ordinal number 4

Ordinal numberFrom Wikipedia, the free encyclopedia

Contents

1 Bijection 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Cardinal number 72.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

i

ii CONTENTS

3 Derived set (mathematics) 153.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Topology in terms of derived sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 CantorBendixson rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Georg Cantor 174.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Youth and studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.2 Teacher and researcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.3 Late years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Mathematical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.1 Number theory, trigonometric series and ordinals . . . . . . . . . . . . . . . . . . . . . . 204.2.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Philosophy, religion and Cantors mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Cantors ancestry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Historiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Hereditary property 355.1 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.1 Monotone property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 In matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Injective function 386.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

CONTENTS iii

6.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Integer 467.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8 Natural number 528.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 558.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 55

8.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9 Order isomorphism 639.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.3 Order types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

iv CONTENTS

9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

10 Order type 6510.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

11 Ordinal number 6711.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6811.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

11.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 7011.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

11.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7111.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

11.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7111.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

11.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

11.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

11.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

12 Partially ordered set 7712.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7812.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7812.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7812.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 7912.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

CONTENTS v

12.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8012.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8012.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8012.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8112.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8112.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8312.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8312.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

13 Set theory 8413.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8513.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8613.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8713.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8713.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8813.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

13.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8913.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8913.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8913.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8913.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9013.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9013.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9013.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9013.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

13.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 9113.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9113.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9113.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9213.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

14 Surjective function 9314.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9414.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9414.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

14.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9514.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9614.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

vi CONTENTS

14.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 9714.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9714.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9714.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9814.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

15 Transitive set 9915.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9915.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9915.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9915.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9915.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10015.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10015.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

16 Trigonometric series 10116.1 The zeros of a trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10116.2 Zygmunds book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10116.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

16.3.1 Reviews of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10216.3.2 Publication history of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 102

17 Well-order 10317.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10317.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

17.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10417.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10417.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

17.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10517.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10517.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10617.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10617.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 107

17.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10717.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11117.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Chapter 1

Bijection

X 1

2

3

4

YD

B

C

A

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

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

1

2 CHAPTER 1. 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 setX to the set Y has an inverse function from Y toX. IfX and Y are nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.

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

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

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

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

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

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

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

1.2 Examples

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

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

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

2. No student was in more than one seat,

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

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

1.3. MORE MATHEMATICAL EXAMPLES AND SOME NON-EXAMPLES 3

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

1.3 More mathematical examples and some non-examples For any set X, the identity function 1X: X X, 1X(x) = x, is bijective. The function f: R R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

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

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

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

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

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

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

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

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


X1

2

3

YD

B

C

A

ZP

Q

R

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

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

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

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

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

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

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

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

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

1.9. GENERALIZATION TO PARTIAL FUNCTIONS 5

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

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

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

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

Multivalued function

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

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

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

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

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

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

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


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

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

1.14 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 theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 2

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 to

measure the cardinality (size) of sets. The cardinality of a nite set is a natural number: the number of elements inthe set. The transnite cardinal numbers describe the sizes of innite sets.Cardinality is dened 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 nite sets, this agrees withthe intuitive notion of size. In the case of innite sets, the behavior is more complex. A fundamental theorem due toGeorg Cantor shows that it is possible for innite sets to have dierent 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 innite set to have the same cardinality as the original set, something that cannot happen with propersubsets of nite sets.There is a transnite sequence of cardinal numbers:

0; 1; 2; 3; : : : ; n; : : : ;@0;@1;@2; : : : ;@; : : : :This sequence starts with the natural numbers including zero (nite cardinals), which are followed by the alephnumbers (innite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under theassumption of the axiom of choice, this transnite sequence includes every cardinal number. If one rejects thataxiom, the situation is more complicated, with additional innite 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.

2.1 HistoryThe notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 18741884. Cardinality can be used to compare an aspect of nite 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 innite 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 innite) sets and they all have the same cardinalnumber. This cardinal number is called@0 , aleph-null. He called the cardinal numbers of these innite sets, transnitecardinal 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 zmay be encoded as a nite sequence of integers which are the coecients

7

8 CHAPTER 2. 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 coecients (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 smallesttransnite cardinal number ( @0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal

(@1;@2;@3; ):

His continuum hypothesis is the proposition that c is the same as @1 . 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.

