeitslehre”. Jahresbericht der DeutscheMathematiker-Vereinigung 1890-1891 1: 75–78.
• Cantor, Georg (1895). “Beiträge zur Begründung der transfiniten Mengenlehre (1)" (PDF). MathematischeAnnalen 46: 481–512. doi:10.1007/bf02124929.
• Cantor, Georg (1897). “Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen49: 207–246.
• Cantor, Georg (1932), Ernst Zermelo, ed., Gesammelte Abhandlungen mathematischen und philosophischeninhalts, Berlin: Springer. Almost everything that Cantor wrote. Includes excerpts of his correspondence withDedekind (p. 443-451) and Fraenkel’s Cantor biography (p. 452-483) in the appendix.
Secondary literature
• Aczel, Amir D. (2000), The Mystery of the Aleph: Mathematics, the Kabbala, and the Search for Infinity, NewYork: Four Walls Eight Windows Publishing. ISBN 0-7607-7778-0. A popular treatment of infinity, in whichCantor is frequently mentioned.
• Dauben, Joseph W. (June 1983), “Georg Cantor and the Origins of Transfinite Set Theory”, Scientific American248 (6): 122–131, doi:10.1038/scientificamerican0683-122
• Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought,Basel, Switzerland: Birkhäuser. ISBN 3-7643-8349-6 Contains a detailed treatment of both Cantor’s andDedekind’s contributions to set theory.
• Halmos, Paul (1998) [1960], Naive Set Theory, New York & Berlin: Springer. ISBN 3-540-90092-6
• Hilbert, David (1926). "Über das Unendliche”. Mathematische Annalen 95: 161–190. doi:10.1007/BF01206605.
• Hill, C. O.; Rosado Haddock, G. E. (2000), Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago:Open Court. ISBN 0-8126-9538-0 Three chapters and 18 index entries on Cantor.
• Meschkowski, Herbert (1983), Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influ-ence, in German), Vieweg, Braunschweig
• Penrose, Roger (2004), The Road to Reality, Alfred A. Knopf. ISBN 0-679-77631-1 Chapter 16 illustrateshow Cantorian thinking intrigues a leading contemporary theoretical physicist.
• Rucker, Rudy (2005) [1982], Infinity and the Mind, Princeton University Press. ISBN 0-553-25531-2 Dealswith similar topics to Aczel, but in more depth.
• Rodych, Victor (2007), "Wittgenstein’s Philosophy of Mathematics", in Edward N. Zalta, The Stanford Ency-clopedia of Philosophy.
9.10. EXTERNAL LINKS 43
9.10 External links• Works by or about Georg Cantor at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., “Georg Cantor”, MacTutor History ofMathematics archive, Universityof St Andrews.
• O'Connor, John J.; Robertson, Edmund F., “A history of set theory”, MacTutor History ofMathematics archive,University of St Andrews. Mainly devoted to Cantor’s accomplishment.
• Georg Cantor at the Mathematics Genealogy Project
• Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech.
• Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gymnasium Halle
• Poem about Georg Cantor
44 CHAPTER 9. GEORG CANTOR
Cantor, ca. 1870.
9.10. EXTERNAL LINKS 45
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An illustration of Cantor’s diagonal argument for the existence of uncountable sets.[39] The sequence at the bottom cannot occuranywhere in the infinite list of sequences above.
Passage of Georg Cantor’s article with his famous set definition
46 CHAPTER 9. GEORG CANTOR
X 1
2
3
4
YD
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A bijective function.
9.10. EXTERNAL LINKS 47
The title on the memorial plaque (in Russian): “In this building was born and lived from 1845 till 1854 the great mathematician andcreator of set theory Georg Cantor”, Vasilievsky Island, Saint-Petersburg.
Chapter 10
Hereditary property
In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the termsubobject depends on the context. These properties are particularly considered in topology and graph theory, but alsoin set theory.
10.1 In topology
In topology, a topological property is said to be hereditary if whenever a topological space has that property, then sodoes every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary.For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compact-ness are weakly hereditary, but not hereditary.[1] Connectivity is not weakly hereditary.If P is a property of a topological space X and every subspace also has property P, then X is said to be “hereditarilyP".
10.2 In graph theory
In graph theory, a hereditary property is a property of a graph which also holds for (is “inherited” by) its inducedsubgraphs.[2] Alternately, a hereditary property is preserved by the removal of vertices. A graph class G is saidhereditary if it is closed under induced subgraphs. Examples of hereditary graph classes are independent graphs(graphs with no edges), which is a special case (with c = 1) of being c-colorable for some number c, being forests,planar, complete, complete multipartite etc.In some cases, the term “hereditary” has been defined with reference to graph minors, but this is more properly calleda minor-hereditary property. The Robertson–Seymour theorem implies that a minor-hereditary property may becharacterized in terms of a finite set of forbidden minors.The term “hereditary” has been also used for graph properties that are closed with respect to taking subgraphs.[3] Insuch a case, properties, that are closed with respect to taking induced subgraphs, are called induced-hereditary. Thisapproach is used by the members of the scientific society Hereditarnia Club. The language of hereditary propertiesand induced-hereditary properties provides a powerful tool for study of structural properties of various types ofgeneralized colourings. The most important result from this area is the Unique Factorisation Theorem.[4]
10.2.1 Monotone property
There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions are:
• Preserved by the removal of edges.[5]
• Preserved by the removal of edges and vertices (i.e., the property should hold for all subgraphs).[2]
48
10.3. IN MODEL THEORY 49
• Preserved by the addition of edges and vertices (i.e., the property should hold for all supergraphs).[6]
• Preserved by the addition of edges.[7] (This meaning is used in the statement of the Aanderaa–Karp–Rosenbergconjecture.)
The complementary property of a property that is preserved by the removal of edges is preserved under the additionof edges. Hence some authors avoid this ambiguity by saying a property A is monotone if A or AC (the complementof A) is monotone.[8] Some authors choose to resolve this by using the term increasing monotone for propertiespreserved under the addition of some object, and decreasing monotone for those preserved under the removal of thesame object.
10.3 In model theory
In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditaryproperty if every substructure of a structure in K is again in K. A variant of this definition is used in connection withFraïssé's theorem: A class K of finitely generated structures has the hereditary property if every finitely generatedsubstructure is again in K. See age.
10.4 In matroid theory
In a matroid, every subset of an independent set is again independent. This is also sometimes called the hereditaryproperty.
10.5 In set theory
Recursive definitions using the adjective “hereditary” are often encountered in set theory.A set is said to be hereditary (or pure) if all of its elements are hereditary sets. It is vacuously true that the emptyset is a hereditary set, and thus the set {∅} containing only the empty set ∅ is a hereditary set, and recursively sois {∅, {∅}} , for example. In formulations of set theory that are intended to be interpreted in the von Neumannuniverse or to express the content of Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort ofobject that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interestingonly in a context in which there may be urelements.A couple of notions are defined analogously:
• A hereditarily finite set is defined as a finite set consisting of zero or more hereditarily finite sets. Equivalently,a set is hereditarily finite if and only if its transitive closure is finite.
• A hereditarily countable set is a countable set of hereditarily countable sets. Assuming the axiom of countablechoice, then a set is hereditarily countable if and only if its transitive closure is countable.
Based on the above, it follows that in ZFC a more general notion can be defined for any predicate Φ(x) . A setx is said to have hereditarily the property Φ(y) if x itself and all members of its transitive closure satisfy Φ(y) ,i.e. x ∪ tc(x) ⊆ {y|Φ(y)} . Equivalently, x hereditarily satisfies Φ(y) iff it is a member of a transitive subset of{y|Φ(y)} .[9][10] A property (of a set) is thus said to be hereditary if is inherited by every subset. For example, beingwell-ordered is a hereditary property, and so it being finite.[11]
If we instantiate in the above schema Φ(x) with "x has cardinality less than κ", we obtain the more general notionof a set being hereditarily of cardinality less than κ, usually denoted by Hκ
[12] or H(κ) . We regain the two simplenotions we introduced above as H(ω) being the set of hereditarily finite sets and H(ω1) being the set of hereditarilycountable sets.[13] ( ω1 is the first uncountable ordinal.)
50 CHAPTER 10. HEREDITARY PROPERTY
10.6 References[1] • Goreham, Anthony, "Sequential Convergence in Topological Spaces
[2] Alon, Noga; Shapira, Asaf (2008). “Every monotone graph property is testable”. SIAM Journal on Computing 38 (2):505–522. doi:10.1137/050633445.
[3] M. Borowiecki, I. Broere, M. Frick, P. Mihók, G. Semanišin :A Survey of Hereditary Properties of Graphs, DiscussionesMathematicae – Graph Theory 17 (1997) 5–50.
[4] A. Farrugia, Peter Mihók, R. Bruce Richter, Gabriel Semanišin: Factorizations and characterizations of induced-hereditaryand compositive properties, Journal of Graph Theory 49(1): 11-27 (2005).
[5] King, R. (1990), A lower bound for the recognition of digraph properties, Combinatorica, vol 10, 53–59
[6] http://www.cs.ucsc.edu/~{}optas/papers/k-col-threshold.pdf
[7] Spinrad, J. (2003), Efficient Graph Representations, AMS Bookstore, ISBN 0-8218-2815-0, p9.
[8] Ashish Goel; Sanatan Rai; Bhaskar Krishnamachari (2003). “Monotone properties of random geometric graphs have sharpthresholds”. Annals of Applied Probability 15 (4): 2535–2552. arXiv:math.PR/0310232. doi:10.1214/105051605000000575.
[9] Azriel Levy (2002), Basic set theory, p. 82
[10] Thomas Forster (2003), Logic, induction and sets, p. 197
[11] Judith Roitman (1990), Introduction to modern set theory, p. 10
[12] Levy (2002), p. 137
[13] Kenneth Kunen (1983), Set theory, p. 131
Chapter 11
Injective function
“Injective” redirects here. For injective modules, see Injective module.In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness:
X1
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An injective non-surjective function (not a bijection)
it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of
51
52 CHAPTER 11. INJECTIVE FUNCTION
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An injective surjective function (bijection)
the function’s codomain is the image of at most one element of its domain. The term one-to-one function must not beconfused with one-to-one correspondence (aka bijective function), which uniquely maps all elements in both domainand codomain to each other, (see figures).Occasionally, an injective function from X to Y is denoted f: X ↣ Y, using an arrow with a barbed tail (U+21A3 ↣rightwards arrow with tail).[1] The set of injective functions from X to Y may be denoted YX using a notation derivedfrom that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, thenumber of injections from X to Y is nm (see the twelvefold way).A function f that is not injective is sometimes called many-to-one. However, this terminology is also sometimes usedto mean “single-valued”, i.e., each argument is mapped to at most one value.A monomorphism is a generalization of an injective function in category theory.
11.1 Definition
For more details on notation, see Function (mathematics) § Notation.
Let f be a function whose domain is a set A. The function f is injective if and only if for all a and b in A, if f(a) =
11.2. EXAMPLES 53
X1
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A non-injective surjective function (surjection not a bijection)
f(b), then a = b; that is, f(a) = f(b) implies a = b. Equivalently, if a ≠ b, then f(a) ≠ f(b).Symbolically,
∀a, b ∈ A, f(a) = f(b) ⇒ a = b
which is logically equivalent to the contrapositive,
∀a, b ∈ A, a ̸= b ⇒ f(a) ̸= f(b)
11.2 Examples
• For any set X and any subset S of X the inclusion map S → X (which sends any element s of S to itself) isinjective. In particular the identity function X → X is always injective (and in fact bijective).
• If the domain X = ∅ or X has only one element, the function X → Y is always injective.
54 CHAPTER 11. INJECTIVE FUNCTION
• The function f : R → R defined by f(x) = 2x + 1 is injective.
• The function g : R→ R defined by g(x) = x2 is not injective, because (for example) g(1) = 1 = g(−1). However,if g is redefined so that its domain is the non-negative real numbers [0,+∞), then g is injective.
• The exponential function exp : R → R defined by exp(x) = ex is injective (but not surjective as no real valuemaps to a negative number).
• The natural logarithm function ln : (0, ∞) → R defined by x ↦ ln x is injective.
• The function g : R → R defined by g(x) = xn − x is not injective, since, for example, g(0) = g(1).
More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph isnever intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.
x
y
Y y
X x
f im
Y X : f x f y
Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X =domain of function, Y = range of function, and im(f) denotes image of f. Every one x in X maps to exactly one unique y in Y. Thecircled parts of the axes represent domain and range sets – in accordance with the standard diagrams above.
11.2. EXAMPLES 55
x
y
2 Y y
2 X x
1 Y y
1 X x
X X , X 2 1
Y Y , Y 2 1
Y X : f x f y
f im
Not an injective function. Here X1 and X2 are subsets of X, Y1 and Y2 are subsets of Y: for two regions where the function is notinjective because more than one domain element can map to a single range element. That is, it is possible for more than one x in Xto map to the same y in Y.
x
y
1 Y y
1 X 1 Y : f
1 X x
f im
x f y
x
y
2 Y y
2 X 2 Y : f
x f y
f im
2 X x
Making functions injective. The previous function f : X → Y can be reduced to one or more injective functions (say) f : X1 → Y1
and f : X2 → Y2, shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule f hasnot changed – only the domain and range. X1 and X2 are subsets of X, Y1 and Y2 are subsets of R: for two regions where theinitial function can be made injective so that one domain element can map to a single range element. That is, only one x in X mapsto one y in Y.
56 CHAPTER 11. INJECTIVE FUNCTION
11.3 Injections can be undone
Functions with left inverses are always injections. That is, given f : X → Y, if there is a function g : Y → X such that,for every x ∈ X
g(f(x)) = x (f can be undone by g)
then f is injective. In this case, g is called a retraction of f. Conversely, f is called a section of g.Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics[2]). Note thatg may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity onY. In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible (bijective).Injections are "reversible" but not always invertible.Although it is impossible to reverse a non-injective (and therefore information-losing) function, one can at least obtaina “quasi-inverse” of it, that is a multiple-valued function.
11.4 Injections may be made invertible
In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace itscodomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective.Indeed, f can be factored as inclJ,Y o g, where inclJ,Y is the inclusion function from J into Y.More generally, injective partial functions are called partial bijections.
11.5 Other properties
• If f and g are both injective, then f o g is injective.
X1
2
3
YD
B
C
A
ZP
Q
R
S
The composition of two injective functions is injective.
11.6. PROVING THAT FUNCTIONS ARE INJECTIVE 57
• If g o f is injective, then f is injective (but g need not be).
• f : X → Y is injective if and only if, given any functions g, h : W → X, whenever f o g = f o h, then g = h. Inother words, injective functions are precisely the monomorphisms in the category Set of sets.
• If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A. Thus, A can be recovered from its imagef(A).
• If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).
• Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. Thisdecomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the rangeh(W) of h as a subset of the codomain Y of h.
• If f : X → Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal numbers.In particular, if, in addition, there is an injection from Y to X , then X and Y have the same cardinal number.(This is known as the Cantor–Bernstein–Schroeder theorem.)
• If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f issurjective (in which case f is bijective).
• An injective function which is a homomorphism between two algebraic structures is an embedding.
• Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a propertyof the graph of the function alone; that is, whether a function f is injective can be decided by only consideringthe graph (and not the codomain) of f.
11.6 Proving that functions are injective
A proof that a function ƒ is injective depends on how the function is presented and what properties the function holds.For functions that are given by some formula there is a basic idea. We use the contrapositive of the definition ofinjectivity, namely that if ƒ(x) = ƒ(y), then x = y.[3]
Here is an example:
ƒ = 2x + 3
Proof: Let ƒ : X → Y. Suppose ƒ(x) = ƒ(y). So 2x + 3 = 2y + 3 => 2x = 2y => x = y. Therefore it follows from thedefinition that ƒ is injective.There are multiple other methods of proving that a function is injective. For example, in calculus if ƒ is differentiableon some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval.In linear algebra, if ƒ is a linear transformation it is sufficient to show that the kernel of ƒ contains only the zerovector. If ƒ is a function with finite domain it is sufficient to look through the list of images of each domain elementand check that no image occurs twice on the list.
11.7 See also• Surjective function
• Bijective function
• Partial function
• Injective module
• Bijection, injection and surjection
• Horizontal line test
• Injective metric space
58 CHAPTER 11. INJECTIVE FUNCTION
11.8 Notes[1] “Unicode” (PDF). Retrieved 2013-05-11.
[2] This principle is valid in conventional mathematics, but may fail in constructive mathematics. For instance, a left inverseof the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the realline to the set {0,1}.
[3] Williams, Peter. “Proving Functions One-to-One”.
11.9 References• Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed.), New York: John Wiley & Sons, ISBN
978-0-471-05464-1, p. 17 ff.
• Halmos, Paul R. (1974), Naive Set Theory, New York: Springer, ISBN 978-0-387-90092-6, p. 38 ff.
11.10 External links• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history
of Injection and related terms.
• Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injectivefunctions
Chapter 12
Integer
This article is about numbers traditionally known as “integers”. For computer representations, see integer (computerscience). For the concept in algebraic number theory, see integral element.
An integer (from the Latin integer meaning “whole”)[note 1] is a number that can be written without a fractionalcomponent. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and √2 are not.The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or countingnumbers,[1] and their additive inverses (the negative integers, i.e. −1, −2, −3, …). This is often denoted by a boldfaceZ ("Z") or blackboard bold Z (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], “numbers”).[2][3]
ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite.The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory,the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. Infact, the (rational) integers are the algebraic integers that are also rational numbers.
12.1 Algebraic properties
Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integersare shown in purple and negative integers in red.
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and productof any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z(unlike the natural numbers) is also closed under subtraction. The integers form a unital ring which is the most basicone, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring.This universal property, namely to be an initial object in the category of rings, characterizes the ring Z.Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fractionwhen the exponent is negative).The following lists some of the basic properties of addition and multiplication for any integers a, b and c.In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is anabelian group. As a group under addition, Z is a cyclic group, since every non-zero integer can be written as a finitesum 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). In fact, Z under addition is the only infinite cyclic group, in the sensethat any infinite cyclic group is isomorphic to Z.The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid.However not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the lefthand side is even, while the right hand side is odd. This means that Z under multiplication is not a group.
59
60 CHAPTER 12. INTEGER
All the rules from the above property table, except for the last, taken together say that Z together with addition andmultiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Onlythose equalities of expressions are true in Z for all values of variables, which are true in any unital commutative ring.Note that certain non-zero integers map to zero in certain rings.At last, the property (*) says that the commutative ring Z is an integral domain. In fact, Z provides the motivationfor defining such a structure.The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z isnot a field. The smallest field with the usual operations containing the integers is the field of rational numbers. Theprocess of constructing the rationals from the integers can be mimicked to form the field of fractions of any integraldomain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integerscan be extracted, which includes Z as its subring.Although ordinary division is not defined onZ, the division “with remainder” is defined on them. It is called Euclideandivision and possesses the following important property: that is, given two integers a and b with b ≠ 0, there existunique integers q and r such that a = q × b + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b. The integerq is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computinggreatest common divisors works by a sequence of Euclidean divisions.Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is aprincipal ideal domain and any positive integer can be written as the products of primes in an essentially unique way.This is the fundamental theorem of arithmetic.
12.2 Order-theoretic properties
Z is a totally ordered set without upper or lower bound. The ordering of Z is given by:
… −3 < −2 < −1 < 0 < 1 < 2 < 3 < …
An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negativenor positive.The ordering of integers is compatible with the algebraic operations in the following way:
1. if a < b and c < d, then a + c < b + d
2. if a < b and 0 < c, then ac < bc.
It follows that Z together with the above ordering is an ordered ring.The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.[4] This isequivalent to the statement that any Noetherian valuation ring is either a field or a discrete valuation ring.
12.3 Construction
In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and thenegations of the natural numbers. However, this style of definition leads to many different cases (each arithmeticoperation needs to be defined on each combination of types of integer) and makes it tedious to prove that these oper-ations obey the laws of arithmetic.[5] Therefore, in modern set-theoretic mathematics a more abstract construction,[6]
which allows one to define the arithmetical operations without any case distinction, is often used instead.[7] Theintegers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).[8]
The intuition is that (a,b) stands for the result of subtracting b from a.[8] To confirm our expectation that 1 − 2 and 4− 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:
(a, b) ∼ (c, d)
12.3. CONSTRUCTION 61
Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at theend of the line.
precisely when
a+ d = b+ c.
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[8]
denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:
[(a, b)] + [(c, d)] := [(a+ c, b+ d)].
[(a, b)] · [(c, d)] := [(ac+ bd, ad+ bc)].
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
−[(a, b)] := [(b, a)].
Hence subtraction can be defined as the addition of the additive inverse:
62 CHAPTER 12. INTEGER
[(a, b)]− [(c, d)] := [(a+ d, b+ c)].
The standard ordering on the integers is given by:
[(a, b)] < [(c, d)] iff a+ d < b+ c.
It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number nis identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sendingn to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a secondtime since −0 = 0.Thus, [(a,b)] is denoted by
{a− b, if a ≥ b
−(b− a), if a < b.
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), thisconvention creates no ambiguity.This notation recovers the familiar representation of the integers as {…, −2, −1, 0, 1, 2, …}.Some examples are:
0 = [(0, 0)] = [(1, 1)] = · · · = [(k, k)]
1 = [(1, 0)] = [(2, 1)] = · · · = [(k + 1, k)]
−1 = [(0, 1)] = [(1, 2)] = · · · = [(k, k + 1)]
2 = [(2, 0)] = [(3, 1)] = · · · = [(k + 2, k)]
−2 = [(0, 2)] = [(1, 3)] = · · · = [(k, k + 2)].
12.4 Computer science
Main article: Integer (computer science)
An integer is often a primitive data type in computer languages. However, integer data types can only represent asubset of all integers, since practical computers are of finite capacity. Also, in the common two’s complement repre-sentation, the inherent definition of sign distinguishes between “negative” and “non-negative” rather than “negative,positive, and 0”. (It is, however, certainly possible for a computer to determine whether an integer value is truly pos-itive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programminglanguages (such as Algol68, C, Java, Delphi, etc.).Variable-length representations of integers, such as bignums, can store any integer that fits in the computer’s memory.Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16,etc.) or a memorable number of decimal digits (e.g., 9 or 10).
12.5 Cardinality
The cardinality of the set of integers is equal to ℵ0 (aleph-null). This is readily demonstrated by the constructionof a bijection, that is, a function that is injective and surjective from Z to N. If N = {0, 1, 2, …} then consider thefunction:
f(x) =
{2|x|, if x ≤ 0
2x− 1, if x > 0.
12.6. SEE ALSO 63
{… (−4,8) (−3,6) (−2,4) (−1,2) (0,0) (1,1) (2,3) (3,5) …}If N = {1, 2, 3, ...} then consider the function:
g(x) =
{2|x|, if x < 0
2x+ 1, if x ≥ 0.
{… (−4,8) (−3,6) (−2,4) (−1,2) (0,1) (1,3) (2,5) (3,7) …}If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of Nand by the definition of cardinal equality the two sets have equal cardinality.
12.6 See also
• 0.999...
• Canonical representation of a positive integer
• Hyperinteger
• Integer-valued function
• Integer lattice
• Integer part
• Integer sequence
• Profinite integer
12.7 Notes
[1] Integer 's first, literal meaning in Latin is “untouched”, from in (“not”) plus tangere (“to touch”). "Entire" derives fromthe same origin via French (see: Evans, Nick (1995). “A-Quantifiers and Scope”. In Bach, Emmon W. Quantification inNatural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. ISBN 0-7923-3352-7.)
12.8 References
[1] Weisstein, Eric W., “Counting Number”, and “Whole Number”, MathWorld.
[2] Miller, Jeff (2010-08-29). “Earliest Uses of Symbols of Number Theory”. Retrieved 2010-09-20.
[3] Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4.
[4] Warner, Seth (2012), Modern Algebra, Dover Books on Mathematics, Courier Corporation, Theorem 20.14, p. 185, ISBN9780486137094.
[5] Mendelson, Elliott (2008), Number Systems and the Foundations of Analysis, Dover Books on Mathematics, Courier DoverPublications, p. 86, ISBN 9780486457925.
[6] Ivorra Castillo: Álgebra
[7] Frobisher, Len (1999), Learning to Teach Number: A Handbook for Students and Teachers in the Primary School, TheStanley Thornes Teaching Primary Maths Series, Nelson Thornes, p. 126, ISBN 9780748735150.
