Finite setFrom Wikipedia, the free encyclopedia

Contents

1 Andrzej Mostowski 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3.1 Book by Mostowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Papers by Mostowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Axiom of determinacy 32.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 32.3 Innite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Classical mathematics 63.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Computable analysis 74.1 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Computable real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.2 Computable real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Constructive analysis 95.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.1 The intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.2 The least upper bound principle and compact sets . . . . . . . . . . . . . . . . . . . . . . 10

i

ii CONTENTS

5.1.3 Uncountability of the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Constructive proof 116.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.1.1 Non-constructive proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.1.2 Constructive proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.2 Brouwerian counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Counting 147.1 Forms of counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Inclusive counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Education and development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Counting in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Domain of discourse 198.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Universe of discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.4 Booles 1854 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Element (mathematics) 219.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10 Empty set 2410.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

CONTENTS iii

10.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11 Equiconsistency 3011.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

12 Existential quantication 3212.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.2.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.2.3 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

12.3 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13 First-order logic 3613.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 43

iv CONTENTS

13.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 4413.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

13.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

13.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

13.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14 Formal language 5714.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.2 Words over an alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.3 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

14.4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.5 Language-specication formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

CONTENTS v

14.6 Operations on languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

14.7.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.7.2 Formal theories, systems and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

14.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

14.9.1 Citation footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.9.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

14.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

15 Free variables and bound variables 6415.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

15.1.1 Variable-binding operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2 Formal explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

15.2.1 Function expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.3 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

16 G space 6916.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

17 Harvey Friedman 7017.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

18 Hereditary set 7218.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.2 In formulations of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

19 Injective function 7319.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7519.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

vi CONTENTS

19.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

20 Isolated point 8120.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

21 Lebesgue measure 8321.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

21.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8721.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

22 Logical conjunction 8822.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

22.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9022.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9022.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9122.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

23 Mathematical analysis 9423.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9523.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

23.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9623.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

23.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CONTENTS vii

23.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.3.4 Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9823.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

23.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9823.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

23.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

23.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10023.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

24 Natural language 10224.1 Dening natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10224.2 Native language learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10224.3 Origins of natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10324.4 Controlled languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10324.5 Constructed languages and international auxiliary languages . . . . . . . . . . . . . . . . . . . . . 10324.6 Modalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

24.6.1 Sign languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.6.2 Written languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

24.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

25 Peano axioms 10625.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

25.2.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.2.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

25.3 First-order theory of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.3.1 Equivalent axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

25.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.4.1 Nonstandard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.4.2 Set-theoretic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.4.3 Interpretation in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

25.5 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11225.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11325.7 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

viii CONTENTS

25.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

26 Per Lindstrm 11626.1 Selected publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

27 Perfect set 11727.1 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.2 Imperfection of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.3 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.4 Connection with other topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.5 Perfect spaces in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

28 Perfect set property 11928.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

29 Pigeonhole principle 12029.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

29.1.1 Sock-picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.1.2 Hand-shaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.1.3 Hair-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.1.4 The birthday problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.1.5 Softball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12229.1.6 Subset sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

29.2 Uses and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12229.3 Alternate formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12229.4 Strong form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.5 Generalizations of the pigeonhole principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.6 Innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

30 Predicate (mathematical logic) 12630.1 Simplied overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

CONTENTS ix

30.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12730.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

31 Primitive recursive arithmetic 12831.1 Language and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.2 Logic-free calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12931.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12931.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13031.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

32 Proof theory 13132.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13132.2 Formal and informal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13132.3 Kinds of proof calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.4 Consistency proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.5 Structural proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.6 Proof-theoretic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.7 Tableau systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.8 Ordinal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.9 Logics from proof analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

33 Property of Baire 13533.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13533.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13533.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

34 Quantier (linguistics) 13634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.2 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13734.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13734.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

35 Quantier (logic) 13835.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13935.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14035.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14035.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

x CONTENTS

35.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14135.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14435.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14435.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14435.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

36 Reverse mathematics 14636.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

36.1.1 Use of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636.2 The big ve subsystems of second order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 147