2.2. MOTIVATION 9

Aleph null, the smallest innite cardinal

2.2 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 identied with the natural numbers beginning with 0. The counting numbers are exactlywhat can be dened formally as the nite cardinal numbers. Innite 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 nite 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 , and we can construct the set {a,b,c} which has3 elements. However when dealing with innite sets it is essential to distinguish between the two the two notionsare in fact dierent for innite 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 denition 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 nite 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.


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 identies 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 innite 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 SchroederBernstein 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 numberofX itself is often dened as the least ordinal awith |a| = |X|. This is called the von Neumann cardinal assignment; forthis denition 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 innite hotel paradox, also called Hilberts paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an innite number of rooms. The hotel is full, and then a new guestarrives. It is possible to t 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 rst and the second has been shown. This motivates the denition of an innite 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 dened above for these innite sets. It happens that it doesn't; by considering the above example we cansee that if some object one greater than innity exists, then it must have the same cardinality as the innite setwe started out with. It is possible to use a dierent 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 nite 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 Cantors diagonal argument; classic questions of cardinality (for instance the continuumhypothesis) are concerned with discovering whether there is some cardinal between some pair of other innite 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.

2.3 Formal denitionFormally, assuming the axiom of choice, the cardinality of a set X is the least ordinal such that there is a bijectionbetween X and . This denition is known as the von Neumann cardinal assignment. If the axiom of choice is not

2.4. CARDINAL ARITHMETIC 11

assumed we need to do something dierent. The oldest denition 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 denition 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 dened as follows: |X| |Y | means that there exists an injectivefunction from X to Y. The CantorBernsteinSchroeder 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-innite if there exists a proper subset Y of X with |X| = |Y |, and Dedekind-nite if such a subsetdoesn't exist. The nite cardinals are just the natural numbers, i.e., a set X is nite if and only if |X| = |n| = n forsome natural number n. Any other set is innite. 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 @0 (aleph null or aleph-0, where alephis the rst letter in the Hebrew alphabet, represented @ ) of the set of natural numbers is the smallest innite cardinal,i.e. that any innite set has a subset of cardinality @0: The next larger cardinal is denoted by @1 and so on. For everyordinal there is a cardinal number @; and this list exhausts all innite cardinal numbers.

2.4 Cardinal arithmeticWe can dene arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers.It can be shown that for nite cardinals these operations coincide with the usual operations for natural numbers.Furthermore, these operations share many properties with ordinary arithmetic.

2.4.1 Successor cardinalFor more details on this topic, see Successor cardinal.

If the axiom of choice holds, every cardinal has a successor + > , and there are no cardinals between and itssuccessor. (Without the axiom of choice, using Hartogs theorem, it can be shown that, for any cardinal number ,there is a minimal cardinal +, so that + : ) For nite cardinals, the successor is simply + 1. For innitecardinals, the successor cardinal diers from the successor ordinal.

2.4.2 Cardinal additionIf 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}.

jXj+ jY j = jX [ Y j:Zero is an additive identity + 0 = 0 + = .Addition is associative ( + ) + = + ( + ).Addition is commutative + = + .Addition is non-decreasing in both arguments:

( ) ! ((+ + ) and ( + + )):Assuming the axiom of choice, addition of innite cardinal numbers is easy. If either or is innite, then

+ = maxf; g :


Subtraction

Assuming the axiom of choice and, given an innite cardinal and a cardinal , there exists a cardinal such that + = if and only if . It will be unique (and equal to ) if and only if < .

2.4.3 Cardinal multiplicationThe product of cardinals comes from the cartesian product.

jXj jY j = jX Y j0 = 0 = 0. = 0 ( = 0 or = 0).One is a multiplicative identity 1 = 1 = .Multiplication is associative () = ().Multiplication is commutative = .Multiplication is non-decreasing in both arguments: ( and ).Multiplication distributes over addition: ( + ) = + and ( + ) = + .Assuming the axiom of choice, multiplication of innite cardinal numbers is also easy. If either or is innite andboth are non-zero, then

= maxf; g:

Division

Assuming the axiom of choice and, given an innite cardinal and a non-zero cardinal , there exists a cardinal such that = if and only if . It will be unique (and equal to ) if and only if < .

2.4.4 Cardinal exponentiationExponentiation is given by

jXjjY j = XY where XY is the set of all functions from Y to X.

0 = 1 (in particular 00 = 1), see empty function.If 1 , then 0 = 0.1 = 1.1 = . + = . = ().() = .

Exponentiation is non-decreasing in both arguments:

(1 and ) ( ) and( ) ( ).