[8] Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. p. 83. ISBN 0-390-16895-5.
64 CHAPTER 12. INTEGER
12.9 Sources• Bell, E.T., Men of Mathematics. New York: Simon and Schuster, 1986. (Hardcover; ISBN 0-671-46400-
0)/(Paperback; ISBN 0-671-62818-6)
• Herstein, I.N., Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1.
• Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999).ISBN 0-8218-1646-2.
• Weisstein, Eric W., “Integer”, MathWorld.
12.10 External links• Hazewinkel, Michiel, ed. (2001), “Integer”, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4
• The Positive Integers — divisor tables and numeral representation tools
• On-Line Encyclopedia of Integer Sequences cf OEIS
This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Chapter 13
Natural number
This article is about “positive integers” and “non-negative integers”. For all the numbers ..., −2, −1, 0, 1, 2, ..., seeInteger.In mathematics, the natural numbers (sometimes called the whole numbers)[1][2][3][4] are those used for counting
(as in “there are six coins on the table”) and ordering (as in “this is the third largest city in the country”). In commonlanguage, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product,where the “natural number” is used only for naming (as distinct from a serial number where the order properties ofthe natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used inmathematics.The natural numbers are the basis from which many other number sets may be built by extension: the integers, byincluding an unresolved negation operation; the rational numbers, by including with the integers an unresolved divi-sion operation; the real numbers by including with the rationals the termination of Cauchy sequences; the complexnumbers, by including with the real numbers the unresolved square root of minus one; the hyperreal numbers, by in-cluding with real numbers the infinitesimal value epsilon; vectors, by including a vector structure with reals; matrices,by having vectors of vectors; the nonstandard integers; and so on.[5][6] Therefore, the natural numbers are canonicallyembedded (identified) in the other number systems.Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in numbertheory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.There is no universal agreement about whether to include zero in the set of natural numbers. Some authors beginthe natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1,corresponding to the positive integers 1, 2, 3, ....[7][8][9][10] This distinction is of no fundamental concern for thenatural numbers as such, since their core construction is the unary operation successor. Including the number 0 justsupplies an identity element for the (binary) operation of addition, which makes up together with the multiplicationthe usual arithmetic in the natural numbers, to be completed within the integers and the rational numbers, only.In common language, for example in primary school, natural numbers may be called counting numbers[11] to dis-tinguish them from the real numbers which are used for measurement.
13.1 History
The most primitive method of representing a natural number is to put down a mark for each object. Later, a set ofobjects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to bedeveloped for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distincthieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, sothat the symbol for sixty was the same as the symbol for one, its value being determined from context.[15]
A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The
65
66 CHAPTER 13. NATURAL NUMBER
Natural numbers can be used for counting (one apple, two apples, three apples, ...)
use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, butthey omitted such a digit when it would have been the last symbol in the number.[16] The Olmec and Maya civilizationsused 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.[17][18]
The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with DionysiusExiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); insteadnulla (or the genitive form nullae) from nullus, the Latin word for “none”, was employed to denote a 0 value.[19]
The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras andArchimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even notas a number at all.[20]
Independent studies also occurred at around the same time in India, China, and Mesoamerica.[21]
13.2. NOTATION 67
13.1.1 Modern definitions
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the naturalnumbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche.Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized “God made the integers, allelse is the work of man”.In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations ofmathematics.[22] In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus statingthey were not really natural but a consequence of definitions. Later, two classes of such formal definitions wereconstructed; later, they were shown to be equivalent in most practical applications.Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as theclass of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead toparadoxes including Russell’s paradox. Therefore, this formalism was modified so that a natural number is defined asa particular set, and any set that can be put into one-to-one correspondence with that set is said to have that numberof elements.[23]
The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic. It is basedon an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zeronatural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory.One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC butcannot be proved using the Peano Axioms include Goodstein’s theorem.[24]
With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including0 is now the common convention among set theorists,[25] and logicians.[26] Other mathematicians also include 0,[10]
although many have kept the older tradition and take 1 to be the first natural number.[27] Conditioned by the earlyhardware settings, which allowed for a most simple check of register contents being zero, Computer scientists typicallystart enumerating items, like loop counters, and string- or array- elements, with zero.[28][29]
13.2 Notation
Mathematicians use N or N (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all naturalnumbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying thatthe cardinal number of the set is aleph-naught (ℵ0) .[30]
To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) “0” is added in the formercase, and a superscript " ∗ " or subscript " 1 " is added in the latter case:
N0 = N0 = {0, 1, 2, . . .}
N∗ = N+ = N1 = N>0 = {1, 2, . . .}.
13.3 Properties
13.3.1 Addition
One can recursively define an addition on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for alla, b. Here S should be read as “successor”. This turns the natural numbers (N, +) into a commutative monoid withidentity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property andcan be embedded in a group (in the mathematical sense of the word group). The smallest group containing the naturalnumbers is the integers.If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
68 CHAPTER 13. NATURAL NUMBER
13.3.2 Multiplication
Analogously, given that addition has been defined, a multiplication × can be defined via a × 0 = 0 and a × S(b) = (a× b) + a. This turns (N*, ×) into a free commutative monoid with identity element 1; a generator set for this monoidis the set of prime numbers.
13.3.3 Relationship between addition and multiplication
Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a ×c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring.Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.The lack of additive inverses, which is equivalent to the fact that N is not closed under subtraction, means that N isnot a ring; instead it is a semiring (also known as a rig).If the natural numbers are taken as “excluding 0”, and “starting at 1”, the definitions of + and × are as above, exceptthat they begin with a + 1 = S(a) and a × 1 = a.
13.3.4 Order
In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations isassumed.A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural numberc with a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c arenatural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that theyare well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets isexpressed by an ordinal number; for the natural numbers this is expressed as ω.
13.3.5 Division
In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations isassumed.While it is in general not possible to divide one natural number by another and get a natural number as result, theprocedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 thereare natural numbers q and r such that
a = bq + r and r < b.
The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r areuniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms(such as the Euclidean algorithm), and ideas in number theory.
13.3.6 Algebraic properties satisfied by the natural numbers
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic prop-erties:
• Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are naturalnumbers.
• Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.• Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.• Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.• Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).• No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.
13.4. GENERALIZATIONS 69
13.4 Generalizations
Two generalizations of natural numbers arise from the two uses:
• A natural number can be used to express the size of a finite set; more generally a cardinal number is a measurefor the size of a set also suitable for infinite sets; this refers to a concept of “size” such that if there is a bijectionbetween two sets they have the same size. The set of natural numbers itself and any other countably infiniteset has cardinality aleph-null ( ℵ0 ).
• Linguistic ordinal numbers “first”, “second”, “third” can be assigned to the elements of a totally ordered finiteset, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. Thiscan be generalized to ordinal numbers which describe the position of an element in a well-ordered set in general.An ordinal number is also used to describe the “size” of a well-ordered set, in a sense different from cardinality:if there is an order isomorphism between two well-ordered sets they have the same ordinal number. The firstordinal number that is not a natural number is expressed as ω ; this is also the ordinal number of the set ofnatural numbers itself.
Many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowestpossible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω .For finite well-ordered sets, there is one-to-one correspondence between ordinal and cardinal numbers; therefore theycan both be expressed by the same natural number, the number of elements of the set. This number can also be usedto describe the position of an element in a larger finite, or an infinite, sequence.A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms)was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed fromthe ordinary natural numbers via the ultrapower construction.Georges Reeb used to claim provocatively that The naïve integers don't fill up N . Other generalizations are discussedin the article on numbers.
13.5 Formal definitions
13.5.1 Peano axioms
Main article: Peano axioms
Many properties of the natural numbers can be derived from the Peano axioms.[31][32]
• Axiom One: 0 is a natural number.
• Axiom Two: Every natural number has a successor.
• Axiom Three: 0 is not the successor of any natural number.
• Axiom Four: If the successor of x equals the successor of y, then x equals y.
• Axiom Five (the Axiom of Induction): If a statement is true of 0, and if the truth of that statement for a numberimplies its truth for the successor of that number, then the statement is true for every natural number.
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axiomshave 1 in place of 0. In ordinary arithmetic, the successor of x is x + 1. Replacing Axiom Five by an axiom schemaone obtains a (weaker) first-order theory called Peano Arithmetic.
13.5.2 Constructions based on set theory
Main article: Set-theoretic definition of natural numbers
70 CHAPTER 13. NATURAL NUMBER
In the area of mathematics called set theory, a special case of the von Neumann ordinal construction [33] defines thenatural numbers as follows:
Set 0 := { }, the empty set,and define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function.(Such sets are said to be `inductive'.) Then the intersection of all inductive sets is defined to be the setof natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.Each natural number is then equal to the set of all natural numbers less than it, so that
• 0 = { }• 1 = {0} = {{ }}• 2 = {0, 1} = {0, {0}} = {{ }, {{ }}}
:*3 = {0, 1, 2} = {0, {0}, {0, {0}}} ={{ }, {{ }}, {{ }, {{ }}}}
• n = {0, 1, 2, ..., n−2, n−1} = {0, 1, 2, ..., n−2} ∪ {n−1} = (n−1) ∪ {n−1} = S(n−1)and so on.
With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m.Also, with this definition, different possible interpretations of notations like Rn (n-tuples versus mappings of n intoR) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists,it is still possible to define any one of these sets.
Other constructions
Although the standard construction is useful, it is not the only possible construction. Zermelo's construction goes asfollows:
one defines 0 = { }and S(a) = {a},producing
• 0 = { }• 1 = {0} ={{ }}• 2 = {1} = {{{ }}}, etc.
Each natural number is then equal to the set of the natural number preceding it.
It is also possible to define 0 = {{ }}
and S(a) = a ∪ {a}producing
• 0 = {{ }}• 1 = {{ }, 0} = {{ }, {{ }}}• 2 = {{ }, 0, 1}, etc.
13.6. SEE ALSO 71
13.6 See also• Integer
• Set-theoretic definition of natural numbers
• Peano axioms
• Canonical representation of a positive integer
• Countable set
• Number#Classification for other number systems (rational, real, complex etc.)
13.7 Notes[1] Weisstein, Eric W., “Whole Number”, MathWorld.
[2] Clapham & Nicholson (2014): "whole numberAn integer, though sometimes it is taken to mean only non-negative integers,or just the positive integers.”
[3] James & James (1992) give definitions of “whole number” under several headwords:INTEGER … Syn. whole number.NUMBER … whole number. A nonnegative integer.WHOLE … whole number.(1) One of the integers 0, 1, 2, 3, … .(2) A positive integer; i.e., a natural number.(3) An integer, positive, negative, or zero.
[4] The Common Core State Standards for Mathematics say: “Whole numbers. The numbers 0, 1, 2, 3, ....” (Glossary, p. 87)(PDF)Definitions from The Ontario Curriculum, Grades 1-8: Mathematics, Ontario Ministry of Education (2005) (PDF)"natural numbers. The counting numbers 1, 2, 3, 4, ....” (Glossary, p. 128)"whole number. Any one of the numbers 0, 1, 2, 3, 4, ....” (Glossary, p. 134)Musser, Peterson & Burger (2013, p. 57): “As mentioned earlier, the study of the set of whole numbers, W = {0, 1, 2, 3,4, ...}, is the foundation of elementary school mathematics.”These pre-algebra books define the whole numbers:
• Szczepanski & Kositsky (2008): “Another important collection of numbers is thewhole numbers, the natural numberstogether with zero.” (Chapter 1: The Whole Story, p. 4). On the inside front cover, the authors say: “We based thisbook on the state standards for pre-algebra in California, Florida, New York, and Texas, ...”
• Bluman (2010): “When 0 is added to the set of natural numbers, the set is called the whole numbers.” (Chapter 1:Whole Numbers, p. 1)
Both books define the natural numbers to be: “1, 2, 3, …".
[5] Mendelson (2008) says: “The whole fantastic hierarchy of number systems is built up by purely set-theoretic means froma few simple assumptions about natural numbers.” (Preface, p. x)
[6] Bluman (2010): “Numbers make up the foundation of mathematics.” (p. 1)
[7] Weisstein, Eric W., “Natural Number”, MathWorld.
[8] “natural number”, Merriam-Webster.com (Merriam-Webster), retrieved 4 October 2014
[9] Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)
[10] Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ... } of all naturalnumbers may be described as follows: ℕ contains an “initial” number 0; ...'. They follow that with their version of thePeano Postulates. (p. 15)
[11] Weisstein, Eric W., “Counting Number”, MathWorld.
[12] Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
72 CHAPTER 13. NATURAL NUMBER
[13] Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
[14] The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of NaturalSciences, Brussels, Belgium. UNESCO's Portal to the Heritage of Astronomy
[15] Georges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3
[16] “A history of Zero”. MacTutor History of Mathematics. Retrieved 2013-01-23. ... a tablet found at Kish ... thought todate from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated fromaround the same time use a single hook for an empty place
[17] Mann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063.
[18] Evans, Brian (2014), “Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations”, The De-velopment of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN9781118853979.
[19] Michael L. Gorodetsky (2003-08-25). “Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius”. Hbar.phys.msu.ru.Retrieved 2012-02-13.
[20] This convention is used, for example, in Euclid’s Elements, see Book VII, definitions 1 and 2.
[21] Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7
[22] “Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations andstructure of the subject.” (Eves 1990, p. 606)
[23] Eves 1990, Chapter 15
[24] L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14(4): 285. doi:10.1112/blms/14.4.285, 1982.
[25] Bagaria, Joan. “Set Theory”. The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).
[26] Goldrei, Derek (1998). “3”. Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]:Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0.
[27] This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner(2000, p. 2).
[28] Brown, Jim (1978). “In Defense of Index Origin 0”. ACMSIGAPLAPLQuote Quad 9 (2): 7 – 7. doi:10.1145/586050.586053.Retrieved 19 January 2015.
[29] Hui, Roger. “Is Index Origin 0 a Hindrance?". http://www.jsoftware.com''. JSoftware / Roger Hui. Retrieved 19 January2015.
[30] Weisstein, Eric W., “Cardinal Number”, MathWorld.
[31] G.E. Mints (originator), “Peano axioms”, Encyclopedia of Mathematics (Springer, in cooperation with the European Math-ematical Society), retrieved 8 October 2014
[32] Hamilton (1988) calls them “Peano’s Postulates” and begins with “1. 0 is a natural number.” (p. 117f)Halmos (1960) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with"(I) 0 ∈ ω (where, of course, 0 = ∅ )" ( ω is the set of all natural numbers). (p. 46)Morash (1991) gives “a two-part axiom” in which the natural numbers begin with 1. (Section 10.1: An Axiomatization forthe System of Positive Integers)
[33] Von Neumann 1923
13.8 References• Bluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.), McGraw-Hill Professional
• Carothers, N.L. (2000), Real analysis, Cambridge University Press, ISBN 0-521-49756-6
• Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics (Fifth ed.),Oxford University Press
13.9. EXTERNAL LINKS 73
• Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0-486-21010-3
• Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-X
• Eves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Thomson, ISBN 978-0-03-029558-4
• Halmos, Paul (1960), Naive Set Theory, Springer Science & Business Media
• Hamilton, A. G. (1988), Logic for Mathematicians (Revised ed.), Cambridge University Press
• James, Robert C.; James, Glenn (1992), Mathematics Dictionary (Fifth ed.), Chapman & Hall
• Landau, Edmund (1966), Foundations of Analysis (Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-X
• Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical Society
• Mendelson, Elliott (2008) [1973], Number Systems and the Foundations of Analysis, Dover Publications
• Morash, Ronald P. (1991), Bridge to Abstract Mathematics: Mathematical Proof and Structures (Second ed.),Mcgraw-Hill College
• Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013), Mathematics for Elementary Teachers: AContemporary Approach (10th ed.), Wiley Global Education, ISBN 978-1118457443
• Szczepanski, Amy F.; Kositsky, Andrew P. (2008), The Complete Idiot’s Guide to Pre-algebra, Penguin Group
• Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008), Elementary Real Analysis (Seconded.), ClassicalRealAnalysis.com, ISBN 9781434843678
• Von Neumann, Johann (1923), “Zur Einführung der transfiniten Zahlen”, Acta litterarum ac scientiarum RagiaeUniversitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum 1: 199–208
• Von Neumann, John (January 2002) [1923], “On the introduction of transfinite numbers”, in Jean vanHeijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), HarvardUniversity Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.
13.9 External links• Hazewinkel, Michiel, ed. (2001), “Natural number”, Encyclopedia of Mathematics, Springer, ISBN 978-1-
55608-010-4
• Axioms and Construction of Natural Numbers
• Essays on the Theory of Numbers by Richard Dedekind at Project Gutenberg
74 CHAPTER 13. NATURAL NUMBER
The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[12][13][14] is believed to have been used 20,000years ago for natural number arithmetic.
13.9. EXTERNAL LINKS 75
The double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).
Chapter 14
Order isomorphism
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that consti-tutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic,they can be considered to be “essentially the same” in the sense that one of the orders can be obtained from the otherjust by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddingsand Galois connections.[1]
14.1 Definition
Formally, given two posets (S,≤S) and (T,≤T ) , an order isomorphism from (S,≤S) to (T,≤T ) is a bijectivefunction f from S to T with the property that, for every x and y in S , x ≤S y if and only if f(x) ≤T f(y) . Thatis, it is a bijective order-embedding.[2]
It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions thatf cover all the elements of T and that it preserve orderings, are enough to ensure that f is also one-to-one, for iff(x) = f(y) then (by the assumption that f preserves the order) it would follow that x ≤ y and y ≤ x , implyingby the definition of a partial order that x = y .Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a mono-tone inverse.[3]
An order isomorphism from a partially ordered set to itself is called an order automorphism.[4]
14.2 Examples• The identity function on any partially ordered set is always an order automorphism.
• Negation is an order isomorphism from (R,≤) to (R,≥) (where R is the set of real numbers and ≤ denotesthe usual numerical comparison), since −x ≥ −y if and only if x ≤ y.[5]
• The open interval (0, 1) (again, ordered numerically) does not have an order isomorphism to or from the closedinterval [0, 1] : the closed interval has a least element, but the open interval does not, and order isomorphismsmust preserve the existence of least elements.[6]
14.3 Order types
If f is an order isomorphism, then so is its inverse function. Also, if f is an order isomorphism from (S,≤S) to(T,≤T ) and g is an order isomorphism from (T,≤T ) to (U,≤U ) , then the function composition of f and g is itselfan order isomorphism, from (S,≤S) to (U,≤U ) .[7]
Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to theother.[8] Identity functions, function inverses, and compositions of functions correspond, respectively, to the three
76
14.4. SEE ALSO 77
defining characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomor-phism is an equivalence relation. The class of partially ordered sets can be partitioned by it into equivalence classes,families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called ordertypes.
14.4 See also• Permutation pattern, a permutation that is order-isomorphic to a subsequence of another permutation
14.5 Notes[1] Block (2011); Ciesielski (1997).
[2] This is the definition used by Ciesielski (1997). For Bloch (2011) and Schröder (2003) it is a consequence of a differentdefinition.
[3] This is the definition used by Bloch (2011) and Schröder (2003).
[4] Schröder (2003), p. 13.
[5] See example 4 of Ciesielski (1997), p. 39., for a similar example with integers in place of real numbers.
[6] Ciesielski (1997), example 1, p. 39.
[7] Ciesielski (1997); Schröder (2003).
[8] Ciesielski (1997).
14.6 References• Bloch, Ethan D. (2011), Proofs and Fundamentals: A First Course in Abstract Mathematics, Undergraduate
Texts in Mathematics (2nd ed.), Springer, pp. 276–277, ISBN 9781441971265.
• Ciesielski, Krzysztof (1997), Set Theory for theWorkingMathematician, London Mathematical Society StudentTexts 39, Cambridge University Press, pp. 38–39, ISBN 9780521594653.
• Schröder, Bernd Siegfried Walter (2003), Ordered Sets: An Introduction, Springer, p. 11, ISBN 9780817641283.
Chapter 15
Order type
Not to be confused with ordered types.
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when theyare order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X →Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in thecorrect order). In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.For example, the set of integers and the set of even integers have the same order type, because the mapping n 7→ 2npreserves the order. But the set of integers and the set of rational numbers (with the standard ordering) are not orderisomorphic, because, even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positiveintegers (which has a least element), and that of negative integers (which has a greatest element). The open interval(0,1) of rationals is order isomorphic to the rationals (since
y =2x− 1
1− |2x− 1|
provides a strictly increasing bijection from the former to the latter); the half-closed intervals [0,1) and (0,1], and theclosed interval [0,1], are three additional order type examples.Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.
15.1 Order type of well-orderings
Every well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be thecanonical representatives of their classes, and so the order type of a well-ordered set is usually identified with thecorresponding ordinal. For example, the order type of the natural numbers is ω.The order type of a well-ordered set V is sometimes expressed as ord(V).[1]
For example, consider the set of even ordinals less than ω·2+7, which is:
V = {0, 2, 4, 6, ...; ω, ω+2, ω+4, ...; ω·2, ω·2+2, ω·2+4, ω·2+6}.
Its order type is:
ord(V) = ω·2+4 = {0, 1, 2, 3, ...; ω, ω+1, ω+2, ...; ω·2, ω·2+1, ω·2+2, ω·2+3}.
Because there are 2 separate lists of counting and 4 in sequence at the end.
78
15.2. RATIONAL NUMBERS 79
15.2 Rational numbers
Any countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way.Any dense countable totally ordered set with no highest and no lowest element can be mapped bijectively onto therational numbers in an order-preserving way.
15.3 Notation
The order type of the rationals is usually denoted η . If a set S has order type σ , the order type of the dual of S (thereversed order) is denoted σ∗ .
15.4 See also• Well-order
15.5 External links• Weisstein, Eric W., “Order Type”, MathWorld.
15.6 References[1] Ordinal Numbers and Their Arithmetic
Chapter 16
Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal num-bers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either byconstructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantornormal form provides a standardized way of writing ordinals. The so-called “natural” arithmetical operations retaincommutativity at the expense of continuity.
16.1 Addition
The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinalwhich results from adding the order-types of S and T. If two well-ordered sets are not already disjoint, then they canbe replaced by order-isomorphic disjoint sets, e.g. replace S by {0} × S and T by {1} × T. This way, the well-orderedset S is written “to the left” of the well-ordered set T, meaning one defines an order on S ∪ T in which every element ofS is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This additionof the order-types is associative and generalizes the addition of natural numbers.The first transfinite ordinal is ω, the set of all natural numbers. For example, the ordinal ω + ω is obtained by twocopies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first.Writing 0' < 1' < 2' < ... for the second copy, ω + ω looks like
0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...
This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0and 0' do not have direct predecessors. As another example, here are 3 + ω and ω + 3:
0 < 1 < 2 < 0' < 1' < 2' < ...0 < 1 < 2 < ... < 0' < 1' < 2'
After relabeling, the former just looks like ω itself, i.e. 3 + ω = ω, while the latter does not: ω + 3 is not equal toω since ω + 3 has a largest element (namely, 2') and ω does not. Hence, this addition is not commutative. In factit is quite rare for α+β to be equal to β+α: this happens if and only if α=γm, β=γn for some ordinal γ and naturalnumbers m and n. Moreover the relation α+β = β+α is an equivalence relation on the set of nonzero ordinals, and allthe equivalence classes are countable infinite.However, addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.The definition of addition can also be given inductively (the following induction is on β):
• α + 0 = α,
• α + (β + 1) = (α + β) + 1 (here, "+ 1” denotes the successor of an ordinal),
• and if β is a limit ordinal then α + β is the limit of the α + δ for all δ < β.
80
16.2. MULTIPLICATION 81
Using this definition, ω + 3 can be seen to be a successor ordinal (it is the successor of ω + 2), whereas 3 + ω is alimit ordinal, namely, the limit of 3 + 0 = 3, 3 + 1 = 4, 3 + 2 = 5, etc., which is just ω.Zero is an additive identity α + 0 = 0 + α = α.Addition is associative (α + β) + γ = α + (β + γ).Addition is strictly increasing and continuous in the right argument:
α < β ⇒ γ + α < γ + β
but the analogous relation does not hold for the left argument; instead we only have:
α < β ⇒ α+ γ ≤ β + γ
Ordinal addition is left-cancellative: if α + β = α + γ, then β = γ. Furthermore, one can define left subtraction forordinals β ≤ α: there is a unique γ such that α = β + γ. On the other hand, right cancellation does not work:
3 + ω = 0 + ω = ω but 3 ̸= 0
Nor does right subtraction, even when β ≤ α: for example, there does not exist any γ such that γ + 42 = ω.If the ordinals less than α are closed under addition and contain 0 then α is occasionally called a γ-number (seeadditively indecomposable ordinal). These are exactly the ordinals of the form ωβ.