36.2.1 The base system RCA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14736.2.2 Weak Knigs lemma WKL0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14836.2.3 Arithmetical comprehension ACA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14936.2.4 Arithmetical transnite recursion ATR0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15036.2.5 11 comprehension 11-CA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

36.3 Additional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15036.4 -models and -models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15136.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15136.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

37 Second-order arithmetic 15237.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

37.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15237.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15337.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15337.1.4 The full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

37.2 Models of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15437.3 Denable functions of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15537.4 Subsystems of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

37.4.1 Arithmetical comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15537.4.2 The arithmetical hierarchy for formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15537.4.3 Recursive comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15637.4.4 Weaker systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15637.4.5 Stronger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

37.5 Projective Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15737.6 Coding mathematics in second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 15737.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15737.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

38 Semantics 159

CONTENTS xi

38.1 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15938.2 Montague grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.3 Dynamic turn in semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.4 Prototype theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16138.5 Theories in semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

38.5.1 Model theoretic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16138.5.2 Formal (or truth-conditional) semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16138.5.3 Lexical and conceptual semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16138.5.4 Lexical semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16138.5.5 Computational semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

38.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16238.6.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16238.6.2 Semantic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

38.7 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16338.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

38.8.1 Linguistics and semiotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16338.8.2 Logic and mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16438.8.3 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16438.8.4 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

38.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16538.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

39 Syntax 16639.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16639.2 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16639.3 Modern theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

39.3.1 Generative grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16739.3.2 Categorial grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16739.3.3 Dependency grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16839.3.4 Stochastic/probabilistic grammars/network theories . . . . . . . . . . . . . . . . . . . . . 16839.3.5 Functionalist grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

39.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16939.4.1 Syntactic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

39.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17339.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17339.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17339.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

40 Topology 17540.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17640.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17740.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

xii CONTENTS

40.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17940.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 18040.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

40.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18040.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18040.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18140.4.3 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18140.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18140.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

40.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18240.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18240.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18240.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18240.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

40.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18240.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18340.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18440.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

41 Universal quantication 18541.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

41.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18641.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

41.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18641.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18741.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18841.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

41.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18841.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18941.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18941.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18941.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

42 Urelement 19042.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19042.2 Urelements in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19042.3 Quine atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19142.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19142.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

43 Well-founded relation 19243.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

CONTENTS xiii

43.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19343.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19343.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19443.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

44 ZermeloFraenkel set theory 19544.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19544.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

44.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19644.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 19644.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19644.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19744.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19744.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19844.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19944.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19944.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

44.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20044.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

44.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20144.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20144.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20244.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20244.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20344.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 204

44.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20444.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21044.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Chapter 1

Andrzej Mostowski

Andrzej Mostowski (1 November 1913 22 August 1975) was a Polish mathematician. He is perhaps best remem-bered for the Mostowski collapse lemma.Born in Lemberg, Austria-Hungary, Mostowski entered University of Warsaw in 1931. He was inuenced byKuratowski, Lindenbaum and Tarski. His Ph.D. came in 1939, ocially directed by Kuratowski but in practicedirected by Tarski who was a young lecturer at that time.He became an accountant after the German invasion of Poland but continued working in the Underground WarsawUniversity. After the Warsaw uprising of 1944 the Nazis tried to put him in a concentration camp. With the help ofsome Polish nurses he escaped to a hospital, choosing to take bread with him rather than his notebook containing hisresearch. Some of this research he reconstructed after the War, however much of it remained lost.This work was largely on recursion theory and undecidability. From 1946 until his death in Vancouver, Canada, heworked at the University of Warsaw. Much of work during that time was on rst order logic and model theory.His son Tadeusz is also a mathematician working on dierential geometry.[1] With Krzysztof Kurdyka and AdamParusinski, Tadeusz Mostowski solved Ren Thom's gradient conjecture in 2000.