2.5. THE CONTINUUM HYPOTHESIS 13

Note that 2|X| is the cardinality of the power set of the set X and Cantors diagonal argument shows that 2|X| > |X| forany set X. This proves that no largest cardinal exists (because for any cardinal , we can always nd a larger cardinal2). 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 and are both nite and greater than 1, and is innite, then = .If is innite and is nite and non-zero, then = .

If 2 and 1 and at least one of them is innite, then:

Max (, 2) Max (2, 2).

Using Knigs theorem, one can prove < cf() and < cf(2) for any innite cardinal , where cf() is the conalityof .

Roots

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 0, the cardinal satisfying = will be .

Logarithms

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 1, there may ormay not be a cardinal satisfying = . However, if such a cardinal exists, it is innite and less than , and anynite cardinality greater than 1 will also satisfy = .The logarithm of an innite cardinal number is dened as the least cardinal number such that 2. Logarithmsof innite cardinals are useful in some elds 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]

2.5 The continuum hypothesisThe continuum hypothesis (CH) states that there are no cardinals strictly between @0 and 2@0 : The latter cardinalnumber is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case2@0 = @1: The generalized continuum hypothesis (GCH) states that for every innite 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).

2.6 See also

2.7 Notes

2.8 ReferencesNotes

[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): 123143. doi:10.1080/01445340903545904.

[3] Enderton, Herbert. Elements of Set Theory, Academic Press Inc., 1977. ISBN 0-12-238440-7


[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "ber das Problem derWohlordnung, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438443, 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 ech, Topological Spaces, revised by Zdenek Frolk 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 Innite, Princeton:Princeton University Press, ISBN 0691-02447-2

Hahn, Hans, Innity, 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).

2.9 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 3

Derived set (mathematics)

In mathematics, more specically 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 S0 .The concept was rst introduced by Georg Cantor in 1872 and he developed set theory in large part to study derivedsets on the real line.

3.1 PropertiesA subset S of a topological space is closed precisely when S0 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 others derived set (though thederived sets don't need to be disjoint from each other).The set S is dened to be a perfect set if S = S0 . 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 CantorBendixson theorem states that any Polish space can be written as the union of a countable set and aperfect set. Because any G subset of a Polish space is again a Polish space, the theorem also shows that any Gsubset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.

3.2 Topology in terms of derived setsBecause 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 2 S =) a 2 (S n fag)

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. ; = ;

15

16 CHAPTER 3. DERIVED SET (MATHEMATICS)

2. S S

3'. S = (S n fag)

4'. (S [ T ) = S [ T

Calling a set S closed if S S will dene a topology on the space in which * is the derived set operator, that is,S = S0 . 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.

3.3 CantorBendixson rankFor ordinal numbers , the -th CantorBendixson derivative of a topological space is dened by transnite in-duction as follows:

X0 = X X+1 = (X)0

X =\

Chapter 4

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (/kntr/ KAN-tor; German: [ek fdinant lutv flp kant];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, dened innite and well-ordered sets, and proved that the realnumbers are more numerous than the natural numbers. In fact, Cantors method of proof of this theorem impliesthe existence of an "innity of innities. He dened the cardinal and ordinal numbers and their arithmetic. Cantorswork is of great philosophical interest, a fact of which he was well aware.[2]

Cantors theory of transnite numbers was originally regarded as so counter-intuitive even shocking that it encoun-tered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincar[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 Cantors work as a challenge to the uniqueness of the absolute innity in the nature of God[6] on one occasion equating the theory of transnite numbers with pantheism[7] a proposition that Cantor vigorouslyrejected.The objections to Cantors work were occasionally erce: Poincar 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 scientic charlatan, a renegade and a corrupter of youth.[9] Kronecker objected to Cantors proofs thatthe algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included ina standard mathematics curriculum. Writing decades after Cantors death, Wittgenstein lamented that mathematicsis ridden through and through with the pernicious idioms of set theory, which he dismissed as utter nonsensethat is laughable and wrong.[10] Cantors 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]

4.1 Life

4.1.1 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 Bhm(17881846) (the violinist Joseph Bhm's brother) was a well-known musician and soloist in a Russian imperialorchestra.[16] Cantors father had been a member of the Saint Petersburg stock exchange; when he became ill, thefamily moved to Germany in 1856, rst 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 Zrich. Afterreceiving a substantial inheritance upon his fathers death in 1863, Cantor shifted his studies to the University of

17

18 CHAPTER 4. GEORG CANTOR

Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of1866 at the University of Gttingen, then and later a center for mathematical research.