16.2 Multiplication
The Cartesian product, S×T, of two well-ordered sets S and T can be well-ordered by a variant of lexicographicalorder that puts the least significant position first. Effectively, each element of T is replaced by a disjoint copy of S.The order-type of the Cartesian product is the ordinal which results from multiplying the order-types of S and T.Again, this operation is associative and generalizes the multiplication of natural numbers.Here is ω·2:
00 < 10 < 20 < 30 < ... < 01 < 11 < 21 < 31 < ...
which has the same order type as ω + ω. In contrast, 2·ω looks like this:
00 < 10 < 01 < 11 < 02 < 12 < 03 < 13 < ...
and after relabeling, this looks just like ω. Thus, ω·2 = ω+ω ≠ ω = 2·ω, showing that multiplication of ordinals isnot commutative. More generally, a natural number greater than 1 never commutes with any infinite ordinal, and twoinfinite ordinals α, β commute if and only if αm = βn for some positive natural numbers m and n. The relation "αcommutes with β" is an equivalence relation on the ordinals greater than 1, and all equivalence classes are countablyinfinite.Distributivity partially holds for ordinal arithmetic: R(S+T) = RS+RT. However, the other distributive law (T+U)R= TR+UR is not generally true: (1+1)·ω = 2·ω = ω while 1·ω+1·ω = ω+ω which is different. Therefore, the ordinalnumbers form a left near-semiring, but do not form a ring.The definition of multiplication can also be given inductively (the following induction is on β):
• α·0 = 0,
• α·(β+1) = (α·β)+α,
• and if β is a limit ordinal then α·β is the limit of the α·δ for δ < β.
82 CHAPTER 16. ORDINAL ARITHMETIC
The main properties of the product are:
• α·0 = 0·α = 0.
• One (1) is a multiplicative identity α·1 = 1·α = α.
• Multiplication is associative (α·β)·γ = α·(β·γ).
• Multiplication is strictly increasing and continuous in the right argument: (α < β and γ > 0) ⇒ γ·α < γ·β
• Multiplication is not strictly increasing in the left argument, for example, 1 < 2 but 1·ω = 2·ω = ω. However,it is (non-strictly) increasing, i.e. α ≤ β ⇒ α·γ ≤ β·γ.
• There is a left cancellation law: If α > 0 and α·β = α·γ, then β = γ.
• Right cancellation does not work, e.g. 1·ω = 2·ω = ω, but 1 and 2 are different.
• α·β = 0 ⇒ α = 0 or β = 0.
• Distributive law on the left: α·(β+γ) = α·β+α·γ
• No distributive law on the right: e.g. (ω+1)·2 = ω+1+ω+1 = ω+ω+1 = ω·2+1 which is not ω·2+2.
• Left division with remainder: for all α and β, if β > 0, then there are unique γ and δ such that α = β·γ+δ and δ< β. (This does not however mean the ordinals are a Euclidean domain, since they are not even a ring, and theEuclidean “norm” is ordinal-valued.)
• Right division does not work: there is no α such that α·ω ≤ ωω ≤ (α+1)·ω.
A δ-number (see additively indecomposable ordinal#Multiplicatively indecomposable) is an ordinal greater than 1such that αδ=δ whenever 0<α<δ. These are exactly the ordinals of the form ωωβ .
16.3 Exponentiation
The definition of ordinal exponentiation for finite exponents is straightforward. If the exponent is a finite number, thepower is the result of iterated multiplication. For instance, ω2 = ω·ω using the operation of ordinal multiplication.Note that ω·ω can be defined using the set of functions from 2 = {0,1} to ω = {0,1,2,...}, ordered lexicographicallywith the least significant position first:
(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...
Here for brevity, we have replaced the function {(0,k), (1,m)} by the ordered pair (k, m).Similarly, for any finite exponent n, ωn can be defined using the set of functions from n (the domain) to the naturalnumbers (the range). These functions can be abbreviated as n-tuples of natural numbers.But for infinite exponents, the definition may not be obvious. A limit ordinal, such as ωω, is the supremum ofall smaller ordinals. It might seem natural to define ωω using the set of all infinite sequences of natural numbers.However, we find that any absolutely defined ordering on this set is not well-ordered. To deal with this issue we canuse the variant lexicographical ordering again. We restrict the set to sequences which are nonzero for only a finitenumber of arguments. This is naturally motivated as the limit of the finite powers of the base (similar to the conceptof coproduct in algebra). This can also be thought of as the infinite union
∪n<ω ωn .
Each of those sequences corresponds to an ordinal less than ωω such as ωn1c1 +ωn2c2 + · · ·+ωnkck and ωω is thesupremum of all those smaller ordinals.The lexicographical order on this set is a well ordering that resembles the ordering of natural numbers written indecimal notation, except with digit positions reversed, and with arbitrary natural numbers instead of just the digits0–9:
16.3. EXPONENTIATION 83
(0,0,0,...) < (1,0,0,0,...) < (2,0,0,0,...) < ... <(0,1,0,0,0,...) < (1,1,0,0,0,...) < (2,1,0,0,0,...) < ... <(0,2,0,0,0,...) < (1,2,0,0,0,...) < (2,2,0,0,0,...)
< ... <
(0,0,1,0,0,0,...) < (1,0,1,0,0,0,...) < (2,0,1,0,0,0,...)
< ...
In general, any ordinal α can be raised to the power of another ordinal β in the same way to get αβ.It is easiest to explain this using Von Neumann’s definition of an ordinal as the set of all smaller ordinals. Then, toconstruct a set of order type αβ consider all functions from β to α such that only a finite number of elements of thedomain β map to a non zero element of α (essentially, we consider the functions with finite support). The order islexicographic with the least significant position first. We find
• 1ω = 1,
• 2ω = ω,
• 2ω+1 = ω·2 = ω+ω.
The definition of exponentiation can also be given inductively (the following induction is on β, the exponent):
• α0 = 1,
• αβ+1 = (αβ)·α, and
• if δ is limit, then αδ is the limit of the αβ for all β < δ.
Properties of ordinal exponentiation:
• α0 = 1.
• If 0 < α, then 0α = 0.
• 1α = 1.
• α1 = α.
• αβ·αγ = αβ + γ.
• (αβ)γ = αβ·γ.
• There are α, β, and γ for which (α·β)γ ≠ αγ·βγ. For instance, (ω·2)2 = ω·2·ω·2 = ω2·2 ≠ ω2·4.
• Ordinal exponentiation is strictly increasing and continuous in the right argument: If γ > 1 and α < β, then γα
< γβ.
• If α < β, then αγ ≤ βγ. Note, for instance, that 2 < 3 and yet 2ω = 3ω = ω.
• If α > 1 and αβ = αγ, then β = γ. If α = 1 or α = 0 this is not the case.
• For all α and β, if β > 1 and α > 0 then there exist unique γ, δ, and ρ such that α = βγ·δ + ρ such that 0 < δ <β and ρ < βγ.
Warning: Ordinal exponentiation is quite different from cardinal exponentiation. For example, the ordinal expo-nentiation 2ω = ω, but the cardinal exponentiation 2ℵ0 is the cardinality of the continuum which is larger than ℵ0 .To avoid confusing ordinal exponentiation with cardinal exponentiation, one can use symbols for ordinals (e.g. ω) inthe former and symbols for cardinals (e.g. ℵ0 ) in the latter.Jacobsthal showed that the only solutions of αβ = βα with α≤β are given by α=β, or α=2 β=4, or α is any limit ordinaland β=εα where ε is an ε-number larger than α.
84 CHAPTER 16. ORDINAL ARITHMETIC
16.4 Cantor normal form
Every ordinal number α can be uniquely written as ωβ1c1 + ωβ2c2 + · · · + ωβkck , where k is a natural number,c1, c2, . . . , ck are positive integers, and β1 > β2 > . . . > βk ≥ 0 are ordinal numbers. This decomposition of αis called the Cantor normal form of α, and can be considered the base-ω positional numeral system. The highestexponent β1 is called the degree of α , and satisfies β1 ≤ α . The equality β1 = α applies if and only if α = ωα .In that case Cantor normal form does not express the ordinal in terms of smaller ones; this can happen as explainedbelow.A minor variation of Cantor normal form, which is usually slightly easier to work with, is to set all the numbers ciequal to 1 and allow the exponents to be equal. In other words, every ordinal number α can be uniquely written asωβ1 + ωβ2 + · · ·+ ωβk , where k is a natural number, and β1 ≥ β2 ≥ . . . ≥ βk ≥ 0 are ordinal numbers.Another variation of the Cantor normal form is the “base δ expansion”, where ω is replaced by any ordinal δ>1, andthe numbers ci are positive ordinals less than δ.The Cantor normal form allows us to uniquely express—and order—the ordinals α that are built from the naturalnumbers by a finite number of arithmetical operations of addition, multiplication and exponentiation base- ω : inother words, assuming β1 < α in the Cantor normal form, we can also express the exponents βi in Cantor normalform, and making the same assumption for the βi as for α and so on recursively, we get a system of notation for theseordinals (for example,
ω
(ω(ω
7·6+ω+42)·1729+ω9+88
)· 3 + ω(ωω) · 5 + 65537
denotes an ordinal).The ordinal ε0 (epsilon nought) is the set of ordinal values α of the finite-length arithmetical expressions of Cantornormal form that are hereditarily non-trivial where non-trivial means β1<α when 0<α. It is the smallest ordinal thatdoes not have a finite arithmetical expression in terms of ω, and the smallest ordinal such that ε0 = ωε0 , i.e. inCantor normal form the exponent is not smaller than the ordinal itself. It is the limit of the sequence
0, 1 = ω0, ω = ω1, ωω, ωωω
, . . . .
The ordinal ε0 is important for various reasons in arithmetic (essentially because it measures the proof-theoreticstrength of the first-order Peano arithmetic: that is, Peano’s axioms can show transfinite induction up to any ordinalless than ε0 but not up to ε0 itself).The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example,one need merely know that
ωβc+ ωβ′c′ = ωβ′
c′ ,
if β′ > β (if β′ = β one can obviously rewrite this as ωβ(c+ c′) , and if β′ < β the expression is already in Cantornormal form); and to compute products, the essential facts are that when 0 < α = ωβ1c1 + · · ·+ωβkck is in Cantornormal form and 0 < β′ , then
αωβ′= ωβ1+β′
and
αn = ωβ1c1n+ ωβ2c2 + · · ·+ ωβkck ,
if n is a non-zero natural number.To compare two ordinals written in Cantor normal form, first compare β1 , then c1 , then β2 , then c2 , etc.. Atthe first difference, the ordinal that has the larger component is the larger ordinal. If they are the same until oneterminates before the other, then the one that terminates first is smaller.
16.5. FACTORIZATION INTO PRIMES 85
16.5 Factorization into primes
Ernst Jacobsthal showed that the ordinals satisfy a form of the unique factorization theorem: every nonzero ordinalcan be written as a product of a finite number of prime ordinals. This factorization into prime ordinals is in generalnot unique, but there is a “minimal” factorization into primes that is unique up to changing the order of finite primefactors (Sierpiński 1958).A prime ordinal is an ordinal greater than 1 that cannot be written as a product of two smaller ordinals. Some of thefirst primes are 2, 3, 5, ... , ω, ω+1, ω2+1, ω3+1, ..., ωω, ωω+1, ωω+1+1, ... There are three sorts of prime ordinals:
• The finite primes 2, 3, 5, ...
• The ordinals of the form ωωα for any ordinal α. These are the prime ordinals that are limits, and are the deltanumbers.
• The ordinals of the form ωα+1 for any ordinal α>0. These are the infinite successor primes, and are thesuccessors of gamma numbers, the additively indecomposable ordinals.
Factorization into primes is not unique: for example, 2×3=3×2, 2×ω=ω, (ω+1)×ω=ω×ω and ω×ωω = ωω. Howeverthere is a unique factorization into primes satisfying the following additional conditions:
• Every limit prime occurs before every successor prime
• If two consecutive primes of the prime factorization are both limits or both finite, then the second one is atmost the first one.
This prime factorization can easily be read off using the Cantor normal form as follows:
• First write the ordinal as a product αβ where α is the smallest power of ω in the Cantor normal form and β isa successor.
• If α=ωγ then writing γ in Cantor normal form gives an expansion of α as a product of limit primes.
• Now look at the Cantor normal form of β. If β = ωλm + ωμn+ smaller terms then β = (ωμn+ smaller terms)(ωλ−μ
+ 1)m is a product of a smaller ordinal and a prime and an integer m. Repeating this and factorizing the integersinto primes gives the prime factorization of β.
So the factorization of the Cantor normal form ordinal
ωα1n1 + · · ·+ ωαknk (with α1 > · · · > αk )
into a minimal product of infinite primes and integers is
ωωβ1 · · ·ωωβmnk(ω
αk−1−αk + 1)nk−1 · · ·n2(ωα1−α2 + 1)n1
where each ni should be replaced by its factorization into a non-increasing sequence of finite primes and
αk = ωβ1 + · · ·+ ωβm with β1 ≥ · · · ≥ βm .
16.6 Large countable ordinals
As discussed above, the Cantor Normal Form of ordinals below ϵ0 can be expressed in an alphabet containing only thefunction symbols for addition, multiplication and exponentiation, as well as constant symbols for each natural numberand for ω . We can do away with the infinitely many numerals by using just the constant symbol 0 and the operation ofsuccessor, S (for example, the integer 4 may be expressed as S(S(S(S(0)))) ). This describes an ordinal notation: asystem for naming ordinals over a finite alphabet. This particular system of ordinal notation is called the collection of
86 CHAPTER 16. ORDINAL ARITHMETIC
arithmetical ordinal expressions, and can express all ordinals below ϵ0 , but cannot express ϵ0 . There are other ordinalnotations capable of capturing ordinals well past ϵ0 , but because there are only countably many strings over any finitealphabet, for any given ordinal notation there will be ordinals below ω1 that are not expressible. Such ordinals areknown as large countable ordinals.The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions,and more general primitive recursive ordinal functions can be used to describe larger ordinals.
16.7 Natural operations
The natural sum and natural product operations on ordinals were defined in 1906 by Gerhard Hessenberg, andare sometimes called the Hessenberg sum (or product) (Sierpinski 1958). These are the same as the addition andmultiplication (restricted to ordinals) of John Conway’s field of surreal numbers. They have the advantage that they areassociative and commutative, and natural product distributes over natural sum. The cost of making these operationscommutative is that they lose the continuity in the right argument which is a property of the ordinary sum and product.The natural sum of α and β is sometimes denoted by α # β, and the natural product by a sort of doubled × sign: αβ. (Other common notation is α ⊕ β and α ⊗ β). To define the natural sum of two ordinals, consider once again thedisjoint union S∪T of two well-ordered sets having these order types. Start by putting a partial order on this disjointunion by taking the orders on S and T separately but imposing no relation between S and T. Now consider the ordertypes of all well-orders on S ∪ T which extend this partial order: the least upper bound of all these ordinals (whichis, actually, not merely a least upper bound but actually a greatest element) is the natural sum.[1] Alternatively, we candefine the natural sum of α and β inductively (by simultaneous induction on α and β) as the smallest ordinal greaterthan the natural sum of α and γ for all γ < β and of γ and β for all γ < α.The natural sum is associative and commutative. It is always greater or equal to the usual sum, but it may be greater.For example, the natural sum of ω and 1 is ω+1 (the usual sum), but this is also the natural sum of 1 and ω.To define the natural product of two ordinals, consider once again the cartesian product S × T of two well-orderedsets having these order types. Start by putting a partial order on this cartesian product by using just the product order(compare two pairs if and only if each of the two coordinates is comparable). Now consider the order types of allwell-orders on S × T which extend this partial order: the least upper bound of all these ordinals (which is, actually, notmerely a least upper bound but actually a greatest element) is the natural product. There is also an inductive definitionof the natural product (by mutual induction), but it is somewhat tedious to write down and we shall not do so (seethe article on surreal numbers for the definition in that context, which, however, uses surreal subtraction, somethingwhich obviously cannot be defined on ordinals).The natural product is associative and commutative and distributes over the natural sum. It is always greater or equalto the usual product, but it may be greater. For example, the natural product of ω and 2 is ω·2 (the usual product),but this is also the natural product of 2 and ω.Yet another way to define the natural sum and product of two ordinals α and β is to use the Cantor normal form:one can find a sequence of ordinals γ1 > … > γ and two sequences (k1, …, k ) and (j1, …, j ) of natural numbers(including zero, but satisfying ki + ji > 0 for all i) such that
α = ωγ1 · k1 + · · ·+ ωγn · kn
β = ωγ1 · j1 + · · ·+ ωγn · jnand defines
α#β = ωγ1 · (k1 + j1) + · · ·+ ωγn · (kn + jn).
Under natural addition, the ordinals can be identified with the elements of the free abelian group with basis the gammanumbers ωα that have non-negative integer coefficients. Under natural addition and multiplication, the ordinals canbe identified with the elements of the (commutative) polynomial ring generated by the delta numbers ωωα that havenon-negative integer coefficients. The ordinals do not have unique factorization into primes under the natural product.While the full polynomial ring does have unique factorization, the subset of polynomials with non-negative coefficientsdoes not: for example, if x is any delta number, then
16.8. NOTES 87
x5 + x4 + x3 + x2 + x+ 1 = (x+ 1)(x4 + x2 + 1) = (x2 + x+ 1)(x3 + 1)
has two incompatible expressions as a natural product of polynomials with non-negative coefficients that cannot bedecomposed further.
16.8 Notes[1] Philip W. Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math.
Soc. 48 (1942), 262–271. See Theorem 1. Available here
16.9 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.
• Sierpiński, Wacław (1958), Cardinal and ordinal numbers., Polska Akademia Nauk Monografie Matematyczne34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787
16.10 External links• Ordinal calculator for download (MS-DOS executable or Borland C++ source)
Chapter 17
Ordinal number
This article is about the mathematical concept. For number words denoting a position in a sequence (“first”, “second”,“third”, etc.), see Ordinal number (linguistics).In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set. They are usually identified
with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and fromcardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.Ordinals were introduced by Georg Cantor in 1883[1] to accommodate infinite sequences and to classify derived sets,which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2]
Two sets S and S' have the same cardinality if there is a bijection between them (i.e. there exists a function f that isboth injective and surjective, that is it maps each element x of S to a unique element y = f(x) of S' and each elementy of S' comes from exactly one such element x of S).If a partial order < is defined on set S, and a partial order <' is defined on set S' , then the posets (S,<) and (S' ,<') areorder isomorphic if there is a bijection f that preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Everywell-ordered set (S,<) is order isomorphic to the set of ordinals less than one specific ordinal number [the order typeof (S,<)] under their natural ordering.The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings ofa finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number ℵ0 .However, in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their orderinformation. Whereas there is only one countably infinite cardinal, namely ℵ0 itself, there are uncountably manycountably infinite ordinals, namely
ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω , …, ε0, ….
Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1 and likewise, 2·ω isω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1, which is identifiedwith the cardinal ℵ1 (next cardinal after ℵ0 ). Well-ordered cardinals are identified with their initial ordinals, i.e. thesmallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals tocardinals.In general, each ordinal α is the order type of the set of ordinals strictly less than the ordinal α itself. This propertypermits every ordinal to be represented as the set of all ordinals less than it. Ordinals may be categorized as: zero,successor ordinals, and limit ordinals (of various cofinalities). Given a class of ordinals, one can identify the α-th member of that class, i.e. one can index (count) them. Such a class is closed and unbounded if its indexingfunction is continuous and never stops. The Cantor normal form uniquely represents each ordinal as a finite sum ofordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referentialrepresentations as ε0 = ωε0 . Larger and larger ordinals can be defined, but they become more and more difficult todescribe. Any ordinal number can be made into a topological space by endowing it with the order topology; thistopology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in theorder topology if and only if either it is cofinite or it does not contain ω as an element.
88
17.1. ORDINALS EXTEND THE NATURAL NUMBERS 89
0
1
2
3
ω
ω+1
ω+2
ω+3
ω·2
ω·3
ω·2+1
ω·2+2
ω·4
ω²ω²+
1ω²+2
ω²+ω
ω²+ω·2
ω²·2
ω²·3ω²·4
ω³
ω³+ω
ω³+ω²ω·5
4
5
ω+4
ωωω⁴
ω³·2
ω·2+3
Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω
17.1 Ordinals extend the natural numbers
A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the sizeof a set, or to describe the position of an element in a sequence. When restricted to finite sets these two conceptscoincide; there is only one way to put a finite set into a linear sequence, up to isomorphism. When dealing withinfinite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion ofposition, which is generalized by the ordinal numbers described here. This is because, while any set has only one size(its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals areintimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some
90 CHAPTER 17. ORDINAL NUMBER
mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (givenany two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasingsequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set hasa least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element beinglabelled 0, the one after that 1, the next one 2, “and so on”) and to measure the “length” of the whole set by the leastordinal that is not a label for an element of the set. This “length” is called the order type of the set.Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals identifieseach ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less thanit, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generallyidentified as the set {0,1,2,…,41}. Conversely, any set (S) of ordinals that is downward-closed—meaning that forany ordinal α in S and any ordinal β < α, β is also in S—is (or can be identified with) an ordinal.There are infinite ordinals as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers(finite ordinals) and that can even be identified with the set of natural numbers (indeed, the set of natural numbers iswell-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associatedwith it, which is exactly how ω is defined).
A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and nare natural numbers.
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they startwith the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and afterthat come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them asnames.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Nowthe set of ordinals formed in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinalassociated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and muchlater on ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continuedindefinitely far (“indefinitely far” is exactly what ordinals are good at: basically every time one says “and so on” whenenumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals,expressed as ω1.
17.2. DEFINITIONS 91
17.2 Definitions
17.2.1 Well-ordered sets
Further information: Ordered set
In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependentchoice, this is equivalent to just saying that the set is totally ordered and there is no infinite decreasing sequence,something perhaps easier to visualize. In practice, the importance of well-ordering is justified by the possibility ofapplying transfinite induction, which says, essentially, that any property that passes on from the predecessors of anelement to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation(computer program or game) can be well-ordered in such a way that each step is followed by a “lower” step, then thecomputation will terminate.It is inappropriate to distinguish between two well-ordered sets if they only differ in the “labeling of their elements”, ormore formally: if the elements of the first set can be paired off the with the elements of the second set such that if oneelement is smaller than another in the first set, then the partner of the first element is smaller than the partner of thesecond element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphismand the two well-ordered sets are said to be order-isomorphic, or similar (obviously this is an equivalence relation).Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: thismakes it quite justifiable to consider the two sets as essentially identical, and to seek a “canonical” representative ofthe isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling ofthe elements of any well-ordered set.Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalenceclass for the equivalence relation of “being order-isomorphic”. There is a technical difficulty involved, however, inthe fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of settheory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class.
17.2.2 Definition of an ordinal as an equivalence class
The original definition of ordinal number, found for example in Principia Mathematica, defines the order type ofa well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, anordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF andrelated systems of axiomatic set theory because these equivalence classes are too large to form a set. However, thisdefinition still can be used in type theory and in Quine’s axiomatic set theory New Foundations and related systems(where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).
17.2.3 Von Neumann definition of ordinals
Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and everywell-ordered set will be order-isomorphic to exactly one ordinal number.The standard definition, suggested by John von Neumann, is: each ordinal is the well-ordered set of all smaller ordinals.In symbols, λ = [0,λ).[3][4] Formally:
A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and everyelement of S is also a subset of S.
Note that the natural numbers are ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals,that is, there is an order preserving bijective function between them.Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element ofT if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are
92 CHAPTER 17. ORDINAL NUMBER
equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes thefact that every set of natural numbers is well-ordered.Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every setof ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union existsregardless of the set’s size, by the axiom of union.The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself,which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinalsis variously called “Ord”, “ON”, or "∞".An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of itssubsets has a maximum.