1.1 See also Mostowski model

1.2 References[1] http://www.mimuw.edu.pl/wydzial/organizacja/pracownicy/tadeusz.mostowski.xml?LANG=en&para=&parb=

1.3 Bibliography

1.3.1 Book by MostowskiAndrzej Mostowski, Sentences Undecidable in Formalized Arithmetic: An Exposition of the Theory of Kurt Godel,North-Holland, Amsterdam, 1952, ISBN 978-0313231513

1.3.2 Papers by Mostowski Andrzej Mostowski, "ber die Unabhngigkeit des Wohlordnungssatzes von Ordnungsprinzip. Fundamenta

Mathematicae Vol. 32, No.1, ss. 201-252, (1939).

Andrzej Mostowski, On denable sets of positive integers, Fundamenta Mathematicae Vol. 34, No. 1, ss.81-112, (1947).

1

2 CHAPTER 1. ANDRZEJ MOSTOWSKI

Andrzej Mostowski, Un thorme sur les nombres cos 2k/n, Colloquium Mathematicae Vol. 1, No. 3, ss.195-196, (1948).

Casimir Kuratowski, Andrzej Mostowski, Sur un problme de la thorie des groupes et son rapport latopologie, Colloquium Mathematicae Vol. 2, No. 3-4, ss. 212-215, (1951).

Andrzej Mostowski, Groups connected with Boolean algebras. (Partial solution of the problem P92)", Collo-quium Mathematicae Vol. 2, No. 3-4, ss. 216-219, (1951).

Andrzej Mostowski, On direct products of theories, Journal of Symbolic Logic, Vol. 17, No. 1, ss. 1-31,(1952).

Andrzej Mostowski, Models of axiomatic systems, Fundamenta Mathematicae Vol. 39, No. 1, ss. 133-158,(1952).

Andrzej Mostowski, On a system of axioms which has no recursively enumerable arithmetic model, Funda-menta Mathematicae Vol. 40, No. 1, ss. 56-61, (1953).

Andrzej Mostowski, A formula with no recursively enumerable model, Fundamenta Mathematicae Vol. 42,No. 1, ss. 125-140, (1955).

Andrzej Mostowski, Examples of sets denable by means of two and three quantiers, Fundamenta Mathe-maticae Vol. 42, No. 2, ss. 259-270, (1955).

Andrzej Mostowski, Contributions to the theory of denable sets and functions, Fundamenta MathematicaeVol. 42, No. 2, ss. 271-275, (1955).

Andrzej Ehrenfeucht, Andrzej Mostowski, Models of Axiomatic Theories Admitting Automorphisms, Fun-damenta Mathematicae, Vol. 43, No. 1, ss. 50-68 (1956).

Andrzej Mostowski, L'oeuvre scientique de Jan ukasiewicz dans le domaine de la logique mathmatique,Fundamenta Mathematicae Vol. 44, No. 1, ss. 1-11, (1957).

AndrzejMostowski, On a generalization of quantiers, FundamentaMathematicaeVol. 44, No. 1, ss. 12-36,(1957).

Andrzej Mostowski, On computable sequences, Fundamenta Mathematicae Vol. 44, No. 1, ss. 37-51,(1957).

Andrzej Grzegorczyk, AndrzejMostowski and Czesaw Ryll-Nardzewski, The classical and -complete arith-metic, Journal of Symbolic Logic Vol. 23, No. 2, ss. 188-206, (1958).

Andrzej Mostowski, On a problem of W. Kinna and K. Wagner, Colloquium Mathematicae Vol. 6, No. 1,ss. 207-208, (1958).

Andrzej Mostowski, A generalization of the incompleteness theorem, Fundamenta Mathematicae Vol. 49,No. 2, ss. 205-232, (1961).

Andrzej Mostowski, Axiomatizability of some many valued predicate calculi, Fundamenta MathematicaeVol. 50, No. 2, ss. 165-190, (1961).

Yoshindo Suzuki, Andrzej Mostowski, On -models which are not -models, Fundamenta MathematicaeVol. 65, No. 1, ss. 83-93, (1969).

1.4 External links Andrzej Mostowski at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., Andrzej Mostowski, MacTutor History of Mathematics archive,University of St Andrews.