4.1.2 Teacher and researcherCantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briey 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 headedmathematics 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 gure within the mathematicalcommunity and Cantors former professor, disagreed fundamentally with the thrust of Cantors work. Kronecker, nowseen as one of the founders of the constructive viewpoint in mathematics, dislikedmuch of Cantors set theory becauseit asserted the existence of sets satisfying certain properties, without giving specic examples of sets whose membersdid indeed satisfy those properties. Cantor came to believe that Kroneckers stance would make it impossible for himever to leave Halle.In 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantors suggestionthat it be oered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair afterbeing oered 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 Dedekinds declining the chair at Halle.[21] Cantor also began another important correspondence, with GstaMittag-Leer in Sweden, and soon began to publish in Mittag-Leers journal Acta Mathematica. But in 1885,Mittag-Leer 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 withMittag-Leer,writing to a third party:

Had Mittag-Leer 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 suered his rst known bout of depression in 1884.[24] Criticism of his work weighed on his mind: every oneof the fty-two letters he wrote to Mittag-Leer in 1884 mentioned Kronecker. A passage from one of these lettersis revealing of the damage to Cantors self-condence:

... I don't know when I shall return to the continuation of my scientic 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 scientically 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 187484.He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreementsand diculties dividing them persisted.In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its rst meeting inHalle in 1891, where he rst introduced his diagonal argument; his reputation was strong enough, despite Kroneckers

4.2. MATHEMATICAL WORK 19

opposition to his work, to ensure he was elected as the rst 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.

4.1.3 Late yearsAfter Cantors 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.[24] Soon afterthat second hospitalization, Cantors 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 Knig at the Third International Congress of Mathematicians. The paper attempted to provethat the basic tenets of transnite 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 Knigs proof had failed, Cantor remained shaken, and momentarily questioning God.[13]Cantor suered from chronic depression for the rest of his life, for which he was excused from teaching on severaloccasions and repeatedly conned 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, Cantors paradox, and Russells paradox) to a meeting of the DeutscheMathematikerVereinigung 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 Cantors 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 suering 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 nal year of his life.

4.2 Mathematical workCantors 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 onlynite sets (which are easy to understand) and the innite (which was considered a topic for philosophical, ratherthan mathematical, discussion). By proving that there are (innitely) many possible sizes for innite 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 rst time, that there exist innite sets of dierent sizes. Hewas also the rst to appreciatethe importance of one-to-one correspondences (hereinafter denoted 1-to-1 correspondence) in set theory. He usedthis concept to dene nite and innite sets, subdividing the latter into denumerable (or countably innite) sets anduncountable sets (nondenumerable innite 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 innite set; this result soon became known as Cantors theorem. Cantor developed an entire theory andarithmetic of innite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. Hisnotation for the cardinal numbers was the Hebrew letter @ (aleph) with a natural number subscript; for the ordinalshe employed the Greek letter (omega). This notation is still in use today.


The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the rst of his twenty-three openproblems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantors work alsoattracted favorable notice beyond Hilberts celebrated encomium.[15] The US philosopher Charles Sanders Peircepraised Cantors set theory, and, following public lectures delivered by Cantor at the rst 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 Cantors religious ideas. Thiswas later published, as were several of his expository works.

4.2.1 Number theory, trigonometric series and ordinals

Cantors rst 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 dicult problem in 1869. It was while working on this problemthat he discovered transnite 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 innite decreasing sequence ofsets S, S1, S2, S3,... formed a limit set, which we would now call S, and then he noticed that S would also haveto have a set of limit points S, and so on. He had examples that went on forever, and so here was a naturallyoccurring innite sequence of innite numbers , + 1, + 2, ...[36]

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper dening irrationalnumbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paperlater that year, in the paper where he rst set out his celebrated denition of real numbers by Dedekind cuts. Whileextending the notion of number bymeans of his revolutionary concept of innite cardinality, Cantor was paradoxicallyopposed to theories of innitesimals 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 innitesimals.[38]

4.2.2 Set theory

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantors 1874 article,[31]"ber eine Eigenschaft des Inbegries aller reellen algebraischen Zahlen ("On a Property of the Collection of AllReal Algebraic Numbers").[40] This article was the rst to provide a rigorous proof that there was more than one kindof innity. Previously, all innite 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 diers fromdiagonal argument that he gave in 1891.[42] Cantors article also contains a newmethod of constructing transcendentalnumbers. Transcendental numbers were rst constructed by Joseph Liouville in 1844.[43]