17.2.4 Other definitions
There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, thefollowing are equivalent for a set x:
• x is an ordinal,
• x is a transitive set, and set membership is trichotomous on x,
• x is a transitive set totally ordered by set inclusion,
• x is a transitive set of transitive sets.
These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to furthermake sure that the definition excludes urelements from appearing in ordinals.
17.3 Transfinite sequence
If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept,a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinarysequence corresponds to the case α = ω.
17.4 Transfinite induction
Main article: Transfinite induction
17.4.1 What is transfinite induction?
Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restatinghere.
Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of allordinals.
That is, if P(α) is true whenever P(β) is true for all β<α, then P(α) is true for all α. Or, more practically: in order toprove a property P for all ordinals α, one can assume that it is already known for all smaller β<α.
17.4. TRANSFINITE INDUCTION 93
17.4.2 Transfinite recursion
Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normallysaid to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a(class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α,one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β). Itthen follows by transfinite induction that there is one and only one function satisfying the recursion formula up to andincluding α.Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function Fby letting F(α) be the smallest ordinal not in the set {F(β) | β < α}, that is, the set consisting of all F(β) for β < α.This definition assumes the F(β) known in the very process of defining F; this apparent vicious circle is exactly whatdefinition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal β < 0, and the set {F(β)| β < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied toF(1) makes sense (it is the smallest ordinal not in the singleton set {F(0)} = {0}), and so on (the and so on is exactlytransfinite induction). It turns out that this example is not very exciting, since provably F(α) = α for all ordinals α,which can be shown, precisely, by transfinite induction.
17.4.3 Successor and limit ordinals
Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is nolargest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successorordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α isα ∪ {α} since its elements are those of α and α itself.[3]
A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinalis indeed the limit in a topological sense of all smaller ordinals (under the order topology).When ⟨αι|ι < γ⟩ is an ordinal-indexed sequence, indexed by a limit γ and the sequence is increasing, i.e. αι < αρ
whenever ι < ρ,its limit is defined the least upper bound of the set {αι|ι < γ},that is, the smallest ordinal (it alwaysexists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexedby itself). Put more directly, it is the supremum of the set of smaller ordinals.Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:
There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξsuch that ζ < ξ < α.
So in the following sequence:
0, 1, 2, ... , ω, ω+1
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (naturalnumber) larger than it, but still less than ω.Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction isimportant, because many definitions by transfinite induction rely upon it. Very often, when defining a function F bytransfinite induction on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinalsδ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as previously explained,or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is thesuccessor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) arecalled continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their secondargument.
17.4.4 Indexing classes of ordinals
Any well-ordered set is similar (order-isomorphic) to a unique ordinal number α , or, in other words, that its elementscan be indexed in increasing fashion by the ordinals less than α . This applies, in particular, to any set of ordinals:
94 CHAPTER 17. ORDINAL NUMBER
any set of ordinals is naturally indexed by the ordinals less than some α . The same holds, with a slight modification,for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any classof ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it inclass-bijection with the class of all ordinals). So the γ -th element in the class (with the convention that the “0-th”is the smallest, the “1-th” is the next smallest, and so on) can be freely spoken of. Formally, the definition is bytransfinite induction: the γ -th element of the class is defined (provided it has already been defined for all β < γ ), asthe smallest element greater than the β -th element for all β < γ .This could be applied, for example, to the class of limit ordinals: the γ -th ordinal, which is either a limit or zero isω · γ (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consider additivelyindecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the γ -thadditively indecomposable ordinal is indexed as ωγ . The technique of indexing classes of ordinals is often useful inthe context of fixed points: for example, the γ -th ordinal α such that ωα = α is written εγ . These are called the"epsilon numbers".
17.4.5 Closed unbounded sets and classes
A classC of ordinals is said to be unbounded, or cofinal, when given any ordinal α , there is a β inC such thatα < β(then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequenceof ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous inthe sense that, for δ a limit ordinal, F (δ) (the δ -th ordinal in the class) is the limit of all F (γ) for γ < δ ; this isalso the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on properclasses, one can demand that the intersection of the class with any given ordinal is closed for the order topology onthat ordinal, this is again equivalent).Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. Forexample, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limitordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the termi-nology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε· ordinals, orthe class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed,and any finite set of ordinals is closed but not unbounded.A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closedunbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are notclosed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals withcountable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersectionof a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may beempty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinalsbelow a given ordinal α : A subset of a limit ordinal α is said to be unbounded (or cofinal) under α provided anyordinal less than α is less than some ordinal in the set. More generally, we can call a subset of any ordinal α cofinalin α provided every ordinal less than α is less than or equal to some ordinal in the set. The subset is said to be closedunder α provided it is closed for the order topology in α , i.e. a limit of ordinals in the set is either in the set or equalto α itself.
17.5 Arithmetic of ordinals
Main article: Ordinal arithmetic
There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be de-fined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operationor by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called“natural” arithmetical operations retain commutativity at the expense of continuity.
17.6 Ordinals and cardinals
17.7. SOME “LARGE” COUNTABLE ORDINALS 95
17.6.1 Initial ordinal of a cardinal
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-orderedset having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as itscardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infiniteordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e.that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initialordinal, and we say that the initial ordinal is a cardinal.Cantor used the cardinality to partition ordinals into classes. He referred to the natural numbers as the first numberclass, the ordinals with cardinality ℵ0 (the countably infinite ordinals) as the second number class and generally,the ordinals with cardinality ℵn−2 as the n-th number class.[5]
The α-th infinite initial ordinal is written ωα . Its cardinality is written ℵα . For example, the cardinality of ω0 = ωis ℵ0 , which is also the cardinality of ω2 or ε0 (all are countable ordinals). So (assuming the axiom of choice) weidentify ω with ℵ0 , except that the notation ℵ0 is used when writing cardinals, and ω when writing ordinals (this isimportant since, for example, ℵ2
0 = ℵ0 whereas ω2 > ω ). Also, ω1 is the smallest uncountable ordinal (to see thatit exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-orderingdefines a countable ordinal, and ω1 is the order type of that set), ω2 is the smallest ordinal whose cardinality is greaterthan ℵ1 , and so on, and ωω is the limit of the ωn for natural numbers n (any limit of cardinals is a cardinal, so thislimit is indeed the first cardinal after all the ωn ).See also Von Neumann cardinal assignment.
17.6.2 Cofinality
The cofinality of an ordinal α is the smallest ordinal δ that is the order type of a cofinal subset of α . Notice that anumber of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any otherwell-ordered set is the cofinality of the order type of that set.Thus for a limit ordinal, there exists a δ -indexed strictly increasing sequence with limit α . For example, the cofinalityof ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally,any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ωω oran uncountable cofinality.The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at leastω .An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinalsis a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom ofChoice, then ωα+1 is regular for each α. In this case, the ordinals 0, 1, ω , ω1 , and ω2 are regular, whereas 2, 3, ωω
, and ωω·₂ are initial ordinals that are not regular.The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinalityof α. So the cofinality operation is idempotent.
17.7 Some “large” countable ordinals
For more details on this topic, see Large countable ordinal.
We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equationωα = α , so it is the limit of the sequence 0, 1, ω , ωω , ωωω , etc. Many ordinals can be defined in such a manneras fixed points of certain ordinal functions (the ι -th ordinal such that ωα = α is called ει , then we could go ontrying to find the ι -th ordinal such that εα = α , “and so on”, but all the subtlety lies in the “and so on”). We cantry to do this systematically, but no matter what system is used to define and construct ordinals, there is always anordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limitsa system of construction in this manner is the Church–Kleene ordinal, ωCK
1 (despite the ω1 in the name, this ordinalis countable), which is the smallest ordinal that cannot in any way be represented by a computable function (this canbe made rigorous, of course). Considerably large ordinals can be defined below ωCK
1 , however, which measure the
96 CHAPTER 17. ORDINAL NUMBER
“proof-theoretic strength” of certain formal systems (for example, ε0 measures the strength of Peano arithmetic).Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.
17.8 Topology and ordinals
For more details on this topic, see Order topology.
Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. See theTopology and ordinals section of the “Order topology” article.
17.9 Downward closed sets of ordinals
A set is downward closed if anything less than an element of the set is also in the set. If a set of ordinals is downwardclosed, then that set is an ordinal—the least ordinal not in the set.Examples:
• The set of ordinals less than 3 is 3 = { 0, 1, 2 }, the smallest ordinal not less than 3.
• The set of finite ordinals is infinite, the smallest infinite ordinal: ω.
• The set of countable ordinals is uncountable, the smallest uncountable ordinal: ω1.
17.10 See also
• Counting
• Ordinal space
17.11 Notes[1] Thorough introductions are given by Levy (1979) and Jech (2003).
[2] Hallett, Michael (1979), “Towards a theory of mathematical research programmes. I”, The British Journal for the Philosophyof Science 30 (1): 1–25, doi:10.1093/bjps/30.1.1, MR 532548. See the footnote on p. 12.
[3] von Neumann 1923
[4] Levy (1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the1920s.
[5] Dauben (1990:97)
17.12 References
• Cantor, G., (1897), Beitrage zur Begrundung der transfinitenMengenlehre. II (tr.: Contributions to the Foundingof the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-246 English translation.
• Conway, J. H. and Guy, R. K. “Cantor’s Ordinal Numbers.” In The Book of Numbers. New York: Springer-Verlag, pp. 266–267 and 274, 1996.
• Dauben, Joseph Warren, (1990), Georg Cantor: his mathematics and philosophy of the infinite. Chapter 5:The Mathematics of Cantor’s Grundlagen. ISBN 0-691-02447-2
17.13. EXTERNAL LINKS 97
• Hamilton, A. G. (1982), Numbers, Sets, and Axioms : the Apparatus of Mathematics, New York: CambridgeUniversity Press, ISBN 0-521-24509-5 See Ch. 6, “Ordinal and cardinal numbers”
• Kanamori, A., Set Theory from Cantor to Cohen, to appear in: Andrew Irvine and John H. Woods (editors),The Handbook of the Philosophy of Science, volume 4, Mathematics, Cambridge University Press.
• Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag Reprinted 2002, Dover. ISBN 0-486-42079-5
• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag
• Sierpiński, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Państwowe WydawnictwoNaukowe. Also defines ordinal operations in terms of the Cantor Normal Form.
• Suppes, P. (1960), Axiomatic Set Theory, D.Van Nostrand Company Inc., ISBN 0-486-61630-4
• von Neumann, Johann (1923), “Zur Einführung der trasfiniten Zahlen”, Acta litterarum ac scientiarum RagiaeUniversitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum 1: 199–208
• von Neumann, John (January 2002) [1923], “On the introduction of transfinite numbers”, in Jean van Hei-jenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard UniversityPress, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.
17.13 External links• Hazewinkel, Michiel, ed. (2001), “Ordinal number”, Encyclopedia of Mathematics, Springer, ISBN 978-1-
55608-010-4
• Weisstein, Eric W., “Ordinal Number”, MathWorld.
• Ordinals at ProvenMath
• Beitraege zur Begruendung der transfiniten Mengenlehre Cantor’s original paper published in MathematischeAnnalen 49(2), 1897
• Ordinal calculator GPL'd free software for computing with ordinals and ordinal notations
• Chapter 4 of Don Monk’s lecture notes on set theory is an introduction to ordinals.
Chapter 18
Partially ordered set
{x,y,z}
{y,z}{x,z}{x,y}
{y} {z}{x}
Ø
The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.
In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]
A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.
98
18.1. FORMAL DEFINITION 99
18.1 Formal definition
A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive,i.e., which satisfies for all a, b, and c in P:
• a ≤ a (reflexivity);
• if a ≤ b and b ≤ a then a = b (antisymmetry);
• if a ≤ b and b ≤ c then a ≤ c (transitivity).
In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as “ordered sets”, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they areincomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-rightfigure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no thirdelement c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. Amore concise definition will be given below using the strict order corresponding to "≤". For example, {x} is coveredby {x,z} in the top-right figure, but not by {x,y,z}.
18.2 Examples
Standard examples of posets arising in mathematics include:
• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).
• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, theset of sequences ordered by subsequence, and the set of strings ordered by substring.
• The set of natural numbers equipped with the relation of divisibility.
• The vertex set of a directed acyclic graph ordered by reachability.
• The set of subspaces of a vector space ordered by inclusion.
• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕif and only if a ≤ b for all n in ℕ, i.e. a componentwise order.
• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ gif and only if f(x) ≤ g(x) for all x in X.
• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...
18.3 Extrema
There are several notions of “greatest” and “least” element in a poset P, notably:
100 CHAPTER 18. PARTIALLY ORDERED SET
• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g.An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest orleast element.
• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a <m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be morethan one maximal element, and similarly for least elements and minimal elements.
• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for eachelement a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is alower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, anda least element is a lower bound of P.
For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements;on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, whichis a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not evenhave any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If thenumber 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting posetdoes not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound(though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in theposet); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.
18.4 Orders on the Cartesian product of partially ordered sets
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesianproduct of two partially ordered sets are (see figures):
• the lexicographical order: (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);
• the product order: (a,b) ≤ (c,d) if a ≤ c and b ≤ d;
• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d)or (a = c and b = d).
All three can similarly be defined for the Cartesian product of more than two sets.Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.See also orders on the Cartesian product of totally ordered sets.
18.5 Sums of partially ordered sets
Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of theunderlying sets X and Y by the order a ≤Z b if and only if:
• a, b ∈ X with a ≤X b, or
• a, b ∈ Y with a ≤Y b, or
• a ∈ X and b ∈ Y.
If two posets are well-ordered, then so is their ordinal sum.[5]
18.6. STRICT AND NON-STRICT PARTIAL ORDERS 101
18.6 Strict and non-strict partial orders
In some contexts, the partial order defined above is called a non-strict (or reflexive, orweak) partial order. In thesecontexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric,i.e. which satisfies for all a, b, and c in P:
• not a < a (irreflexivity),
• if a < b and b < c then a < c (transitivity), and
• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).
There is a 1-to-1 correspondence between all non-strict and strict partial orders.If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:
a < b if a ≤ b and a ≠ b
Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closuregiven by:
a ≤ b if a < b or a = b.
This is the reason for using the notation "≤".Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b forany c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): everystrict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.
18.7 Inverse and order dual
The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partialorder relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of apartially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > isto ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to eachother: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is onethat rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Thenatural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitudewhereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; wecould for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is notordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitudeyields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolutemagnitude but are not equal, violating antisymmetry.
18.8 Mappings between partially ordered sets
Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone,or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S→ T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f:S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving andorder-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective,since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that Scan be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the
102 CHAPTER 18. PARTIALLY ORDERED SET
partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams(cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘gyields the identity function on S and T, respectively, then S and T are order-isomorphic. [7]
For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set ofnatural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Itis order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neitherinjective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Takinginstead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number tothe set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and itsisomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to awide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".
18.9 Number of partial orders
Partially ordered set of set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.
Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).
18.10 Linear extension
A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and yof X, whenever x ≤ y, it is also the case that x ≤* y. A linear extension is an extension that is also a linear (i.e., total)order. Every partial order can be extended to a total order (order-extension principle).[8]
In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability ordersof directed acyclic graphs) are called topological sorting.
18.11. IN CATEGORY THEORY 103
18.11 In category theory
Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element.More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets areequivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initialobject, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset.Finally, every subcategory of a poset is isomorphism-closed.
18.12 Partial orders in topological spaces
Main article: Partially ordered space
If P is a partially ordered set that has also been given the structure of a topological space, then it is customary toassume that {(a, b) : a ≤ b} is a closed subset of the topological product space P ×P . Under this assumption partialorder relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.[9]
18.13 Interval
For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains atleast the elements a and b.Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a< x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers isempty since there are no integers i such that 1 < i < 2.Sometimes the definitions are extended to allow a > b, in which case the interval is empty.The half-open intervals [a,b) and (a,b] are defined similarly.A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order-ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = {a,b}.This concept of an interval in a partial order should not be confused with the particular class of partial orders knownas the interval orders.
18.14 See also
• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets
• causal set
• comparability graph
• complete partial order
• directed set
• graded poset
• incidence algebra
• lattice
• locally finite poset
• Möbius function on posets
• ordered group
104 CHAPTER 18. PARTIALLY ORDERED SET
• poset topology, a kind of topological space that can be defined from any poset
• Scott continuity - continuity of a function between two partial orders.
• semilattice
• semiorder
• series-parallel partial order
• stochastic dominance
• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive.
• Zorn’s lemma
18.15 Notes[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.
p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hassediagram...
[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory,Partial Orders, Combinatorics. Springer. ISBN 9781848002012.
[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895
[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18
[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.
[6] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.
[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). NewYork: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.
[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.
[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1):144–161. doi:10.1090/S0002-9939-1954-0063016-5
18.16 References• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American Mathematical
Society 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7.
• Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.
• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam-bridge University Press. ISBN 0-521-66351-2.
18.17 External links• A001035: Number of posets with n labeled elements in the OEIS
• A000112: Number of posets with n unlabeled elements in the OEIS
Chapter 19
Set theory
This article is about the branch of mathematics. For musical set theory, see Set theory (music).Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Al-
A BA∩B
A Venn diagram illustrating the intersection of two sets.
though any type of object can be collected into a set, set theory is applied most often to objects that are relevant tomathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discoveryof paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which theZermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics inits own right, with an active research community. Contemporary research into set theory includes a diverse collectionof topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
105
106 CHAPTER 19. SET THEORY
19.1 History
Georg Cantor
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, how-
19.2. BASIC CONCEPTS AND NOTATION 107
ever, was founded by a single paper in 1874 by Georg Cantor: “On a Characteristic Property of All Real AlgebraicNumbers”.[1][2]
Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathemati-cians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of BernardBolzano in the first half of the 19th century.[3] Modern understanding of infinity began in 1867–71, with Cantor’swork on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor’s thinking andculminated in Cantor’s 1874 paper.Cantor’s work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supportedCantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theoryeventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence amongsets, his proof that there are more real numbers than integers, and the “infinity of infinities” ("Cantor’s paradise")resulting from the power set operation. This utility of set theory led to the article “Mengenlehre” contributed in 1898by Arthur Schoenflies to Klein’s encyclopedia.The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gaverise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independentlyfound the simplest and best known paradox, now called Russell’s paradox: consider “the set of all sets that are notmembers of themselves”, which leads to a contradiction since it must be a member of itself, and not a member ofitself. In 1899 Cantor had himself posed the question “What is the cardinal number of the set of all sets?", andobtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in hisThe Principles of Mathematics.In 1906 English readers were treated to Theory of Sets of Points[4] by William Henry Young and his wife GraceChisholm Young, published by Cambridge University Press.The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work ofZermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonlyused set of axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematicalutility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonlyused as a foundational system, although in some areas category theory is thought to be a preferred foundation.
19.2 Basic concepts and notation
Main articles: Set (mathematics) and Algebra of sets
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element)of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of setA are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , andso is {2} but {1,4} is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes torule this out, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B,but B is not a subset of A. Note also that 1 and 2 and 3 are members (elements) of set {1,2,3} , but are not subsets,and the subsets in turn are not as such members of the set.Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:
• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. Theunion of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B.The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
• Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The setdifference {1,2,3} \ {2,3,4} is {1} , while, conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A isa subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice ofU is clear from the context, the notation Ac is sometimes used instead of U \ A, particularly if U is a universalset as in the study of Venn diagrams.
108 CHAPTER 19. SET THEORY
• Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all objects that are a member ofexactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3}and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection,(A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).
• Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b)where a is a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1,red), (1, white), (2, red), (2, white)}.
• Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1,2} is { {}, {1}, {2}, {1,2} } .
Some basic sets of central importance are the empty set (the unique set containing no elements), the set of naturalnumbers, and the set of real numbers.
19.3 Some ontology
Main article: von Neumann universeA set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set
{{}} containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention tothe von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize thepure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentiallyall mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into acumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in thishierarchy is assigned (by transfinite recursion) an ordinal number α, known as its rank. The rank of a pure set Xis defined to be the least upper bound of all successors of ranks of members of X. For example, the empty set isassigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set Vαis defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.
19.4 Axiomatic set theory
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venndiagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfyingany particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which areRussell’s paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of suchparadoxes.[5]
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systemscome in two flavors, those whose ontology consists of:
• Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), whichincludes the axiom of choice. Fragments of ZFC include:
• Zermelo set theory, which replaces the axiom schema of replacement with that of separation;• General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;• Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the
axiom schemata of separation and replacement.
• Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strengthas ZFC for theorems about sets alone, and Morse-Kelley set theory and Tarski–Grothendieck set theory, bothof which are stronger than ZFC.
The above systems can be modified to allow urelements, objects that can be members of sets but that are not them-selves sets and do not have any members.The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulativehierarchy. NF and NFU include a “set of everything,” relative to which every set has a complement. In these systemsurelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.
19.5. APPLICATIONS 109
V0
V1
V2
V3
V4
V5
Vω
Vω+1
Vω+2
V2ω
V3ω
Vω*ω
...
...
...
An initial segment of the von Neumann hierarchy.
Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead ofclassical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These includerough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relationis not simply True or False. The Boolean-valued models of ZFC are a related subject.An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.
19.5 Applications
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematicalstructures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (ax-iomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematicalrelations can be described in set theory.Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volumeof Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using anaptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For
110 CHAPTER 19. SET THEORY
example, properties of the natural and real numbers can be derived within set theory, as each number system can beidentified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is like-wise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from therelevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems fromset theory have been formally verified, however, because such formal derivations are often much longer than thenatural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer‐verified derivations of more than 12,000 theorems starting from ZFC set theory, first order logicand propositional logic.
19.6 Areas of study
Set theory is a major area of research in mathematics, with many interrelated subfields.
19.6.1 Combinatorial set theory
Main article: Infinitary combinatorics
Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study ofcardinal arithmetic and the study of extensions of Ramsey’s theorem such as the Erdős–Rado theorem.
19.6.2 Descriptive set theory
Main article: Descriptive set theory
Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It beginswith the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as theprojective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but provingthese properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightfacepointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive settheory have effective versions; in some cases, new results are obtained by proving the effective version first and thenextending (“relativizing”) it to make it more broadly applicable.A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations.This has important applications to the study of invariants in many fields of mathematics.
19.6.3 Fuzzy set theory
Main article: Fuzzy set theory
In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not.In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set,a number between 0 and 1. For example, the degree of membership of a person in the set of “tall people” is moreflexible than a simple yes or no answer and can be a real number such as 0.75.
19.6.4 Inner model theory
Main article: Inner model theory
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfiesall the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that
19.6. AREAS OF STUDY 111
the study of inner models is of interest is that it can be used to prove consistency results. For example, it can beshown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, theinner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and theaxiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together withthese two principles is consistent.The study of inner models is common in the study of determinacy and large cardinals, especially when consideringaxioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theorysatisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, theexistence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (andthus not satisfying the axiom of choice).[6]
19.6.5 Large cardinals
Main article: Large cardinal property
A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessiblecardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be verylarge, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.
19.6.6 Determinacy
Main article: Determinacy
Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect informationare determined from the start in the sense that one player must have a winning strategy. The existence of thesestrategies has important consequences in descriptive set theory, as the assumption that a broader class of games isdetermined often implies that a broader class of sets will have a topological property. The axiom of determinacy(AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets ofthe real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to provethat the Wadge degrees have an elegant structure.
19.6.7 Forcing
Main article: Forcing (mathematics)
Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesisfails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additionalsets in order to create a larger model with properties determined (i.e. “forced”) by the construction and the originalmodel. For example, Cohen’s construction adjoins additional subsets of the natural numbers without changing any ofthe cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency byfinitistic methods, the other method being Boolean-valued models.
19.6.8 Cardinal invariants
Main article: Cardinal invariant
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studiedinvariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These areinvariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant.Many cardinal invariants have been studied, and the relationships between them are often complex and related toaxioms of set theory.
112 CHAPTER 19. SET THEORY
19.6.9 Set-theoretic topology
Main article: Set-theoretic topology
Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advancedmethods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axiomsfor their proof. A famous problem is the normal Moore space question, a question in general topology that was thesubject of intense research. The answer to the normal Moore space question was eventually proved to be independentof ZFC.