Chapter 2

Axiom of determinacy

The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski andHugo Steinhaus in 1962. It refers to certain two-person games of length with perfect information. AD states thatevery such game in which both players choose natural numbers is determined; that is, one of the two players has awinning strategy.The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies thatall subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property.The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the samecardinality as the full set of reals).Furthermore, AD implies the consistency of ZermeloFraenkel set theory (ZF). Hence, as a consequence of theincompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

2.1 Types of game that are determinedNot all games require the axiom of determinacy to prove them determined. Games whose winning sets are closed aredetermined. These correspond to many naturally dened innite games. It was shown in 1975 by Donald A. Martinthat games whose winning set is a Borel set are determined. It follows from the existence of sucient large cardinalsthat all games with winning set a projective set are determined (see Projective determinacy), and that AD holds inL(R).

2.2 Incompatibility of the axiom of determinacy with the axiom of choiceThe set S1 of all rst player strategies in an -game G has the same cardinality as the continuum. The same is trueof the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in Gis also the continuum. Let A be the subset of SG of all sequences which make the rst player win. With the axiomof choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portiondoes not have the cardinality of the continuum. We create a counterexample by transnite induction on the set ofstrategies under this well ordering:We start with the set A undened. Let T be the time whose axis has length continuum. We need to consider allstrategies {s1(T)} of the rst player and all strategies {s2(T)} of the second player to make sure that for every strategythere is a strategy of the other player that wins against it. For every strategy of the player considered we will generatea sequence which gives the other player a win. Let t be the time whose axis has length 0 and which is used duringeach game sequence.

1. Consider the current strategy {s1(T)} of the rst player.2. Go through the entire game, generating (together with the rst players strategy s1(T)) a sequence {a(1), b(2),

a(3), b(4),...,a(t), b(t+1),...}.3. Decide that this sequence does not belong to A, i.e. s1(T) lost.

3

4 CHAPTER 2. AXIOM OF DETERMINACY

4. Consider the strategy {s2(T)} of the second player.

5. Go through the next entire game, generating (together with the second players strategy s2(T)) a sequence{c(1), d(2), c(3), d(4),...,c(t), d(t+1),...}, making sure that this sequence is dierent from {a(1), b(2), a(3),b(4),...,a(t), b(t+1),...}.

6. Decide that this sequence belongs to A, i.e. s2(T) lost.

7. Keep repeating with further strategies if there are any, making sure that sequences already considered do notbecome generated again. (We start from the set of all sequences and each time we generate a sequence andrefute a strategy we project the generated sequence onto rst player moves and onto second player moves, andwe take away the two resulting sequences from our set of sequences.)

8. For all sequences that did not come up in the above consideration arbitrarily decide whether they belong to A,or to the complement of A.

Once this has been done we have a game G. If you give me a strategy s1 then we considered that strategy at sometime T = T(s1). At time T, we decided an outcome of s1 that would be a loss of s1. Hence this strategy fails. Butthis is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.

2.3 Innite logic and the axiom of determinacyMany dierent versions of innitary logic were proposed in the late 20th century. One reason that has been given forbelieving in the axiom of determinacy is that it can be written as follows (in a version of innite logic):8G Seq(S) :8a 2 S : 9a0 2 S : 8b 2 S : 9b0 2 S : 8c 2 S : 9c0 2 S::: : (a; a0; b; b0; c; c0:::) 2 G OR9a 2 S : 8a0 2 S : 9b 2 S : 8b0 2 S : 9c 2 S : 8c0 2 S::: : (a; a0; b; b0; c; c0:::) /2 GNote: Seq(S) is the set of all ! -sequences of S. The sentences here are innitely long with a countably innite list ofquantiers where the ellipses appear.In an innitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantiersthat are true for nite formulas, such as 8a : 9b : 8c : 9d : R(a; b; c; d) OR 9a : 8b : 9c : 8d : :R(a; b; c; d) .

2.4 Large cardinals and the axiom of determinacyThe consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinalaxioms. By a theorem of Woodin, the consistency of ZermeloFraenkel set theory without choice (ZF) together withthe axiom of determinacy is equivalent to the consistency of ZermeloFraenkel set theory with choice (ZFC) togetherwith the existence of innitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD isconsistent, then so are an innity of inaccessible cardinals.Moreover, if to the hypothesis of an innite set of Woodin cardinals is added the existence of a measurable cardinallarger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable thatthe axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

2.5 See also Axiom of real determinacy (ADR) AD+, a variant of the axiom of determinacy formulated by Woodin Axiom of quasi-determinacy (ADQ) Martin measure

2.6. REFERENCES 5

2.6 References Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

Kanamori, Akihiro (2000). The Higher Innite (2nd ed.). Springer. ISBN 3-540-00384-3. Martin, Donald A.; Steel, John R. (Jan 1989). A Proof of Projective Determinacy. Journal of the American

Mathematical Society 2 (1): 71125. doi:10.2307/1990913. JSTOR 1990913. Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Mycielski, Jan; Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice. Bulletin

de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques 10: 13.ISSN 0001-4117. MR 0140430.