Cantor established these results using two constructions. His rst 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 Liouvilles theo-rem: Every interval contains innitely many transcendental numbers.[45] Cantors 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 inMathematische Annalen that together formed anintroduction to his set theory. At the same time, there was growing opposition to Cantors ideas, led by Kronecker,who admitted mathematical concepts only if they could be constructed in a nite number of steps from the naturalnumbers, which he took as intuitively given. For Kronecker, Cantors hierarchy of innities was inadmissible, sinceaccepting the concept of actual innity would open the door to paradoxes which would challenge the validity of

4.2. MATHEMATICAL WORK 21

mathematics as a whole.[47] Cantor also introduced the Cantor set during this period.The fth 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 Cantors reply to his critics and showed how the transnite numbers were a systematicextension of the natural numbers. It begins by dening well-ordered sets. Ordinal numbers are then introduced asthe order types of well-ordered sets. Cantor then denes 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 Cantors 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 innite sets, and of the cardinal and ordinalarithmetic that Cantor had dened. His argument is fundamental in the solution of the Halting problem and the proofof Gdels rst incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894.In 1895 and 1897, Cantor published a two-part paper inMathematische Annalen under Felix Klein's editorship; thesewere his last signicant papers on set theory.[49] The rst paper begins by dening 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 Schrder had stated this theorem a bit earlier, but his proof,as well as Cantors, was awed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the nameCantorBernsteinSchroeder theorem.

One-to-one correspondence

Main article: BijectionCantors 1874 Crelle paper was the rst 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 Crelles Journal, in which he dened 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 denedcountable 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 innite 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 rst 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 diculty Cantor had in proving the continuum hypothesis has been underscored by later developments in theeld of mathematics: a 1940 result by Gdel and a 1963 one by Paul Cohen together imply that the continuum


hypothesis can neither be proved nor disproved using standard ZermeloFraenkel 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 Cantors set theory program.[53] In an 1897 paper on an unrelated topic, CesareBurali-Forti set out the rst 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 Cantors 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 Russells 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]

4.3 Philosophy, religion and Cantors mathematicsThe concept of the existence of an actual innity 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 Cantors.[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 innite and the philosophical one.[57] To Cantor, his mathematicalviews were intrinsically linked to their philosophical and theological implications he identied the Absolute Innitewith God,[58] and he considered his work on transnite 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 innity. Some held to the view that innity was an abstraction which was not mathematically legitimate,and denied its existence.[59] Mathematicians from three major schools of thought (constructivism and its two o-shoots, intuitionism and nitism) opposed Cantors theories in this matter. For constructivists such as Kronecker,this rejection of actual innity stems from fundamental disagreement with the idea that nonconstructive proofs suchas Cantors diagonal argument are sucient proof that something exists, holding instead that constructive proofs arerequired. Intuitionism also rejects the idea that actual innity is an expression of any sort of reality, but arrive at thedecision via a dierent route than constructivism. Firstly, Cantors argument rests on logic to prove the existence oftransnite 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 innity asan expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an in-nite set.[60] Mathematicians such as Brouwer and especially Poincar adopted an intuitionist stance against Cantorswork. Citing the paradoxes of set theory as an example of its fundamentally awed nature, Poincar held that mostof the ideas of Cantorian set theory should be banished from mathematics once and for all.[8] Finally, Wittgenstein'sattacks were nitist: he believed that Cantors diagonal argument conated the intension of a set of cardinal or realnumbers with its extension, thus conating the concept of rules for generating a set with an actual set.[10]

Some Christian theologians saw Cantors work as a challenge to the uniqueness of the absolute innity in the nature ofGod.[6] In particular, Neo-Thomist thinkers saw the existence of an actual innity that consisted of something otherthan God as jeopardizing Gods exclusive claim to supreme innity.[61] Cantor strongly believed that this view was

4.4. CANTORS ANCESTRY 23

a misinterpretation of innity, and was convinced that set theory could help correct this mistake:[62]

... the transnite species are just as much at the disposal of the intentions of the Creator and Hisabsolute boundless will as are the nite numbers.[63]

Cantor also believed that his theory of transnite 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 transnite numbers with pantheism.[7]Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[62]

Cantors philosophy on the nature of numbers led him to arm 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 denitions, 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 ercely opposed to innitesimals, describing them as both an abomination and thecholera bacillus of mathematics.Cantors 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 innite 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 freely

ordinal number 4

Documents

external links

number theory

algebraic properties

cardinal number

cardinal addition

cardinal multiplication

model theory

graph theory