19.7 Objections to set theory as a foundation for mathematics
From set theory’s inception, some mathematicians have objected to it as a foundation for mathematics. The mostcommon objection to set theory, one Kronecker voiced in set theory’s earliest years, starts from the constructivistview that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets,both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computableeven in principle.Ludwig Wittgenstein condemned set theory. He wrote that “set theory is wrong”, since it builds on the “nonsense” offictitious symbolism, has “pernicious idioms”, and that it is nonsensical to talk about “all numbers”.[7] Wittgenstein’sviews about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and investigatedby Crispin Wright, among others.Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory caninterpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.[8]
Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well asproviding the framework for pointless topology and Stone spaces.[9]
An active area of research is the univalent foundations arising from homotopy type theory. Here, sets may be definedas certain kinds of types, with universal properties of sets arising from higher inductive types. Principles such as theaxiom of choice and the law of the excluded middle appear in a spectrum of different forms, some of which can beproven, others which correspond to the classical notions; this allows for a detailed discussion of the effect of theseaxioms on mathematics.[10][11]
19.8 See also
• Glossary of set theory
• Category theory
• List of set theory topics
• Relational model – borrows from set theory
19.9 Notes
[1] Cantor, Georg (1874), “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen”, J. Reine Angew. Math.77: 258–262, doi:10.1515/crll.1874.77.258
[2] Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6
[3] Bolzano, Bernard (1975), Berg, Jan, ed., Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre,Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich From-mann Verlag, p. 152, ISBN 3-7728-0466-7
[4] William Henry Young & Grace Chisholm Young (1906) Theory of Sets of Points, link from Internet Archive
19.10. FURTHER READING 113
[5] In his 1925, John von Neumann observed that “set theory in its first, “naive” version, due to Cantor, led to contradictions.These are the well-known antinomies of the set of all sets that do not contain themselves (Russell), of the set of all transfinteordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard).” He goes on to observe that two“tendencies” were attempting to “rehabilitate” set theory. Of the first effort, exemplified by Bertrand Russell, Julius König,Hermann Weyl and L. E. J. Brouwer, von Neumann called the “overall effect of their activity . . . devastating”. Withregards to the axiomatic method employed by second group composed of Zermelo, Abraham Fraenkel and Arthur MoritzSchoenflies, von Neumann worried that “We see only that the known modes of inference leading to the antinomies fail, butwho knows where there are not others?" and he set to the task, “in the spirit of the second group”, to “produce, by meansof a finite number of purely formal operations . . . all the sets that we want to see formed” but not allow for the antinomies.(All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976), “From Frege to Gödel:A Source Book in Mathematical Logic, 1979–1931”, Harvard University Press, Cambridge MA, ISBN 0-674-32449-8(pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann’s1925.
[6] Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York:Springer-Verlag, p. 642, ISBN 978-3-540-44085-7, Zbl 1007.03002
[7] Wittgenstein, Ludwig (1975). Philosophical Remarks, §129, §174. Oxford: Basil Blackwell. ISBN 0631191305.
[8] Ferro, A.; Omodeo, E. G.; Schwartz, J. T. (1980), “Decision procedures for elementary sublanguages of set theory. I.Multi-level syllogistic and some extensions”, Comm. Pure Appl. Math. 33 (5): 599–608, doi:10.1002/cpa.3160330503
[9] Saunders Mac Lane and Ieke Moerdijk (1992) Sheaves in Geometry and Logic: a First Introduction to Topos Theory.Springer Verlag.
[10] homotopy type theory in nLab
[11] Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Ad-vanced Study.
19.10 Further reading• Devlin, Keith, 1993. The Joy of Sets (2nd ed.). Springer Verlag, ISBN 0-387-94094-4
• Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics.Basel, Birkhäuser. ISBN 978-3-7643-8349-7
• Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0-87150-154-6
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. North-Holland, ISBN 0-444-85401-0.
• Potter, Michael, 2004. Set Theory and Its Philosophy: A Critical Introduction. Oxford University Press.
• Tiles, Mary, 2004 (1989). The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise.Dover Publications. ISBN 978-0-486-43520-6
19.11 External links• Foreman, Matthew, Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys
some aspect of contemporary research in set theory. Does not cover established elementary set theory, onwhich see Devlin (1993).
• Hazewinkel, Michiel, ed. (2001), “Axiomatic set theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Hazewinkel, Michiel, ed. (2001), “Set theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Jech, Thomas (2002). "Set Theory", Stanford Encyclopedia of Philosophy.
• Schoenflies, Arthur (1898). Mengenlehre in Klein’s encyclopedia.
• Online books, and library resources in your library and in other libraries about set theory
Chapter 20
Surjective function
“Onto” redirects here. For other uses, see wikt:onto.In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y
X1
2
3
4
YD
B
C
A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value off(x) for at least one point x in the domain.
114
20.1. DEFINITION 115
has a corresponding element x in X such that f(x) = y. The function f may map more than one element of X to thesame element of Y.The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,[1] the pseudonymfor a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition ofmodern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the factthat the image of the domain of a surjective function completely covers the function’s codomain.
20.1 Definition
For more details on notation, see Function (mathematics) § Notation.
A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domainX and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y . Surjections aresometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ rightwards two headed arrow),[2] as in f : X ↠Y.Symbolically,
If f : X → Y , then f is said to be surjective if
∀y ∈ Y, ∃x ∈ X, f(x) = y
20.2 Examples
For any set X, the identity function idX on X is surjective.The function f : Z → {0,1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1)is surjective.The function f : R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number ywe have an x such that f(x) = y: an appropriate x is (y − 1)/2.The function f : R → R defined by f(x) = x3 − 3x is surjective, because the pre-image of any real number y is thesolution set of the cubic polynomial equation x3 − 3x − y = 0 and every cubic polynomial with real coefficients has atleast one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)The function g : R → R defined by g(x) = x2 is not surjective, because there is no real number x such that x2 = −1.However, the function g : R → R0
+ defined by g(x) = x2 (with restricted codomain) is surjective because for every yin the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y.The natural logarithm function ln : (0,+∞) → R is a surjective and even bijective mapping from the set of positivereal numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the setof positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponentialis not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as amap from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertiblematrices. Under this definition the matrix exponential is surjective for complex matrices, although still not surjectivefor real matrices.The projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty.In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function.
20.3 Properties
A function is bijective if and only if it is both surjective and injective.
116 CHAPTER 20. SURJECTIVE FUNCTION
X Y
f(x)
f : X → Y
x
A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. Thisfunction is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process:First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue.The function f is surjective only if there are no blue points.
If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, butrather a relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of thegraph of the function alone.
20.3.1 Surjections as right invertible functions
The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can beundone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identityfunction on the domain Y of g. The function g need not be a complete inverse of f because the composition in theother order, g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g,but cannot necessarily be reversed by it.Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has aright inverse is equivalent to the axiom of choice.If f : X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimagef −1(B).For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f suchthat f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters that f “reverses” g.
• Another surjective function. (This one happens to be a bijection)
• A non-surjective function. (This one happens to be an injection)
• Surjective composition: the first function need not be surjective.
20.3. PROPERTIES 117
x
y
x
y
1 X 1 Y : f 2 X 2 Y : f
2 X x 1 X x
x f y
f im Y y
Y y
f im
Y X : f x f y
X x
Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain offunction, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. Theremay be a number of domain elements which map to the same range element. That is, every y in Y is mapped from an element x inX, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domainsX1 and X2 are shown.
x
y
X x 0
Y X : f x f y
X x
Y y
f im
Y y 0
x
y
X x 1
Y y 2
Y y 1
X x 2
X x 3
Y y 3
X x
1 X 1 Y : f 2 X 2 Y : f
Y y
f im
2 X x 1 X x
x f y
Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do havea value x in X such that y = f(x), some parts are not. Left: There is y0 in Y, but there is no x0 in X such that y0 = f(x0). Right:There are y1, y2 and y3 in Y, but there are no x1, x2, and x3 in X such that y1 = f(x1), y2 = f(x2), and y3 = f(x3).
20.3.2 Surjections as epimorphisms
A function f : X → Y is surjective if and only if it is right-cancellative:[3] given any functions g,h : Y → Z, whenever go f = h o f, then g = h. This property is formulated in terms of functions and their composition and can be generalizedto the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms arecalled epimorphisms. Specifically, surjective functions are precisely the epimorphisms in the category of sets. The
118 CHAPTER 20. SURJECTIVE FUNCTION
prefix epi is derived from the Greek preposition ἐπί meaning over, above, on.Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of amorphism f is called a section of f. A morphism with a right inverse is called a split epimorphism.
20.3.3 Surjections as binary relations
Any function with domain X and codomain Y can be seen as a left-total and right-unique binary relation between Xand Y by identifying it with its function graph. A surjective function with domain X and codomain Y is then a binaryrelation between X and Y that is right-unique and both left-total and right-total.
20.3.4 Cardinality of the domain of a surjection
The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (Theproof appeals to the axiom of choice to show that a function g : Y → X satisfying f(g(y)) = y for all y in Y exists. gis easily seen to be injective, thus the formal definition of |Y | ≤ |X| is satisfied.)Specifically, if both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only iff is injective.
20.3.5 Composition and decomposition
The composite of surjective functions is always surjective: If f and g are both surjective, and the codomain of gis equal to the domain of f, then f o g is surjective. Conversely, if f o g is surjective, then f is surjective (but g,the function applied first, need not be). These properties generalize from surjections in the category of sets to anyepimorphisms in any category.Any function can be decomposed into a surjection and an injection: For any function h : X → Z there exist a surjectionf : X → Y and an injection g : Y → Z such that h = g o f. To see this, define Y to be the sets h −1(z) where z is inZ. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carrieseach element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, andg is injective by definition.
20.3.6 Induced surjection and induced bijection
Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijectiondefined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, everysurjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalenceclasses of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set ofall preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class[x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).
20.4 See also• Bijection, injection and surjection
• Cover (algebra)
• Covering map
• Enumeration
• Fiber bundle
• Index set
• Section (category theory)
20.5. NOTES 119
20.5 Notes[1] “Injection, Surjection and Bijection”, Earliest Uses of Some of the Words of Mathematics, Tripod |first1= missing |last1= in
Authors list (help).
[2] “Arrows – Unicode” (PDF). Retrieved 2013-05-11.
[3] Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic (Revised ed.). Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-25.
20.6 References• Bourbaki, Nicolas (2004) [1968]. Theory of Sets. Springer. ISBN 978-3-540-22525-6.
Chapter 21
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, alongwith a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. Thedefinition of a topological space relies only upon set theory and is the most general notion of a mathematical spacethat allows for the definition of concepts such as continuity, connectedness, and convergence.[1] Other spaces, such asmanifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being sogeneral, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.The branch of mathematics that studies topological spaces in their own right is called point-set topology or generaltopology.
21.1 Definition
Main article: Characterizations of the category of topological spaces
The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure.Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, isthat in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: Avariety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.
21.1.1 Neighbourhoods definition
This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though theycan be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X anon-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect toN (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below[2] aresatisfied; and then X with N is called a topological space.
1. If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of itsneighbourhoods.
2. If N is a subset of X and contains a neighbourhood of x, then N is a neighbourhood of x. I.e., every supersetof a neighbourhood of a point x in X is again a neighbourhood of x.
3. The intersection of two neighbourhoods of x is a neighbourhood of x.
4. Any neighbourhood N of x contains a neighbourhood M of x such that N is a neighbourhood of each point ofM.
The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in thestructure of the theory, that of linking together the neighbourhoods of different points of X.
120
21.1. DEFINITION 121
A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to bea neighbourhood of a real number x if there is an open interval containing x and contained in N.Given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. It is aremarkable fact that the open sets then satisfy the elegant axioms given below, and that, given these axioms, we canrecover the neighbourhoods satisfying the above axioms by defining N to be a neighbourhood of x if N contains anopen set U such that x ∈ U.[3]
21.1.2 Open sets definition
1 2 3 1 2 3
1 2 3 1 2 3
1 2 3 1 2 3
Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology becausethe union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and{2,3} [i.e. {2}], is missing.
A topological space is then a set X together with a collection of subsets of X, called open sets and satisfying thefollowing axioms:[4]
1. The empty set and X itself are open.
2. Any union of open sets is open.
3. The intersection of any finite number of open sets is open.
The collection τ of open sets is then also called a topology on X, or, if more precision is needed, an open set topology.The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of X may be neitherclosed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.
122 CHAPTER 21. TOPOLOGICAL SPACE
Examples
1. X = {1, 2, 3, 4} and collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forma topology, the trivial topology (indiscrete topology).
2. X = {1, 2, 3, 4} and collection τ = {{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}} of six subsets of X formanother topology.
3. X = {1, 2, 3, 4} and collection τ = P(X) (the power set of X) form a third topology, the discrete topology.
4. X =Z, the set of integers, and collection τ equal to all finite subsets of the integers plusZ itself is not a topology,because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is notin τ .
21.1.3 Closed sets definition
Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets:
1. The empty set and X are closed.
2. The intersection of any collection of closed sets is also closed.
3. The union of any pair of closed sets is also closed.
Using these axioms, another way to define a topological space is as a set X together with a collection τ of closedsubsets of X. Thus the sets in the topology τ are the closed sets, and their complements in X are the open sets.
21.1.4 Other definitions
There are many other equivalent ways to define a topological space: in other words, the concepts of neighbourhoodor of open respectively closed set can be reconstructed from other starting points and satisfy the correct axioms.Another way to define a topological space is by using the Kuratowski closure axioms, which define the closed sets asthe fixed points of an operator on the power set of X.A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X theset of its accumulation points is specified.
21.2 Comparison of topologies
Main article: Comparison of topologies
A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also ina topology τ2 and τ1 is a subset of τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. A proof that reliesonly on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies onlyon certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used inplace of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with littleagreement on the meaning, so one should always be sure of an author’s convention when reading.The collection of all topologies on a given fixed set X forms a complete lattice: if F = {τα| α in A} is a collectionof topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of alltopologies on X that contain every member of F.
21.3 Continuous functions
A function f : X→ Y between topological spaces is called continuous if for all x ∈ X and all neighbourhoods N of f(x)there is a neighbourhood M of x such that f(M) ⊆ N. This relates easily to the usual definition in analysis. Equivalently,
21.4. EXAMPLES OF TOPOLOGICAL SPACES 123
f is continuous if the inverse image of every open set is open.[5] This is an attempt to capture the intuition that thereare no “jumps” or “separations” in the function. A homeomorphism is a bijection that is continuous and whose inverseis also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From thestandpoint of topology, homeomorphic spaces are essentially identical.In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functionsas morphisms is one of the fundamental categories in category theory. The attempt to classify the objects of thiscategory (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homologytheory, and K-theory etc.
21.4 Examples of topological spaces
A given set may have many different topologies. If a set is given a different topology, it is viewed as a differenttopological space. Any set can be given the discrete topology in which every subset is open. The only convergentsequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence andnet in this topology converges to every point of the space. This example shows that in general topological spaces,limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limitpoints are unique.There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generatedby the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every openset is a union of some collection of sets from the base. In particular, this means that a set is open if there exists anopen interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be givena topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complexnumbers, and Cn have a standard topology in which the basic open sets are open balls.Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is thesame for all norms.Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying whena particular sequence of functions converges to the zero function.Any local field has a topology native to it, and this can be extended to vector spaces over that field.Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicialcomplex inherits a natural topology from Rn.The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, theclosed sets of the Zariski topology are the solution sets of systems of polynomial equations.A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices andedges.The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of com-putation and semantics.There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spacesare sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complementis finite. This is the smallest T1 topology on any infinite set.Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complementis countable. When the set is uncountable, this topology serves as a counterexample in many situations.The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b).This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in thistopology if and only if it converges from above in the Euclidean topology. This example shows that a set may havemany distinct topologies defined on it.If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, Γ) where a and b are elements of Γ.
124 CHAPTER 21. TOPOLOGICAL SPACE
21.5 Topological constructions
Every subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in finite products, a basis for the product topology consists of all products of open sets. Forinfinite products, there is the additional requirement that in a basic open set, all but finitely many of its projectionsare the entire space.A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. Inother words, the quotient topology is the finest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is defined on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, isgenerated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consistingof all subsets of the union of the Ui that have non-empty intersections with each Ui.
21.6 Classification of topological spaces
Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topologicalproperty is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not home-omorphic it is sufficient to find a topological property not shared by them. Examples of such properties includeconnectedness, compactness, and various separation axioms.See the article on topological properties for more details and examples.
21.7 Topological spaces with algebraic structure
For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuousfunctions. For any such structure that is not finite, we often have a natural topology compatible with the algebraicoperations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topologicalgroups, topological vector spaces, topological rings and local fields.
21.8 Topological spaces with order structure
• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).
• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if andonly if cl{x} ⊆ cl{y}.
21.9 Specializations and generalizations
The following spaces and algebras are either more specialized or more general than the topological spaces discussedabove.
• Proximity spaces provide a notion of closeness of two sets.
• Metric spaces embody a metric, a precise notion of distance between points.
• Uniform spaces axiomatize ordering the distance between distinct points.
• A topological space in which the points are functions is called a function space.
21.10. SEE ALSO 125
• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general settingfor studying completions.
• Convergence spaces capture some of the features of convergence of filters.
• Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object.Sites are a general setting for defining sheaves.
21.10 See also
• Space (mathematics)
• Kolmogorov space (T0)
• accessible/Fréchet space (T1)
• Hausdorff space (T2)
• Completely Hausdorff space and Urysohn space (T₂½)
• Regular space and regular Hausdorff space (T3)
• Tychonoff space and completely regular space (T₃½)
• Normal Hausdorff space (T4)
• Completely normal Hausdorff space (T5)
• Perfectly normal Hausdorff space (T6)
• Quasitopological space
• Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is acomplete Heyting algebra.
21.11 Notes[1] Schubert 1968, p. 13
[2] Brown 2006, section 2.1.
[3] Brown 2006, section 2.2.
[4] Armstrong 1983, definition 2.1.
[5] Armstrong 1983, theorem 2.6.
21.12 References
• Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate texts in mathematics. Springer. ISBN0-387-90839-0.
• Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17,1997). ISBN 0-387-97926-3.
• Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).
• Brown, Ronald, Topology and groupoids, Booksurge (2006) ISBN 1-4196-2722-8 (3rd edition of differentlytitled books) (order from amazon.com).
• Čech, Eduard; Point Sets, Academic Press (1969).
126 CHAPTER 21. TOPOLOGICAL SPACE
• Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5,1997). ISBN 0-387-94327-7.
• Lipschutz, Seymour; Schaum’s Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN0-07-037988-2.
• Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
• Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.
• Schubert, Horst (1968), Topology, Allyn and Bacon
• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.
• Vaidyanathaswamy, R. (1960). Set Topology. Chelsea Publishing Co. ISBN 0486404560.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
21.13 External links• Hazewinkel, Michiel, ed. (2001), “Topological space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-
55608-010-4
• Topological space at PlanetMath.org.
Chapter 22
Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denotedby infix ≤) on some set X which is transitive, antisymmetric, and total. A set paired with a total order is called a totallyordered set, a linearly ordered set, a simply ordered set, or a chain.If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:
If a ≤ b and b ≤ a then a = b (antisymmetry);If a ≤ b and b ≤ c then a ≤ c (transitivity);a ≤ b or b ≤ a (totality).
Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a.[1] A relation having the property of“totality” means that any pair of elements in the set of the relation are comparable under the relation. This also meansthat the set can be diagrammed as a line of elements, giving it the name linear.[2] Totality also implies reflexivity, i.e.,a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (Itrequires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extensionof that partial order.
22.1 Strict total order
For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict totalorder, which can equivalently be defined in two ways:
• a < b if and only if a ≤ b and a ≠ b
• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)
Properties:
• The relation is transitive: a < b and b < c implies a < c.• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.• The relation is a strict weak order, where the associated equivalence is equality.
We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤can equivalently be defined in two ways:
• a ≤ b if and only if a < b or a = b
• a ≤ b if and only if not b < a
Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whetherwe are talking about the non-strict or the strict total order.
127
128 CHAPTER 22. TOTAL ORDER
22.2 Examples
• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.
• Any subset of a totally ordered set, with the restriction of the order on the whole set.
• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).
• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on Xby setting x1 < x2 if and only if f(x1) < f(x2).
• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, isitself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as asubset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol tothe alphabet (and defining a space to be less than any letter).
• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hencealso the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique(to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A isthe smallest with a certain property if whenever B has the property, there is an order isomorphism from A to asubset of B):
• The natural numbers comprise the smallest totally ordered set with no upper bound.• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. The
definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there isa 'q' in the rational numbers such that 'a' < 'q' < 'b'.
• The real numbers comprise the smallest unbounded totally ordered set that is connected in the ordertopology (defined below).
• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers.
22.3 Further concepts
22.3.1 Chains
While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset ofsome partially ordered set. The latter definition has a crucial role in Zorn’s lemma.For example, consider the set of all subsets of the integers partially ordered by inclusion. Then the set { In : n is anatural number}, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally orderedunder inclusion: If n≤k, then In is a subset of Ik.
22.3.2 Lattice theory
One may define a totally ordered set as a particular kind of lattice, namely one in which we have
{a ∨ b, a ∧ b} = {a, b} for all a, b.
We then write a ≤ b if and only if a = a ∧ b . Hence a totally ordered set is a distributive lattice.
22.3. FURTHER CONCEPTS 129
22.3.3 Finite total orders
A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subsetthereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observingthat every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphicto an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements inducesa bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with ordertype ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
22.3.4 Category theory
Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being mapswhich respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.
22.3.5 Order topology
For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b},(a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, theorder topology.When more than one order is being used on a set one talks about the order topology induced by a particular order.For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology onN induced by < and the order topology on N induced by > (in this case they happen to be identical but will not ingeneral).The order topology induced by a total order may be shown to be hereditarily normal.
22.3.6 Completeness
A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upperbound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.There are a number of results relating properties of the order topology to the completeness of X:
• If the order topology on X is connected, X is complete.
• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is twopoints a and b in X with a < b such that no c satisfies a < c < b.)
• X is complete if and only if every bounded set that is closed in the order topology is compact.
A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervalsof real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line).There are order-preserving homeomorphisms between these examples.
22.3.7 Sums of orders
For any two disjoint total orders (A1,≤1) and (A2,≤2) , there is a natural order ≤+ on the set A1 ∪ A2 , which iscalled the sum of the two orders or sometimes just A1 +A2 :
For x, y ∈ A1 ∪A2 , x ≤+ y holds if and only if one of the following holds:
1. x, y ∈ A1 and x ≤1 y
2. x, y ∈ A2 and x ≤2 y
3. x ∈ A1 and y ∈ A2
130 CHAPTER 22. TOTAL ORDER
Intutitively, this means that the elements of the second set are added on top of the elements of the first set.More generally, if (I,≤) is a totally ordered index set, and for each i ∈ I the structure (Ai,≤i) is a linear order,where the sets Ai are pairwise disjoint, then the natural total order on
∪i Ai is defined by
For x, y ∈∪
i∈I Ai , x ≤ y holds if:1. Either there is some i ∈ I with x ≤i y
2. or there are some i < j in I with x ∈ Ai , y ∈ Aj
22.4 Orders on the Cartesian product of totally ordered sets
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product oftwo totally ordered sets are:
• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of
the corresponding strict total orders). This is also a partial order.
All three can similarly be defined for the Cartesian product of more than two sets.Applied to the vector space Rn, each of these make it an ordered vector space.See also examples of partially ordered sets.A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding totalpreorder on that subset.
22.5 Related structures
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.A group with a compatible total order is a totally ordered group.There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientationresults in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both dataresults in a separation relation.[3]
22.6 See also• Order theory• Well-order• Suslin’s problem• Countryman line
22.7 Notes[1] Nederpelt, Rob (2004). “Chapter 20.2: Ordered Sets. Orderings”. Logical Reasoning: A First Course. Texts in Computing
3 (3rd, Revised ed.). King’s College Publications. p. 325. ISBN 0-9543006-7-X.
[2] Nederpelt, Rob (2004). “Chapter 20.3: Ordered Sets. Linear orderings”. Logical Reasoning: A First Course. Texts inComputing 3 (3rd, Revisied ed.). King’s College Publications. p. 330. ISBN 0-9543006-7-X.
[3] Macpherson, H. Dugald (2011), “A survey of homogeneous structures” (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011
22.8. REFERENCES 131
22.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN
0-7167-0442-0
• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
Chapter 23
Transitive set
In set theory, a set A is transitive, if and only if
• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,• whenever x ∈ A, and x is not an urelement, then x is a subset of A.
Similarly, a class M is transitive if every element of M is a subset of M.
23.1 Examples
Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarilytransitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals).Any of the stages Vα and Lα leading to the construction of the von Neumann universe V and Gödel’s constructibleuniverse L are transitive sets. The universes L and V themselves are transitive classes.
23.2 Properties
A set X is transitive if and only if∪X ⊆ X , where
∪X is the union of all elements of X that are sets,
∪X = {y |
(∃x ∈ X)y ∈ x} . If X is transitive, then∪X is transitive. If X and Y are transitive, then X∪Y∪{X,Y} is transitive.
In general, if X is a class all of whose elements are transitive sets, then X ∪∪X is transitive.
A set X which does not contain urelements is transitive if and only if it is a subset of its own power set, X ⊂ P(X).The power set of a transitive set without urelements is transitive.
23.3 Transitive closure
The transitive closure of a set X is the smallest (with respect to inclusion) transitive set which contains X. Supposeone is given a set X, then the transitive closure of X is
∪{X,
∪X,
∪∪X,
∪∪∪X,
∪∪∪∪X, . . .}.
Note that this is the set of all of the objects related to X by the transitive closure of the membership relation.
23.4 Transitive models of set theory
Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models.The reason is that properties defined by bounded formulas are absolute for transitive classes.
132
23.5. SEE ALSO 133
A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system.Transitivity is an important factor in determining the absoluteness of formulas.In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity.[1]
23.5 See also• End extension
• Transitive relation
• Supertransitive class
23.6 References[1] Goldblatt (1998) p.161
• Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society StudentTexts 39, Cambridge: Cambridge University Press, ISBN 0-521-59441-3, Zbl 0938.03067
• Goldblatt, Robert (1998), Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Textsin Mathematics 188, New York, NY: Springer-Verlag, ISBN 0-387-98464-X, Zbl 0911.03032
• Jech, Thomas (2008) [originally published in 1973], The Axiom of Choice, Dover Publications, ISBN 0-486-46624-8, Zbl 0259.02051
23.7 External links• Weisstein, Eric W., “Transitive”, MathWorld.
• Weisstein, Eric W., “Transitive Closure”, MathWorld.
• Weisstein, Eric W., “Transitive Reduction”, MathWorld.
Chapter 24
Trigonometric series
A trigonometric series is a series of the form:
A0 +∞∑
n=1
(An cosnx+Bn sinnx).
It is called a Fourier series if the terms An and Bn have the form:
An =1
π
∫ 2π
0
f(x) cosnx dx (n = 0, 1, 2, 3 . . . )
Bn =1
π
∫ 2π
0
f(x) sinnx dx (n = 1, 2, 3, . . . )
where f is an integrable function.
24.1 The zeros of a trigonometric series
The uniqueness and the zeros of trigonometric series was an active area of research in the 19th century Europe. First,Georg Cantor proved that if a trigonometric series is convergent to a function f(x) on the interval [0, 2π] , which isidentically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series areall zero. But almost half a millennium back the Indian Mathematicians, notably from Kerala school of astronomyand mathematics like Madhava of Sangamagrama and Neelakanta Somayaji had already created the whole basis ofthe same theory. Due to the imperialism that occurred in India most of the information was hidden from the outsideworld.Later Cantor proved that even if the set S on which f is nonzero is infinite, but the derived set S' of S is finite, then thecoefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. Ifthere is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if thereis a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor’s work onthe uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts αin Sα .[1]
24.2 Zygmund’s book
Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many differ-ent aspects of these series, which we will not attempt to discuss here.The first edition was a single volume, publishedin 1935 (under the slightly different title “trigonometrical series”). The second edition of 1959 was greatly expanded,taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is sim-ilar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, inparticular Carleson’s theorem about almost everywhere pointwise convergence for square integrable functions.
134
24.3. REFERENCES 135
24.3 References[1] Cooke, Roger (1993), “Uniqueness of trigonometric series and descriptive set theory, 1870–1985”, Archive for History of
Exact Sciences 45 (4): 281, doi:10.1007/BF01886630.
24.3.1 Reviews of Trigonometric Series
• Kahane, Jean-Pierre (2004), “Book review: Trigonometric series, Vols. I, II”, Bulletin of the American Math-ematical Society 41: 377–390, doi:10.1090/s0273-0979-04-01013-4, ISSN 0002-9904
• Salem, Raphael (1960), “Book Review: Trigonometric series”, Bulletin of the American Mathematical Society66 (1): 6–12, doi:10.1090/S0002-9904-1960-10362-X, ISSN 0002-9904, MR 1566029
• Tamarkin, J. D. (1936), “Zygmund on Trigonometric Series”, Bull. Amer. Math. Soc. 42 (1): 11–13,doi:10.1090/s0002-9904-1936-06235-x
24.3.2 Publication history of Trigonometric Series
• Zygmund, Antoni (1935), Trigonometrical series., Monogr. Mat. 5, Warszawa, Lwow: Subwencji FunduszKultury Narodowej., Zbl 0011.01703
• Zygmund, Antoni (1952), Trigonometrical series, New York: Chelsea Publishing Co., MR 0076084
• Zygmund, Antoni (1955), Trigonometrical series, Dover Publications, New York, MR 0072976
• Zygmund, Antoni (1959), Trigonometric series Vols. I, II (2nd ed.), Cambridge University Press, MR 0107776
• Zygmund, Antoni (1968), Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and someadditions (2nd ed.), Cambridge University Press, MR 0236587
• Zygmund, Antoni (1977), Trigonometric series. Vol. I, II, Cambridge University Press, ISBN 978-0-521-07477-3, MR 0617944
• Zygmund, Antoni (1988), Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge Uni-versity Press, ISBN 978-0-521-35885-9, MR 933759
• Zygmund, Antoni (2002), Fefferman, Robert A., ed., Trigonometric series. Vol. I, II, Cambridge MathematicalLibrary (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498
Chapter 25
Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-orderedset U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely:
|U | = card(U) = inf{α ∈ ON | α =c U},
where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinalsis well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so everyset has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily foundto coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers.
25.1 Initial ordinal of a cardinal
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-orderedset having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as itscardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infiniteordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e.that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initialordinal, and we say that the initial ordinal is a cardinal.The α-th infinite initial ordinal is written ωα . Its cardinality is written ℵα (the α-th aleph number). For example,the cardinality of ω0 = ω is ℵ0, which is also the cardinality of ω2, ωω, and ε0 (all are countable ordinals). So(assuming the axiom of choice) we identify ωα with ℵα, except that the notation ℵα is used for writing cardinals, andωα for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals,for example ℵα2 = ℵα whereas ωα2 > ωα. Also, ω1 is the smallest uncountable ordinal (to see that it exists, considerthe set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countableordinal, and ω1 is the order type of that set), ω2 is the smallest ordinal whose cardinality is greater than ℵ1, and soon, and ωω is the limit of ωn for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed thefirst cardinal after all the ωn).Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, α < ωᵦ implies α+ωᵦ = ωᵦ, and 1 ≤ α < ωᵦ impliesα·ωᵦ = ωᵦ, and 2 ≤ α < ωᵦ implies αωᵦ = ωᵦ. Using the Veblen hierarchy, β ≠ 0 and α < ωᵦ imply φα(ωβ) = ωβ
and Γωᵦ = ωᵦ. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strongkind of limit.
25.2 See also
• Aleph number
136
25.3. REFERENCES 137
25.3 References• Y.N. Moschovakis Notes on Set Theory (1994 Springer) p. 198
Chapter 26
Well-order
In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that everynon-empty subset of S has a least element in this ordering. The set S together with the well-order relation is thencalled awell-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellingswellorder,wellordered, and wellordering.Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatestelement, has a unique successor (next element), namely the least element of the subset of all elements greater thans. There may be elements besides the least element which have no predecessor (see Natural numbers below for anexample). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the leastelement of the subset of all upper bounds of T in S.If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it isa well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since theyare easily interconvertible.Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can bewell-ordered. If a set is well-ordered (or even if it merely admits a wellfounded relation), the proof technique oftransfinite induction can be used to prove that a given statement is true for all elements of the set.The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called thewell-ordering principle (for natural numbers).
26.1 Ordinal numbers
Main article: Ordinal number
Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of afinite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object witha particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (numberof elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically startsfrom one, so it assigns to each object the size of the initial segment with that object as last element. Note that thesenumbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equalto the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-thelement” of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-thelement” where β can also be an infinite ordinal, it will typically count from zero.For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particularcardinality can have many different order types. For a countably infinite set, the set of possible order types is evenuncountable.
138
26.2. EXAMPLES AND COUNTEREXAMPLES 139
26.2 Examples and counterexamples
26.2.1 Natural numbers
The standard ordering ≤ of the natural numbers is a well-ordering and has the additional property that every non-zeronatural number has a unique predecessor.Another well-ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers,and the usual ordering applies within the evens and the odds:
0 2 4 6 8 ... 1 3 5 7 9 ...
This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Twoelements lack a predecessor: 0 and 1.
26.2.2 Integers
Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well-ordering,since, for example, the set of negative integers does not contain a least element.The following relation R is an example of well-ordering of the integers: x R y if and only if one of the followingconditions holds:
1. x = 0
2. x is positive, and y is negative
3. x and y are both positive, and x ≤ y
4. x and y are both negative, and |x| ≤ |y|
This relation R can be visualized as follows:
0 1 2 3 4 ... −1 −2 −3 ...
R is isomorphic to the ordinal number ω + ω.Another relation for well-ordering the integers is the following definition: x ≤ y iff (|x| < |y| or (|x| = |y| and x ≤ y)).This well-order can be visualized as follows:
0 −1 1 −2 2 −3 3 −4 4 ...
This has the order type ω.
26.2.3 Reals
The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0,1) does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can showthat there is a well-order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuumhypothesis) imply the axiom of choice and hence a well-order of the reals. Nonetheless, it is possible to show thatthe ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order ofthe reals.[1] However it is consistent with ZFC that a definable well-ordering of the reals exists—for example, it isconsistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well-orders the reals, or indeedany set.An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: Suppose X is a subsetof R well-ordered by ≤. For each x in X, let s(x) be the successor of x in ≤ ordering on X (unless x is the last elementof X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains
140 CHAPTER 26. WELL-ORDER
at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (exceptpossibly for a last element of X which could be mapped to zero later). And it is well known that there is an injectionfrom Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to thenatural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may ormay not be a well-order with the standard "≤".
• The natural numbers are a well-order.• The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order.
Examples of well-orders:
• The set of numbers { − 2−n | 0 ≤ n < ω } has order type ω.• The set of numbers { − 2−n − 2−m−n | 0 ≤ m,n < ω } has order type ω². The previous set is the set of limit
points within the set. Within the set of real numbers, either with the ordinary topology or the order topology,0 is also a limit point of the set. It is also a limit point of the set of limit points.
• The set of numbers { − 2−n | 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 isa limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers itis not.
26.3 Equivalent formulations
If a set is totally ordered, then the following are equivalent to each other:
1. The set is well-ordered. That is, every nonempty subset has a least element.2. Transfinite induction works for the entire ordered set.3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming
the axiom of dependent choice).4. Every subordering is isomorphic to an initial segment.
26.4 Order topology
Every well-ordered set can be made into a topological space by endowing it with the order topology.With respect to this topology there can be two kinds of elements:
• isolated points - these are the minimum and the elements with a predecessor.• limit points - this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets
without limit point are the sets of order type ω, for example N.
For subsets we can distinguish:
• Subsets with a maximum (that is, subsets which are bounded by themselves); this can be an isolated point or alimit point of the whole set; in the latter case it may or may not be also a limit point of the subset.
• Subsets which are unbounded by themselves but bounded in the whole set; they have no maximum, but asupremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hencealso of the whole set; if the subset is empty this supremum is the minimum of the whole set.
• Subsets which are unbounded in the whole set.
A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is alsomaximum of the whole set.A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal toω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.
26.5. SEE ALSO 141
26.5 See also• Tree (set theory), generalization
• Well-ordering theorem
• Ordinal number
• Well-founded set
• Well partial order
• Prewellordering
• Directed set
26.6 References[1] S. Feferman: “Some Applications of the Notions of Forcing and Generic Sets”, Fundamenta Mathematicae, 56 (1964)
325-345
• Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Pure and applied math-ematics (2nd ed.). John Wiley & Sons. pp. 4–6, 9. ISBN 978-0-471-31716-6.
142 CHAPTER 26. WELL-ORDER
26.7 Text and image sources, contributors, and licenses
26.7.1 Text• Axiom of dependent choice Source: https://en.wikipedia.org/wiki/Axiom_of_dependent_choice?oldid=646578198 Contributors: Zun-
dark, Gabbe, Schneelocke, Charles Matthews, Dysprosia, Aleph4, Tobias Bergemann, Jao, Oleg Alexandrov, Salix alba, R.e.b., Sango123,Mathbot, Trovatore, JRSpriggs, CBM, LegitimateAndEvenCompelling, Ttwo, Linkracer, Addbot, RJGray, Juniuswikiae, RedBot, Emaus-Bot, Frietjes, Brirush and Anonymous: 7
• Bijection Source: https://en.wikipedia.org/wiki/Bijection?oldid=670090850 Contributors: Damian Yerrick, AxelBoldt, Tarquin, Jan Hid-ders, XJaM, Toby Bartels, Michael Hardy, Wshun, TakuyaMurata, GTBacchus, Karada, Александър, Glenn, Poor Yorick, Rob Hooft,Pizza Puzzle, Hashar, Hawthorn, Charles Matthews, Dcoetzee, Dysprosia, Hyacinth, David Shay, Ed g2s, Bevo, Robbot, Fredrik, Ben-wing, Bkell, Salty-horse, Tobias Bergemann, Giftlite, Jorge Stolfi, Alberto da Calvairate~enwiki, MarkSweep, Tsemii, Vivacissama-mente, Guanabot, Guanabot2, Quistnix, Paul August, Ignignot, MisterSheik, Nickj, Kevin Lamoreau, Obradovic Goran, Pearle, Hashar-Bot~enwiki, Dallashan~enwiki, ABCD, Schapel, Palica, MarSch, Salix alba, FlaBot, VKokielov, RexNL, Chobot, YurikBot, MichaelSlone, Member, SmackBot, RDBury, Mmernex, Octahedron80, Mhym, Bwoodacre, Dreadstar, Davipo, Loadmaster, Mets501, Drefty-mac, Hilverd, Johnfuhrmann, Bill Malloy, Domitori, JRSpriggs, CmdrObot, Gregbard, Yaris678, Sam Staton, Panzer raccoon!, Kilva,AbcXyz, Escarbot, Salgueiro~enwiki, JAnDbot, David Eppstein, Martynas Patasius, Paulnwatts, Cpiral, GaborLajos, Policron, Diegovb,UnicornTapestry, Yomcat, Wykypydya, Bongoman666, SieBot, Paradoctor, Paolo.dL, Smaug123, MiNombreDeGuerra, JackSchmidt,I Spel Good~enwiki, Peiresc~enwiki, Classicalecon, Adrianwn, Biagioli, Watchduck, Hans Adler, Humanengr, Neuralwarp, Baudway,FactChecker1199, Kal-El-Bot, Subversive.sound, Tanhabot, Glane23, PV=nRT, Meisam, Legobot, Luckas-bot, Yobot, Ash4Math, Shva-habi, Omnipaedista, RibotBOT, Thehelpfulbot, FrescoBot, MarcelB612, CodeBlock, MastiBot, FoxBot, Duoduoduo, Xnn, EmausBot,Hikaslap, TuHan-Bot, Cobaltcigs, Wikfr, Karthikndr, Anita5192, Wcherowi, Widr, Strike Eagle, PhnomPencil, Knwlgc, Dhoke san-ket, Victor Yus, Cerabot~enwiki, JPaestpreornJeolhlna, Yardimsever, CasaNostra, KoriganStone, Whamsicore, JMP EAX, Sweepy andAnonymous: 89
• Cardinal number Source: https://en.wikipedia.org/wiki/Cardinal_number?oldid=668048521Contributors: AxelBoldt, Tobias Hoevekamp,Derek Ross, Calypso, Mav, Bryan Derksen, Zundark, The Anome, Iwnbap, Andre Engels, Danny, XJaM, Christian List, Gritchka, TobyBartels, Olivier, Patrick, JohnOwens, Michael Hardy, Wshun, Nixdorf, Rp, TakuyaMurata, Mpagano, Poor Yorick, Jiang, Vargenau,Schneelocke, Charles Matthews, Dysprosia, Jitse Niesen, Maximus Rex, Hyacinth, Gutsul, Mosesklein, Bloodshedder, Dmytro, Hmackier-nan, Robbot, Fredrik, Sverdrup, Henrygb, Choni, Paul Murray, Fuelbottle, Wile E. Heresiarch, Tobias Bergemann, Giftlite, Ævar ArnfjörðBjarmason, Lethe, Fropuff, Snowdog, Ajgorhoe, Mboverload, Mobius, Fuzzy Logic, Beland, Pmanderson, Karl-Henner, Sam Hocevar,Neutrality, Splatty, Paul August, Dmr2, Grick, Randall Holmes, PWilkinson, Minghong, Jumbuck, Sligocki, Caesura, Spambit, DV8 2XL,Oleg Alexandrov, Needmoredinosaur, Linas, Miaow Miaow, Aaron McDaid, Dionyziz, Graham87, ArchonOFages, Qwertyus, GoldenEternity, Salix alba, FlaBot, Chobot, Algebraist, YurikBot, Alpt, Jacques Antoine, Gaius Cornelius, Trovatore, PAStheLoD, Expensivehat,Figaro, Arthur Rubin, Clair de Lune, Fram, Teply, Bo Jacoby, SmackBot, YellowMonkey, InverseHypercube, Although, Jagged 85, TimPierce, Nakon, Dreadstar, Mmehdi.g, Lambiam, John H, Morgan, Loadmaster, JHunterJ, Gabn1, Jason.grossman, Joseph Solis in Aus-tralia, Easwaran, JRSpriggs, The Letter J, Jafet, CRGreathouse, CBM, Jokes Free4Me, Myasuda, FilipeS, Ntsimp, KarolS, Pcu123456789,Fenrisulfr, Hannes Eder, JAnDbot, Four Dog Night, LittleOldMe, Althai, LookingGlass, R'n'B, Ttwo, Policron, Szipucsu, Anony-mous Dissident, SieBot, J-puppy, Randomblue, ClueBot, DFRussia, MarceloTapiaGaete~enwiki, J567o678e789l, Thingg, XLinkBot,Чръный человек, Addbot, AnnaFrance, Tassedethe, Luckas-bot, Yobot, Fraggle81, TaBOT-zerem, TechKid, AnomieBOT, Rubinbot,Bsea, AdjustShift, Csigabi, Xqbot, Gonzalcg, Лев Дубовой, FrescoBot, TheLaeg, Kismalac, Distortiondude, RjwilmsiBot, УральскийКот, Beyond My Ken, EmausBot, Set theorist, Tanner Swett, Josve05a, Quondum, Wmayner, L Kensington, ClueBot NG, Wcherowi,Demonsquirrel, Helpful Pixie Bot, Joydeep, Micenine, Khalpern, ChrisGualtieri, Jochen Burghardt, Mark viking, IORIUS, Weux082690,Nigellwh, Monkbot and Anonymous: 123
• Cardinality Source: https://en.wikipedia.org/wiki/Cardinality?oldid=663424629Contributors: AxelBoldt, Zundark, LapoLuchini, MichaelHardy, Revolver, Charles Matthews, Dcoetzee, Hyacinth, Gutsul, Robbot, Tobias Bergemann, Schmmd, Tosha, Giftlite, Ævar Arn-fjörð Bjarmason, Waltpohl, Manuel Anastácio, TedPavlic, Cjrs 79, Paul August, Pmcm, Kwamikagami, Bobo192, Larry V, Jumbuck,EvenT, Oleg Alexandrov, Tbsmith, E=MC^2, Ruud Koot, Tabletop, Ralfipedia, Dionyziz, Dpv, Rjwilmsi, Bhadani, Mathbot, Chobot,Jersey Devil, Surgo, Gaius Cornelius, SEWilcoBot, Trovatore, Muu-karhu, Eltwarg, Arthur Rubin, Pred, Goetz, SmackBot, Betacom-mand, Mhss, Silly rabbit, Octahedron80, Gala.martin, BalthaZar, Dreadstar, Keithjones, SashatoBot, MvH, Jim.belk, Mets501, DanGluck, Newone, 'Ff'lo, JRSpriggs, CRGreathouse, CmdrObot, Sam Staton, Xantharius, Kilva, Rlupsa, JAnDbot, MER-C, Olaf, Con-normah, JamesBWatson, Ling.Nut, Usien6, David Eppstein, Ttwo, Trumpet marietta 45750, GaborLajos, UnicornTapestry, Mistercup-cake, Lradrama, PaulTanenbaum, Wikiisawesome, SieBot, Gerakibot, Yintan, Paolo.dL, Svick, Classicalecon, Justin W Smith, Marc vanLeeuwen, WikHead, Noctibus, Legobot, Luckas-bot, Yobot, KamikazeBot, , 9258fahsflkh917fas, Materialscientist, RJGray, Gonzalcg,Erik9bot, Dataffer9, FrescoBot, HRoestBot, Dinamik-bot, WildBot, EmausBot, RenamedUser01302013, AvicBot, Perremba, NGPriest,Maschen, EdoBot, ClueBot NG, Snotbot, Helpful Pixie Bot, Northamerica1000, Wiki13, Derschueler, J'odore, Khalpern, ChrisGualtieri,Theboombody, Jochen Burghardt, Mark viking, Transonic Crayon, Nigellwh and Anonymous: 93
• Cofiniteness Source: https://en.wikipedia.org/wiki/Cofiniteness?oldid=587573439Contributors: Michael Hardy, Revolver, Charles Matthews,Aleph4, MathMartin, Choni, Saforrest, Tobias Bergemann, Fropuff, Alberto da Calvairate~enwiki, Paul August, ABCD, Burn, Linas,Mathbot, Lenthe, Trovatore, Kompik, SmackBot, Nbarth, Mets501, Neelix, Headbomb, RobHar, JJ Harrison, Sullivan.t.j, LordAnu-bisBOT, Kyle the bot, JackSchmidt, Mr. Granger, Addbot, Legobot, Luckas-bot, Amirobot, Artem M. Pelenitsyn, Chricho, ZéroBot,Josve05a, ChrisGualtieri, Dsou and Anonymous: 3
• Derived set (mathematics) Source: https://en.wikipedia.org/wiki/Derived_set_(mathematics)?oldid=582494758 Contributors: MichaelHardy, MathMartin, Tobias Bergemann, Gauge, Crisófilax, Tsirel, NeoUrfahraner, Roboto de Ajvol, Irishguy, Kompik, Mhss, Load-master, Pietro KC, Easwaran, Esobocinski, Vaughan Pratt, CBM, Erzbischof, Stebulus, Scorwin, Netsettler, Bluestarlight37, Alexan-dar.R.~enwiki, ClueBot, Bob1960evens, Hans Adler, DumZiBoT, Addbot, Ptbotgourou, Rubinbot, Citation bot, Churchill17, Dan-nyAsher, Money is tight, FrescoBot, Louperibot, EmausBot, Aengle1429 and Anonymous: 16
• Discrete space Source: https://en.wikipedia.org/wiki/Discrete_space?oldid=649343661 Contributors: AxelBoldt, Toby Bartels, Patrick,Michael Hardy, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Jose Ramos, MathMartin, Ancheta Wis, Tosha, BenFrantzDale,DefLog~enwiki, Baba, Paul August, Haham hanuka, Ricky81682, Oleg Alexandrov, Linas, Marudubshinki, Jshadias, Hairy Dude, Reyk,Arundhati bakshi, Curpsbot-unicodify, Ilmari Karonen, Poulpy, KnightRider~enwiki, Diegotorquemada, Mhss, Dreadstar, Mwtoews, Ar-glebargleIV, Mets501, Jaybe~enwiki, JRSpriggs, CRGreathouse, Equendil, Cydebot, Thijs!bot, Salgueiro~enwiki, Albmont, VolkovBot,
26.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 143
TXiKiBoT, Anonymous Dissident, AlleborgoBot, He7d3r, Marc van Leeuwen, Addbot, Topology Expert, Download, Luckas-bot, Yobot,Denispir, Ciphers, DannyAsher, FrescoBot, EmausBot, TuHan-Bot, Yukuairoy, Glmccolm, Qmv, Koertefa, Brad7777, Deltahedron andAnonymous: 24
• First uncountable ordinal Source: https://en.wikipedia.org/wiki/First_uncountable_ordinal?oldid=663920033 Contributors: AxelBoldt,Michael Hardy, Aleph4, Tobias Bergemann, Touriste, RussBot, SmackBot, JRSpriggs, YohanN7, Addbot, Yobot, Kenilworth Terrace,Helpful Pixie Bot, Andyhowlett and Anonymous: 4
• Georg Cantor Source: https://en.wikipedia.org/wiki/Georg_Cantor?oldid=670706441 Contributors: AxelBoldt, Mav, Zundark, TheAnome, Andre Engels, Eclecticology, Danny, XJaM, Michael Hardy, Dominus, Wapcaplet, Chinju, Anonymous56789, Card~enwiki,Ahoerstemeier, DavidWBrooks, Den fjättrade ankan~enwiki, LouI, Poor Yorick, Andres, Madir, Ideyal, Hashar, Loren Rosen, Re-named user 4, Charles Matthews, Dcoetzee, Dysprosia, Jogloran, Markhurd, Hyacinth, Taxman, Dimadick, Robbot, Astronautics~enwiki,Jaredwf, Fredrik, Chocolateboy, Baldhur, MathMartin, GregorBrand, Cholling, Timrollpickering, JackofOz, Lupo, Tobias Bergemann,Donald j axel, Giftlite, Curps, Carlo.Ierna, Andris, Andycjp, OverlordQ, MisfitToys, Bob.v.R, Vina, Pmanderson, Icairns, Gscshoyru,Vivacissamamente, Thorwald, Lucidish, D6, Discospinster, Zaheen, Rich Farmbrough, Leibniz, Gadykozma, Pjacobi, Ascánder, Byrial,Paul August, Bender235, Swid, MBisanz, Cedders, Kwamikagami, Vanished user kjij32ro9j4tkse, Shoujun, Bobo192, Obradovic Goran,Knucmo2, Jumbuck, JYolkowski, TheParanoidOne, SnowFire, Riana, Mark.howison, Mrholybrain, Noosphere, VivaEmilyDavies, Cey-ockey, Oleg Alexandrov, Tariqabjotu, Woohookitty, Mindmatrix, FeanorStar7, E=MC^2, Twthmoses, FoxInShoes, Macaddct1984, SDC,LogicalDash, Graham87, Rjwilmsi, Koavf, Lockley, R.e.b., Olessi, FlaBot, RobertG, Jak123, Gurch, Russavia, Chobot, Gwernol, Yurik-Bot, Wavelength, RobotE, Chaser, Gaius Cornelius, Jugander, KSchutte, NawlinWiki, Zhaladshar, Trovatore, Jpbowen, Raven4x4x, KyleBarbour, Black Falcon, Wknight94, Tuckerresearch, Emijrp, Zzuuzz, Mike Dillon, BorgQueen, JoanneB, T. Anthony, ArielGold, Teply,Finell, Attilios, SmackBot, Nihonjoe, Malkinann, C.Fred, David Shear, SaxTeacher, Ohnoitsjamie, Hmains, Andy M. Wang, DavidPetry, Tosqueira, Hibbleton, MalafayaBot, Darth Panda, Tim Pierce, Newport, Homestarmy, Savidan, Akriasas, Aditsu, Hgilbert, Run-corn, Tesseran, Byelf2007, SashatoBot, Lambiam, Eliyak, Ser Amantio di Nicolao, Cronholm144, Mgiganteus1, JLeander, Yvesnimmo,Optakeover, Waggers, SandyGeorgia, Doczilla, AdultSwim, Phuzion, Asyndeton, Christian Roess, Quaeler, Dan Gluck, BranStark, Ba-nanaFiend, Criticofpurereason, CzarB, Newone, Twas Now, Cbrown1023, Gil Gamesh, Flubeca, Ioannes Pragensis, CBM, Smallpond,Brownlee, Gregbard, Doctormatt, Cydebot, WillowW, MC10, Jack O'Lantern, Gogo Dodo, A Softer Answer, Michael C Price, Dumb-BOT, Kozuch, Eubulide, Thijs!bot, Biruitorul, Pajz, LaGrange, Mojo Hand, Headbomb, Woody, James086, EdJohnston, Wadsa, Aadal,AntiVandalBot, Akradecki, Linguistus~enwiki, AaronY, VectorPosse, Jj137, Genpark, JAnDbot, Deflective, Avaya1, Clementvidal,WolfmanSF, Georgian, JamesBWatson, Ling.Nut, Generalstudent, Bellbird, David Eppstein, DerHexer, Rickard Vogelberg, Martin-Bot, Genghiskhanviet, Bus stop, CommonsDelinker, J.delanoy, Pharaoh of the Wizards, DrKiernan, Ttwo, Maurice Carbonaro, Owlgo-rithm, BobEnyart, Mathglot, Salih, AKA MBG, R613vlu, NewEnglandYankee, Policron, Robertgreer, Cometstyles, Tellerman, Treisijs,David H. Flint, Alan U. Kennington, Indubitably, Fences and windows, Alex.petralia, Philip Trueman, TXiKiBoT, Rei-bot, The Tetrast,SelketBot, LeaveSleaves, Geometry guy, Gilisa, J Casanova, Arcfrk, GirasoleDE, SieBot, Mikemoral, Jauerback, Parhamr, Simul8, Re-natops, Scientor, Monegasque, Mikeblew, Lightmouse, The Cheaster, Vojvodaen, CharlesGillingham, Ghpicard, Randomblue, RS1900,Mattyclam, ClueBot, DFRussia, MikeVitale, JuPitEer, Binudigitaleye, Timberframe, Vsenderov, UKoch, MasterEditorDXK, Vincen-thashimoto, Pica559, Monkey999999999, Monkeydog2, Solar-Wind, Albinofawn, Excirial, Eeekster, Cenarium, Hans Adler, Wikidok-man, Bletchley, Luwilt, Addbot, Tcncv, TutterMouse, AndersBot, FiriBot, TheMightyPeanut, LemmeyBOT, 5 albert square, Feketekave,Squandermania, AgadaUrbanit, Tide rolls, Zorrobot, Yobot, Ptbotgourou, TaBOT-zerem, Mmxx, AnomieBOT, Zhieaanm, Kingpin13,Materialscientist, Citation bot, Xqbot, Capricorn42, Nfr-Maat, TheWeakWilled, RJGray, RedKiteUK, Srich32977, Omnipaedista, Bran-don5485, Auréola, Lefschetz, Green Cardamom, Totemtotemo, Edgars2007, Immutable92, Phila 033, Citation bot 1, Chenopodiaceous,Tkuvho, Tinton5, RedBot, RazielZero, BasvanPelt, Tim1357, TobeBot, Trappist the monk, Lotje, Viktor Laszlo, TFTD, Patriot8790,RjwilmsiBot, Salvio giuliano, EmausBot, RenamedUser01302013, Gimmetoo, Elee, TuHan-Bot, Italia2006, Kkm010, Daonguyen95,Isarra, DailyShipper, DASHBotAV, Redhorn3, ClueBot NG, SusikMkr, Widr, Telpardec, Lawrentia, MKar, MusikAnimal, Amp71,Davidiad, Saint Johann, JHobson2, Mathbird, Derschueler, DouglasKlein, Maryester, Achowat, BattyBot, Ling.Nut3, Pratyya Ghosh,ChrisGualtieri, Dexbot, Stephan Kulla, Jochen Burghardt, The Mol Man, Nimetapoeg, Monochrome Monitor, CVQT, Hunterkingcool,Monkbot, Jonarnold1985, Patelsandip82, KasparBot, Prabaha and Anonymous: 356
• Hereditary property Source: https://en.wikipedia.org/wiki/Hereditary_property?oldid=646351980 Contributors: Populus, McKay, Al-tenmann, Tobias Bergemann, Giftlite, Paul August, Rjwilmsi, R.e.b., Ott2, Nick Number, David Eppstein, Radagast3, Cenarium, HansAdler, Addbot, LatitudeBot, Yobot, Citation bot, Citation bot 1, RobinK, Junior Wrangler, Tijfo098, CocuBot, Helpful Pixie Bot, Joze-fgajdos, Anrnusna and Anonymous: 3
• Injective function Source: https://en.wikipedia.org/wiki/Injective_function?oldid=670090869 Contributors: AxelBoldt, Tarquin, XJaM,Toby Bartels, Michael Hardy, Wshun, Dominus, Chinju, Dcljr, TakuyaMurata, Karada, Cyp, Александър, Glenn, Hawthorn, CharlesMatthews, Dysprosia, Ed g2s, PuzzletChung, Aleph4, Robbot, Tobias Bergemann, Giftlite, Peruvianllama, Jorge Stolfi, Daniel Brock-man, Edcolins, Keeyu, AHM, MarkSweep, Bob.v.R, Tsemii, Karl Dickman, Naku~enwiki, Vivacissamamente, Cfailde, Quistnix, PaulAugust, MisterSheik, Kevin Lamoreau, Obradovic Goran, Haham hanuka, ABCD, Schapel, MIT Trekkie, Bookandcoffee, Oleg Alexan-drov, Imaginatorium, Tomhab, MarSch, Donotresus, FlaBot, Chobot, Algebraist, RussBot, Gaius Cornelius, Dbfirs, LarryLACa, Pred,InverseHypercube, BiT, SMP, MalafayaBot, Octahedron80, Cassivs, Javalenok, Dreadstar, Acdx, Saippuakauppias, Rohit math, Bio-Tube, 16@r, Beefyt, Mike Fikes, CBM, Sam Staton, Xantharius, Jj137, LucPereira, JAnDbot, David Eppstein, Falcor84, TechnoFaye,Pbroks13, IPonomarev, Numbo3, Austinflorida, Tendays, Dubhe.sk, The enemies of god, VolkovBot, AlleborgoBot, Da Joe, Hawk777,Paolo.dL, OKBot, Skeptical scientist, Classicalecon, WikiBotas, Adrianwn, Ficbot, Rohit nit, Watchduck, Marc van Leeuwen, Redclock2,Kal-El-Bot, Addbot, Tanhabot, Download, LaaknorBot, PV=nRT, Jarble, Legobot, Luckas-bot, Yobot, Nallimbot, Citation bot, Xqbot,Shvahabi, Ubcule, Omnipaedista, Mark Schierbecker, RibotBOT, Atlantia, EmausBot, PrisonerOfIce, TuHan-Bot, Josve05a, Wikfr,Maschen, Anita5192, Jack Greenmaven, Cispyre, K1jacobson, JPaestpreornJeolhlna, Yardimsever, Abitslow, Assuredlonewolf, GildedSnail, JMP EAX and Anonymous: 97
• Integer Source: https://en.wikipedia.org/wiki/Integer?oldid=671475050 Contributors: AxelBoldt, Brion VIBBER, Bryan Derksen, Zun-dark, Andre Engels, Youssefsan, XJaM, Arvindn, Christian List, Dwheeler, Stevertigo, Michael Hardy, TakuyaMurata, Eric119, Ellywa,Ahoerstemeier, Darkwind, Salsa Shark, Ciphergoth, Nikai, Andres, Panoramix, Rob Hooft, Charles Matthews, Dysprosia, Jake Nelson,Hyacinth, Elwoz, Robbot, Moriori, Fredrik, Chris 73, Altenmann, Lowellian, Henrygb, Rholton, Jfire, OmegaMan, Hippietrail, Fuelbot-tle, Jleedev, Tobias Bergemann, Giftlite, Dbenbenn, Christopher Parham, Pretzelpaws, Lupin, Markus Kuhn, Arnejohs, Bovlb, Alanl,Leonard Vertighel, Knutux, Antandrus, Oneiros, Gauss, Elroch, Quota, Brianjd, Discospinster, Rich Farmbrough, Guanabot, Vsmith,Paul August, El C, Kwamikagami, Bendono, Bobo192, Dreish, .:Ajvol:., Sasquatch, Jojit fb, Deryck Chan, Obradovic Goran, DanielArteaga~enwiki, Jumbuck, Msh210, Alansohn, Arthena, Olegalexandrov, Wtmitchell, Velella, RJFJR, VoluntarySlave, Oleg Alexandrov,Saeed, Linas, Georgia guy, Camw, Splintax, MattGiuca, Kadri~enwiki, MONGO, Pufferfish101, PhilippWeissenbacher, Graham87,
144 CHAPTER 26. WELL-ORDER
Chun-hian, Mendaliv, Jshadias, Josh Parris, Sjö, TheRingess, Salix alba, Mike Segal, Alejo2083, FlaBot, Mathbot, Jrtayloriv, Tardis,King of Hearts, Chobot, NSR, Roboto de Ajvol, YurikBot, Sceptre, Michael Slone, KSmrq, Stephenb, Gaius Cornelius, Wimt, Nawl-inWiki, Trovatore, Darkmeerkat, Wknight94, Analoguedragon, Saric, Theda, Arthur Rubin, Pb30, GrinBot~enwiki, Finell, A13ean,Amalthea, SmackBot, Sticky Parkin, Moeron, Incnis Mrsi, InverseHypercube, KnowledgeOfSelf, Unyoyega, AndyZ, Edgar181, Gilliam,Skizzik, Bluebot, Optikos, Jprg1966, Raja Hussain, Miquonranger03, Akanemoto, DHN-bot~enwiki, Darth Panda, Can't sleep, clown willeat me, Timothy Clemans, Vegard, Grover cleveland, Aldaron, Cybercobra, Nakon, Jiddisch~enwiki, Dreadstar, Tanyakh, DMacks, Dku-sic~enwiki, BrianH123, Ace ETP, SashatoBot, Lambiam, Kuru, FrozenMan, Bjankuloski06en~enwiki, Jim.belk, IronGargoyle, 16@r,Vanished user 8ij3r8jwefi, Stwalkerster, Mets501, Quaeler, Iridescent, Igoldste, Cls14, Blehfu, Az1568, Jh12, Spindled, InvisibleK,Cxw, Wafulz, WeggeBot, SuperMidget, Kanags, Julian Mendez, Odie5533, Xantharius, JodyB, Thijs!bot, Epbr123, Koeplinger, Kahri-man~enwiki, N5iln, Marek69, Kathovo, Escarbot, Mentifisto, AntiVandalBot, Seaphoto, Fnerchei, Jj137, Dryke, Karadimos, JAnDbot,GromXXVII, Davexia, Hut 8.5, Dricherby, Bongwarrior, VoABot II, JamesBWatson, ALostIguana, David Eppstein, Lukeelms, DerHexer,Esanchez7587, Seba5618, Gwern, Ksero, MartinBot, Kawehi 65, J.delanoy, Ttwo, Extransit, Guardian72, McSly, DJ1AM, NewEng-landYankee, SJP, Policron, DavidCBryant, Ja 62, Lights, X!, 28bytes, VolkovBot, Pleasantville, AlnoktaBOT, Philip Trueman, DoorsA-jar, TXiKiBoT, Moogwrench, Drake Redcrest, Qxz, Imasleepviking, Metatron’s Cube, Corvus cornix, Digby Tantrum, JhsBot, Fnenu,Wolfrock, Synthebot, Seresin, AlleborgoBot, Tvinh, Katzmik, Omerks, Deconstructhis, Demmy100, SieBot, Dirtylittlesecerts, Ger-akibot, RJaguar3, Triwbe, Keilana, PookeyMaster, Flyer22, Radon210, JSpung, Sbowers3, Oxymoron83, Faradayplank, JackSchmidt,Macy, OKBot, Denisarona, Sasha Callahan, Troy 07, Explicit, Blockofwood, SLSB, Loren.wilton, Elassint, ClueBot, Rumping, PipepBot,Snigbrook, The Thing That Should Not Be, GreekHouse, Jan1nad, Gaia Octavia Agrippa, P0mbal, Boing! said Zebedee, CounterVan-dalismBot, Harland1, LizardJr8, Liempt, ChandlerMapBot, Gakusha, DragonBot, -Midorihana-, Robbie098, Sillychiva593, Sun Cre-ator, ParisianBlade, Sin Harvest, Computer97, La Pianista, Aitias, SoxBot III, Apparition11, BarretB, AgnosticPreachersKid, Marc vanLeeuwen, ZooFari, Airplaneman, Gggh, Wyatt915, Addbot, Amyx231, Jncraton, Fieldday-sunday, Davidw1985, Leszek Jańczuk, Fluffer-nutter, Download, LaaknorBot, CarsracBot, 5 albert square, Numbo3-bot, Ehrenkater, Zorrobot, Legobot, Luckas-bot, DB.Gerry, TheFlying Spaghetti Monster, AnomieBOT, Hairhorn, Jim1138, 9258fahsflkh917fas, Kingpin13, Jarmiz, Materialscientist, Xqbot, Capri-corn42, Johnferrer, Jeffrey Mall, Jsharpminor, Almabot, Zarcillo, Ivan Shmakov, Shadowjams, Hakunamenta, Aaron Kauppi, Aram-mozuob, Wikipe-tan, GT1345, Citation bot 1, Maggyero, MacMed, Pinethicket, I dream of horses, 10metreh, Riitoken, Jumpytoo, Al-tinBotak, Merlion444, Jauhienij, Jonkerz, Isaac909, 777sms, Clader2, Tbhotch, Hornlitz, Mean as custard, TomdFr, Shafaet, TomT0m,LayZeeDK, DASHBot, EmausBot, Gfoley4, Notinlist, RenamedUser01302013, MartinThoma, Solarra, Scgtrp, Wikipelli, K6ka, Tele-ComNasSprVen, Fayimora, Mb5576, Lucas Thoms, ZéroBot, Fæ, Bollyjeff, Duperman01, KuduIO, Quondum, D.Lazard, Tolly4bolly,L Kensington, Donner60, Chewings72, ChuispastonBot, DASHBotAV, Anita5192, ClueBot NG, Wcherowi, Ypnypn, MelbourneStar,Rtucker913, Frietjes, Cntras, O.Koslowski, Lolimakethingsfun, Widr, Stapler9124, Theopolisme, Helpful Pixie Bot, BG19bot, Lann123,Dan653, Mark Arsten, Rm1271, Bishopgraeme, Jrobbinz1, Sweetnessman, Celceus, Mathenaire, DarafshBot, Ducknish, Lugia2453,Frosty, Soda drinker, CsDix, Redd Foxx 1991, Khwartz, Dtw45, DavidLeighEllis, Galactic Citizen 299495038858569, JustinJustin-JustinJustin, Mynameisrichard, Someone not using his real name, Sam Sailor, AddWittyNameHere, Jamesh1998, Catsweeds, BethNaught,Calvin.stone99, Rasheed49, This is a fake user name, UnixDaemon, This is another fake user name, GeoffreyT2000, Eduvgugvuv, Sam-vandervlies, Jj 1213 wiki, Esquivalience, Glitterstars23, Hgccbjhfsh, Kkk2000, SteakMuncher, Phuasien, KasparBot, Kafishabbir andAnonymous: 675
• Natural number Source: https://en.wikipedia.org/wiki/Natural_number?oldid=669940142 Contributors: AxelBoldt, Brion VIBBER,Bryan Derksen, Zundark, The Anome, Koyaanis Qatsi, AstroNomer~enwiki, Ed Poor, XJaM, Toby Bartels, Patrick, Infrogmation,TeunSpaans, Michael Hardy, Wshun, Wapcaplet, TakuyaMurata, Ellywa, Stevenj, Angela, Den fjättrade ankan~enwiki, Александър,Nikai, Andres, Panoramix, YishayMor, Revolver, Charles Matthews, Berteun, Crissov, Dcoetzee, Dysprosia, Jitse Niesen, Daniel Quin-lan, Markhurd, VeryVerily, Paul-L~enwiki, Mosesklein, Bevo, Shizhao, Elwoz, Qianfeng, Pakaran, Daran, RadicalBender, Sewing,Jni, Skeetch, Robbot, Murray Langton, Fredrik, Altenmann, Peak, Henrygb, Bkell, Moink, Wikibot, Aetheling, Fuelbottle, TobiasBergemann, Tosha, Giftlite, Dbenbenn, Vfp15, Ævar Arnfjörð Bjarmason, Dissident, Wwoods, Elias, Joe Kress, Dmmaus, Siroxo,Chameleon, DemonThing, Knutux, MarkSweep, Bob.v.R, Gauss, Bumm13, Pmanderson, Arcturus, Gscshoyru, Petershank, Joyous!,Fanghong~enwiki, Mormegil, Perey, Discospinster, Liso, Mani1, Paul August, Khalid, ZeroOne, EmilJ, Grick, Randall Holmes, Reiny-day, .:Ajvol:., Brim, Jojit fb, Obradovic Goran, Juanpabl, Jumbuck, JohnyDog, Aisaac, Kuratowski’s Ghost, Msh210, Jeltz, AzaToth,Mystyc1, Gbeeker, Woodstone, Talkie tim, Alem Dain, Oleg Alexandrov, Linas, LOL, StradivariusTV, JonH, MFH, GregorB, Xiong,Graham87, SixWingedSeraph, Roger McCoy, Mendaliv, Jshadias, Drbogdan, Rjwilmsi, MarSch, Salix alba, FlaBot, Nihiltres, Lmatt,Haonhien, Chobot, DVdm, Bgwhite, Algebraist, YurikBot, RobotE, Michael Slone, Sasuke Sarutobi, KSmrq, Stephenb, Tenebrae, GaiusCornelius, Ihope127, Rsrikanth05, Wimt, B-Con, Rick Norwood, Grafen, Trovatore, Zwobot, Martinwilke1980, Lt-wiki-bot, ArthurRubin, QmunkE, Katieh5584, That Guy, From That Show!, Yvwv, SmackBot, Travuun, Tom Lougheed, Unyoyega, Bomac, Eskimbot,BiT, Xaosflux, Gilliam, Hmains, JAn Dudík, PJTraill, Raja Hussain, MalafayaBot, SchfiftyThree, Akanemoto, Alink, DHN-bot~enwiki,Ladislav Mecir, Can't sleep, clown will eat me, Jamnik~enwiki, SundarBot, Grover cleveland, Khoikhoi, JackSlash, Jiddisch~enwiki,Dreadstar, RandomP, BullRangifer, Jon Awbrey, SashatoBot, Lambiam, Nishkid64, IronGargoyle, John H, Morgan, 16@r, Loadmaster,Mets501, Asyndeton, Stephen B Streater, Quaeler, Hrumph, Tawkerbot2, JRSpriggs, Heikobot, JForget, CRGreathouse, Ale jrb, Cxw,Simian1k, CBM, SuperMidget, Doctormatt, Cream147, Chasingsol, SimenH, Odie5533, Thijs!bot, Epbr123, Kahriman~enwiki, Com-positeFan, Escarbot, AntiVandalBot, Majorly, Opelio, JAnDbot, Husond, Martinkunev, Hut 8.5, Dricherby, .anacondabot, Hurmata,Io Katai, Magioladitis, Bongwarrior, VoABot II, JNW, JamesBWatson, Kajasudhakarababu, Animum, David Eppstein, SlamDiego, Jo-ergenB, Wdflake, Patstuart, Darksniperdragon, Catmoongirl, Ttwo, Maproom, Being blunt, Mahewa, Gill110951, Rommels, Policron,Fylwind, Vinsfan368, Xiahou, Idioma-bot, Vlma111, Ramanujam first, TXiKiBoT, Thomas1617, Abdullais4u, Meters, Synthebot, Alle-borgoBot, Demmy100, SieBot, BotMultichill, ToePeu.bot, Vanished User 8a9b4725f8376, Keilana, Prestonmag, OKBot, Angielaj, An-chor Link Bot, TheCatalyst31, Sfan00 IMG, ClueBot, PipepBot, Snigbrook, The Thing That Should Not Be, Cliff, Bballgrl42351, Drmies,Cp111, Pallida Mors, Boing! said Zebedee, Joshwashere10, CounterVandalismBot, Excirial, Da rulz07, He7d3r, Sun Creator, Cenar-ium, Jotterbot, Leohenrique0908~enwiki, Hans Adler, Principianewton, InternetMeme, Aaron north, Marc van Leeuwen, Aoeuidhtns,Pichpich, SilvonenBot, Paulginz, Airplaneman, Addbot, Proofreader77, WardenWalk, Fluffernutter, Download, Debresser, LinkFA-Bot,Numbo3-bot, Tide rolls, Muiranec, Gail, MuZemike, Ben Ben, Legobot, Luckas-bot, Yobot, Denispir, Caracho, KamikazeBot, IW.HG,AnomieBOT, DemocraticLuntz, Fullmetalactor, Neptune5000, ArthurBot, FactSpewer, MauritsBot, Xqbot, Bdmy, Txebixev, TechBot,InsérerNombreHere, RibotBOT, DosDIS 778, Aaron Kauppi, A. di M., Carl cuthbert, Phoebe alison, Auclairde, FrescoBot, HJ Mitchell,Machine Elf 1735, Maggyero, Tkuvho, Pinethicket, Ebony Jackson, Robertas.Vilkas, Achim1999, December21st2012Freak, FoxBot,TobeBot, SepIHw, Mrs.Barbera, Tsunhimtse, Lotje, Ajb1947, PleaseStand, Jesse V., DARTH SIDIOUS 2, Shafaet, BCtl, EmausBot,RenamedUser01302013, Paul Martyn-Smith, Wham Bam Rock II, Tommy2010, Ornithikos, Velpaedia Jenkuklordanus, Tubalubalu,ZéroBot, Alpha Quadrant (alt), Quondum, D.Lazard, SporkBot, Wayne Slam, Staszek Lem, Stephanos21, Matsievsky, Chuispaston-Bot, Davey2010, Anita5192, ResearchRave, ClueBot NG, Raiden10, Jack Greenmaven, Wcherowi, Widr, Helpful Pixie Bot, BG19bot,
26.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 145
Max Longint, Rasheeq1, Calagahan, Drift chambers, Writ Keeper, Ezhu94, ThatOtherJacob, Mdann52, Aqw400, Comatmebro, Dexbot,TheKing44, Brirush, Salon Alure, FrigidNinja, Eyesnore, Peter162534, Galactic Citizen 299495038858569, Aureomarginata, Mjolnir-Pants, Taowa, Erin.Annette.Brown, Peiffers, Mark D. Marquez, EvilLair, SarahTehCat, Purgy Purgatorio, Girlyhorsegirl, Spmishrag,Degenerate prodigy, Gov vj, Citogenitor, Kameronchia1234 and Anonymous: 473
• Order isomorphism Source: https://en.wikipedia.org/wiki/Order_isomorphism?oldid=593436771Contributors: LC~enwiki, The Anome,Jan Hidders, Michael Hardy, Altenmann, Giftlite, Markus Krötzsch, Jorend, Qwertyus, Crasshopper, Mhss, MisterHand, CBM, MetsBot,David Eppstein, Daniel5Ko, Addbot, PV=nRT, Pcap, Xqbot, Erik9bot, LucienBOT, Kangyh9659 and Anonymous: 4
• Order type Source: https://en.wikipedia.org/wiki/Order_type?oldid=617119064 Contributors: The Anome, Patrick, Dominus, TobiasBergemann, Dbenbenn, Francis Davey, Arthena, Kazvorpal, Mike Segal, SmackBot, Melchoir, BiT, Mhss, Schaef, Octahedron80, Lam-biam, JRSpriggs, Vaughan Pratt, Pjoef, Iamthedeus, Hccrle, DumZiBoT, Addbot, Yobot, Pelotom, EmausBot, Makecat-bot, JamesWaddington and Anonymous: 11
• Ordinal arithmetic Source: https://en.wikipedia.org/wiki/Ordinal_arithmetic?oldid=659870296 Contributors: Patrick, Mountain, To-bias Bergemann, Giftlite, EmilJ, Anders Kaseorg, Sligocki, Iannigb, OneWeirdDude, R.e.b., John Baez, Chobot, Grubber, Trovatore,Bota47, Arthur Rubin, PhS, SmackBot, Acipsen, JoshuaZ, Deadcode, Loadmaster, Gandalfxviv, Stotr~enwiki, JRSpriggs, CmdrObot,Sniffnoy, CBM, Michael C Price, Robertinventor, Crabula, Vanish2, MetsBot, Enoksrd, Popopp, Anchor Link Bot, PixelBot, Addbot,Luckas-bot, Yobot, Ht686rg90, AnomieBOT, VladimirReshetnikov, MarcelB612, Lclem, Paolo Lipparini, ChrisGualtieri, Mogism,Jochen Burghardt, Mark viking and Anonymous: 28
• Ordinal number Source: https://en.wikipedia.org/wiki/Ordinal_number?oldid=668709653 Contributors: AxelBoldt, Mav, Bryan Derk-sen, Zundark, The Anome, Iwnbap, LA2, Christian List, B4hand, Olivier, Stevertigo, Patrick, Michael Hardy, Llywrch, Jketola, Chinju,TakuyaMurata, Karada, Docu, Vargenau, Revolver, Charles Matthews, Dysprosia, Malcohol, Owen, Rogper~enwiki, Hmackiernan, Bald-hur, Adhemar, Fuelbottle, Tobias Bergemann, Giftlite, Markus Krötzsch, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Gro-Tsen, That-tommyhall, Jorend, Siroxo, Wmahan, Beland, Joeblakesley, Elroch, 4pq1injbok, Luqui, Silence, Paul August, EmilJ, Babomb, RandallHolmes, Wood Thrush, Robotje, Blotwell, Crust, Jumbuck, Sligocki, SidP, DV8 2XL, Jim Slim, Oleg Alexandrov, Warbola, Linas, MiaowMiaow, Graham87, Grammarbot, Jorunn, Rjwilmsi, Bremen, Salix alba, Mike Segal, R.e.b., FlaBot, Jak123, Chobot, YurikBot, Wave-length, RobotE, Hairy Dude, CanadianCaesar, Archelon, Gaius Cornelius, Trovatore, Crasshopper, DeadEyeArrow, Pooryorick~enwiki,Hirak 99, Closedmouth, Arthur Rubin, PhS, GrinBot~enwiki, Brentt, Nicholas Jackson, SmackBot, Pokipsy76, KocjoBot~enwiki, Blue-bot, AlephNull~enwiki, Jiddisch~enwiki, Dreadstar, Mmehdi.g, Lambiam, Khazar, Minna Sora no Shita, Bjankuloski06en~enwiki, 16@r,Loadmaster, Limaner, Quaeler, Jason.grossman, Joseph Solis in Australia, Easwaran, Zero sharp, Tawkerbot2, JRSpriggs, Vaughan Pratt,CRGreathouse, CBM, Gregbard, FilipeS, HdZ, Pcu123456789, Lyondif02, Odoncaoa, Jj137, Hannes Eder, Shlomi Hillel, JAnDbot,Agol, BrentG, Smartcat, Bongwarrior, Swpb, David Eppstein, Jondaman21, R'n'B, IPonomarev, RockMFR, Ttwo, It Is Me Here, Policron,VolkovBot, Dommedagsprofet, Hotfeba, Jeff G., LokiClock, PMajer, Alphaios~enwiki, Cremepuff222, Wikithesource, Arcfrk, SieBot,Mrw7, J-puppy, TheCatalyst31, ClueBot, DFRussia, DanielDeibler, DragonBot, Hans Adler, StevenDH, Lacce, Against the current,Dthomsen8, Addbot, Dyaa, Mathemens, Unzerlegbarkeit, Luckas-bot, Yobot, Utvik old, THEN WHO WAS PHONE?, KamikazeBot,AnomieBOT, Angry bee, Citation bot, Nexx892, Twri, Xqbot, Freebirth Toad, Capricorn42, RJGray, GrouchoBot, VladimirReshetnikov,SassoBot, Citation bot 1, RedBot, Burritoburritoburrito, TheStrayCat, Raiden09, EmausBot, Fly by Night, Jens Blanck, SporkBot, Clue-Bot NG, Frietjes, Rezabot, Helpful Pixie Bot, BG19bot, Anthony.de.almeida.lopes, Jochen Burghardt, Mark viking, Pop-up casket, JoseBrox, The Horn Blower, Dustin V. S., George8211, Dconman2, Garfield Garfield, Lalaloopsy1234, SoSivr, Eth450, Neposner, MirceaBRT, Divad42, Smwrd, Wilsonator5000 and Anonymous: 139
• Partially ordered set Source: https://en.wikipedia.org/wiki/Partially_ordered_set?oldid=670822320 Contributors: Bryan Derksen, Zun-dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus,Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De-fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, OlegAlexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, JosephSilverman, Sanguinity,Modify, RDBury, Incnis Mrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki,Mike Fikes, A. Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE,Tbleher, A3nm, David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielE-merson, TXiKiBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush,Megaloxantha, Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead,Watchduck, ComputerGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz,Legobot, Kilom691, AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ri-cardo Ferreira de Oliveira, Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot,Chharvey, The man who was Friday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jak-shap, Paolo Lipparini, ElphiBot, Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987,Victor Lesyk, Some1Redirects4You and Anonymous: 78
• Set theory Source: https://en.wikipedia.org/wiki/Set_theory?oldid=670017193 Contributors: AxelBoldt, Bryan Derksen, Zundark, TheAnome, Christian List, Toby Bartels, Enchanter, Michael Hardy, Karada, William M. Connolley, Plaudite~enwiki, Andres, Evercat, Mar-cosantonio, Dysprosia, ThomasStrohmann~enwiki, HappyDog, Hyacinth, VeryVerily, Robbot, Fredrik, Peak, Romanm, Tobias Berge-mann, Giftlite, Gene Ward Smith, Bogdanb, Mintleaf~enwiki, Lethe, Siroxo, Python eggs, Andycjp, Crawdaddio, Sam Hocevar, Marcos,Mubor, David Sneek, EugeneZelenko, Discospinster, Zaheen, Vsmith, Mani1, Harriv, Paul August, Bender235, *drew, Rgdboer, RandallHolmes, Mike Schwartz, Giraffedata, Obradovic Goran, Msh210, Tablizer, Pinar, Max Naylor, Dirac1933, Itsmine, Feezo, Slgrand-son, Chun-hian, Mendaliv, Search4Lancer, Rjwilmsi, Koavf, Salix alba, R.e.b., JonnyR, Gurch, BMF81, Chobot, Pip2andahalf, RussBot,CarlHewitt, Rick Norwood, Joth, Dysmorodrepanis~enwiki, Trovatore, PrologFan, Jpbowen, Crasshopper, Wheelybrook, Haemo, Googl,Arthur Rubin, Modify, Sardanaphalus, SmackBot, Unschool, Jagged 85, Brick Thrower, Skizzik, Dugas, Jprg1966, MalafayaBot, DarthPanda, MyNameIsVlad, Nick Levine, Glloq, KevM, Lesnail, Grover cleveland, Jiddisch~enwiki, Chrylis, Wvbailey, JorisvS, Rundquist,Bjankuloski06en~enwiki, Jim.belk, Dfass, Mr. Vernon, Loadmaster, Mets501, Iridescent, Quantum Burrito, Clarityfiend, Tophtucker,RekishiEJ, Supertigerman, Mjohnrussell, Spdegabrielle, CRGreathouse, CBM, MarsRover, Gregbard, Dragonflare82, Thijs!bot, J. W.Love, RichardVeryard, Klausness, Urdutext, Escarbot, AntiVandalBot, Emeraldcityserendipity, LibLord, Salgueiro~enwiki, Perelaar,Knotwork, JAnDbot, MER-C, Magioladitis, Bongwarrior, Soulbot, Animum, David Eppstein, Kope, Khalid Mahmood, Musictheorist,Pharaoh of the Wizards, Maurice Carbonaro, NewEnglandYankee, Policron, Cometstyles, Tellerman, DorganBot, Treisijs, Idioma-bot,Alan U. Kennington, ABF, JohnBlackburne, LokiClock, AlnoktaBOT, TXiKiBoT, Red Act, Rei-bot, Anonymous Dissident, Saibod,Digby Tantrum, Kmitchell19, Synthebot, VanishedUserABC, AlleborgoBot, Jtarr, YohanN7, Dogah, SieBot, BotMultichill, Gerakibot,
146 CHAPTER 26. WELL-ORDER
Viskonsas, Disneyfreek, Mnlkpo, MacPerry, Billyg, Yoda of Borg, CBM2, Neurophysics, B J Bradford, ClueBot, Smithpith, Cliff, Drag-onBot, BearMachine, Cenarium, Njardarlogar, 7&6=thirteen, Hans Adler, Palnot, BodhisattvaBot, JinJian, Multipundit, Deineka, Addbot,Manuel Trujillo Berges, Blake de doosten, Betterusername, Renamed user 5, Topology Expert, Ronhjones, NjardarBot, MrOllie, Down-load, Dyaa, Ozob, Bob K31416, Numbo3-bot, Tide rolls, TeH nOmInAtOr, Rrmsjp, DaveChild, Ben Ben, Legobot, Tartarus, Yobot, Frag-gle81, TaBOT-zerem, Pcap, Synchronism, Galoubet, Faizulhasan, OllieFury, Tbvdm, Xqbot, RJGray, Qualitydemise, Solphusion~enwiki,Point-set topologist, WissensDürster, La Mejor Ratonera, IShadowed, Red van man, Joxemai, Bekus, FrescoBot, Tobby72, Mark Re-nier, Dagme, Citation bot 1, FirmBenevolence, Tkuvho, Boulaur, Jschnur, Foobarnix, MPeterHenry, JumpDiscont, DARTH SIDIOUS2, EmausBot, Set theorist, Jmencisom, Kimartz, Wikipelli, Midas02, Wayne Slam, Computationalverb, ChuispastonBot, JonRichfield,ClueBot NG, Wcherowi, SusikMkr, Movses-bot, The Master of Mayhem, MerlIwBot, Ftonti, Helpful Pixie Bot, MKar, Hallows AG,Xosé Antonio, Ingmar.lippert, Knwlgc, Nisse Phelsum, YatharthROCK, Brad7777, Sofia karampataki, Spasoev, Justincheng12345-bot,Alfasst, Deltahedron, Nandanchoudhary05, Mark viking, Flimflam97, William2001, Howicus, Tentinator, NeapleBerlina, SakeUPenn,K401sTL3, Mario Castelán Castro, Galileo9, Sashasct, Fahim Of Wiki and Anonymous: 209
• Surjective function Source: https://en.wikipedia.org/wiki/Surjective_function?oldid=670090862Contributors: AxelBoldt, Tarquin, Amil-lar, XJaM, Toby Bartels, Michael Hardy, Wshun, Pit~enwiki, Karada, Александър, Glenn, Jeandré du Toit, Hashar, Hawthorn, CharlesMatthews, Dysprosia, David Shay, Ed g2s, Phil Boswell, Aleph4, Robbot, Fredrik, Tobias Bergemann, Giftlite, Lethe, Jason Quinn,Jorge Stolfi, Matt Crypto, Keeyu, Rheun, MarkSweep, AmarChandra, Tsemii, TheObtuseAngleOfDoom, Vivacissamamente, Rich Farm-brough, Quistnix, Paul August, Bender235, Nandhp, Kevin Lamoreau, Larry V, Obradovic Goran, Dallashan~enwiki, ABCD, Schapel,Oleg Alexandrov, Tbsmith, Mindmatrix, LOL, Rjwilmsi, MarSch, FlaBot, Chobot, Manscher, Algebraist, Angus Lepper, Ksnortum,Rick Norwood, Sbyrnes321, SmackBot, Rotemliss, Bluebot, TedE, Soapergem, Dreadstar, Saippuakauppias, MickPurcell, 16@r, Inquisi-tus, CBM, MatthewMain, Gregbard, Marqueed, Sam Staton, Pjvpjv, Prolog, Salgueiro~enwiki, JAnDbot, JamesBWatson, JJ Harrison,Martynas Patasius, MartinBot, TechnoFaye, Malerin, Dubhe.sk, Theabsurd, UnicornTapestry, Eliuha gmail.com, Anonymous Dissident,SieBot, SLMarcus, Paolo.dL, Peiresc~enwiki, Classicalecon, UKoch, Watchduck, Bender2k14, SchreiberBike, Neuralwarp, Petru Dim-itriu, Matthieumarechal, Kal-El-Bot, Addbot, Download, PV=nRT, ,ماني Zorrobot, Jarble, Legobot, Luckas-bot, Yobot, Fraggle81, IIMusLiM HyBRiD II, Xqbot, TechBot, Shvahabi, Raffamaiden, Omnipaedista, Applebringer, Erik9bot, LucienBOT, Tbhotch, Xnn, Jowafan, EmausBot, PrisonerOfIce, WikitanvirBot, GoingBatty, Sasuketiimer, Maschen, Mjbmrbot, Anita5192, ClueBot NG, Helpful PixieBot, BG19bot, Cispyre, Lfahlberg, JPaestpreornJeolhlna, TranquilHope and Anonymous: 87
• Topological space Source: https://en.wikipedia.org/wiki/Topological_space?oldid=669621236Contributors: AxelBoldt, Zundark, XJaM,Toby Bartels, Olivier, Patrick, Michael Hardy, Wshun, Kku, Dineshjk, Karada, Hashar, Zhaoway~enwiki, Revolver, Charles Matthews,Dcoetzee, Dysprosia, Kbk, Taxman, Phys, Robbot, Nizmogtr, Fredrik, Saaska, MathMartin, P0lyglut, Tobias Bergemann, Giftlite, GeneWard Smith, Lethe, Fropuff, Dratman, DefLog~enwiki, Rhobite, Luqui, Paul August, Dolda2000, Elwikipedista~enwiki, Tompw, Aude,SgtThroat, Tsirel, Marc van Woerkom, Varuna, Kuratowski’s Ghost, Msh210, Keenan Pepper, Danog, Sligocki, Spambit, Oleg Alexan-drov, Woohookitty, Graham87, BD2412, Grammarbot, FlaBot, Sunayana, Tillmo, Chobot, Algebraist, YurikBot, Wavelength, HairyDude, NawlinWiki, Rick Norwood, Bota47, Stefan Udrea, Hirak 99, Arthur Rubin, Lendu, JoanneB, Eaefremov, RonnieBrown, Sar-danaphalus, SmackBot, Maksim-e~enwiki, Sciyoshi~enwiki, DHN-bot~enwiki, Tsca.bot, Tschwenn, LkNsngth, Vriullop, Arialblack, Iri-descent, Devourer09, Mattbr, Andrew Delong, Kupirijo, Roccorossi, Xantharius, Thijs!bot, Konradek, Odoncaoa, Escarbot, Salgueiro~enwiki,JAnDbot, YK Times, Jakob.scholbach, Bbi5291, Wdevauld, J.delanoy, Pharaoh of the Wizards, Maurice Carbonaro, The Mudge, Jma-jeremy, Policron, TXiKiBoT, Anonymous Dissident, Plclark, Aaron Rotenberg, Jesin, Arcfrk, SieBot, MiNombreDeGuerra, JerroldPease-Atlanta, JackSchmidt, Failure.exe, Egmontaz, Palnot, SilvonenBot, Addbot, CarsracBot, AnnaFrance, ChenzwBot, Luckas-bot, Yobot,SwisterTwister, AnomieBOT, Ciphers, Materialscientist, Citation bot, DannyAsher, FlordiaSunshine342, J04n, Point-set topologist, Fres-coBot, Jschnur, Jeroen De Dauw, TobeBot, Seahorseruler, Skakkle, Cstanford.math, ZéroBot, Chharvey, Wikfr, Orange Suede Sofa,Liuthar, ClueBot NG, Wcherowi, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Gaurav Nirala, Tom.hirschowitz, Pacerier,Cpatra1984, Brad7777, Minsbot, LoganFromSA, MikeHaskel, Acer4666, Freeze S, Mark viking, Epicgenius, Kurt Artindagi, Improba-ble keeler, Amonk1962, KasparBot and Anonymous: 109
• Total order Source: https://en.wikipedia.org/wiki/Total_order?oldid=666410207 Contributors: Damian Yerrick, AxelBoldt, Zundark,XJaM, Fritzlein, Patrick, Michael Hardy, Dori, AugPi, Dysprosia, Jitse Niesen, Greenrd, Zoicon5, Hyacinth, VeryVerily, Fibonacci,McKay, Aleph4, Gandalf61, MathMartin, Rursus, Tobias Bergemann, Giftlite, Mshonle~enwiki, Markus Krötzsch, Lethe, Waltpohl,DefLog~enwiki, Alberto da Calvairate~enwiki, Quarl, Elroch, Paul August, Susvolans, Army1987, Func, Cmdrjameson, Msh210, Pion,Joriki, MattGiuca, Yurik, OneWeirdDude, Salix alba, VKokielov, Mathbot, Margosbot~enwiki, Wastingmytime, Chobot, YurikBot,Hede2000, Tetracube, Rdore, Melchoir, Gelingvistoj, Mhss, Chris the speller, Bazonka, Jdthood, Javalenok, Michael Kinyon, Loadmas-ter, Mets501, George100, CRGreathouse, CBM, Thomasmeeks, Oryanw~enwiki, VectorPosse, JAnDbot, David Eppstein, Infovarius,Osquar F, PaulTanenbaum, SieBot, Ceroklis, Anchor Link Bot, Heinzi.at, WurmWoode, Universityuser, Palnot, Marc van Leeuwen,Addbot, Tanhabot, AsphyxiateDrake, Luckas-bot, Yobot, Charlatino, White gecko, 1exec1, Infvwl, GrouchoBot, Jsjunkie, Quondum,D.Lazard, SporkBot, CocuBot, BG19bot, YumOooze, YFdyh-bot, Austinfeller, Mark viking, नितीश् चन्द्र and Anonymous: 47
• Transitive set Source: https://en.wikipedia.org/wiki/Transitive_set?oldid=659275410 Contributors: Edward, Charles Matthews, JitseNiesen, Tobias Bergemann, Lethe, EmilJ, Oleg Alexandrov, Salix alba, Arthur Rubin, Mhss, Keithdunwoody, JRSpriggs, Vaughan Pratt,CBM, Gregbard, Roches, Ttwo, Franklin.vp, Addbot, Barak Sh, Luckas-bot, Xqbot, Erik9bot, Tkuvho, Wikielwikingo, EmausBot,ZéroBot, Deltahedron, Pastisch and Anonymous: 13
• Trigonometric series Source: https://en.wikipedia.org/wiki/Trigonometric_series?oldid=667300975 Contributors: Ubiquity, MichaelHardy, Zhen Lin, PWilkinson, Oleg Alexandrov, Rjwilmsi, Lockley, R.e.b., InverseHypercube, Melchoir, Spireguy, BetacommandBot,Futurebird, Thenub314, R'n'B, Ideal gas equation, Alexbot, Kiensvay, Pichpich, Kal-El-Bot, Addbot, AnomieBOT, Rubinbot, FrescoBot,Trappist the monk, Chiranga87, Jcyangzh, Psgs123xyz, TwoTwoHello, Nigellwh and Anonymous: 15
• VonNeumann cardinal assignment Source: https://en.wikipedia.org/wiki/Von_Neumann_cardinal_assignment?oldid=607168257Con-tributors: The Anome, Dysprosia, Choni, Tobias Bergemann, Giftlite, Lethe, Fropuff, Sligocki, Oleg Alexandrov, Joriki, Btyner, Trova-tore, Arthur Rubin, SmackBot, RDBury, Mhss, Henning Makholm, Loadmaster, JRSpriggs, Yobot, RJGray, Auclairde, KLBot2 andAnonymous: 1
• Well-order Source: https://en.wikipedia.org/wiki/Well-order?oldid=662556656 Contributors: AxelBoldt, Mav, Zundark, Josh Grosse,Patrick, Michael Hardy, David Martland, Dominus, TakuyaMurata, Andres, Vargenau, Revolver, Charles Matthews, Timwi, Populus,Aleph4, R3m0t, MathMartin, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Ian Maxwell, Lethe, Arturus~enwiki, Jorend, Karl-Henner,Rich Farmbrough, Luqui, Paul August, Sligocki, RJFJR, Eyu100, Salix alba, FlaBot, Margosbot~enwiki, Chobot, YurikBot, Hairy Dude,Trovatore, Obey, Bota47, Arthur Rubin, MullerHolk, Ghazer~enwiki, GrinBot~enwiki, KnightRider~enwiki, Alan McBeth, Gelingvistoj,Mhss, Nbarth, Loodog, Jim.belk, Loadmaster, JRSpriggs, CBM, WeggeBot, Myasuda, Thijs!bot, Nadav1, Escarbot, Albmont, Odexios,
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