Woodin,W.Hugh (1988). Supercompact cardinals, sets of reals, andweakly homogeneous trees. Proceedingsof the National Academy of Sciences of theUnited States of America 85 (18): 65876591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

2.7 Further reading Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, Universityof Bonn, Germany, 2001

Telgrsky, R.J. Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J.Math. 17 (1987), pp. 227276. (3.19 MB)

Chapter 3

Classical mathematics

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

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

3.1 See also Constructivism (mathematics) Finitism Intuitionism Non-classical analysis Traditional mathematics Ultranitism Philosophy of Mathematics

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

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

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

6

Chapter 4

Computable analysis

In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspec-tive of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carriedout in a computable manner. The eld is closely related to constructive analysis and numerical analysis.

4.1 Basic constructions

4.1.1 Computable real numbersMain article: Computable number

Computable numbers are the real numbers that can be computed to within any desired precision by a nite, terminatingalgorithm. They are also known as the recursive numbers or the computable reals.

4.1.2 Computable real functionsMain article: Computable real function

A function f : R ! R is sequentially computable if, for every computable sequence fxig1i=1 of real numbers, thesequence ff(xi)g1i=1 is also computable.

4.2 Basic resultsThe computable real numbers form a real closed eld. The equality relation on computable real numbers is notcomputable, but for unequal computable real numbers the order relation is computable.Computable real functions map computable real numbers to computable real numbers. The composition of com-putable real functions is again computable. Every computable real function is continuous.

4.3 See also Specker sequence

4.4 References Oliver Aberth (1980), Computable analysis, McGraw-Hill, 1980.

7

8 CHAPTER 4. COMPUTABLE ANALYSIS

Marian Pour-El and Ian Richards, Computability in Analysis and Physics, Springer-Verlag, 1989. Stephen G. Simpson (1999), Subsystems of second-order arithmetic. Klaus Weihrauch (2000), Computable analysis, Springer, 2000.

4.5 External links Computability and Complexity in Analysis Network

Chapter 5

Constructive analysis

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructivemathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according tothe (ordinary) principles of classical mathematics.Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application toseparable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classicaltheorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms willbe valid in constructive analysis, which uses intuitionistic logic.

5.1 Examples

5.1.1 The intermediate value theorem

For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT says that, given anycontinuous function f from a closed interval [a,b] to the real line R, if f(a) is negative while f(b) is positive, thenthere exists a real number c in the interval such that f(c) is exactly zero. In constructive analysis, this does not hold,because the constructive interpretation of existential quantication (there exists) requires one to be able to constructthe real number c (in the sense that it can be approximated to any desired precision by a rational number). But if fhovers near zero during a stretch along its domain, then this cannot necessarily be done.However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to theusual form in classical analysis, but not in constructive analysis. For example, under the same conditions on f as inthe classical theorem, given any natural number n (no matter how large), there exists (that is, we can construct) a realnumber cn in the interval such that the absolute value of f(cn) is less than 1/n. That is, we can get as close to zero aswe like, even if we can't construct a c that gives us exactly zero.Alternatively, we can keep the same conclusion as in the classical IVT a single c such that f(c) is exactly zero while strengthening the conditions on f. We require that f be locally non-zero, meaning that given any point x in theinterval [a,b] and any natural number m, there exists (we can construct) a real number y in the interval such that |y -x| < 1/m and |f(y)| > 0. In this case, the desired number c can be constructed. This is a complicated condition, butthere are several other conditions which imply it and which are commonly met; for example, every analytic functionis locally non-zero (assuming that it already satises f(a) < 0 and f(b) > 0).For another way to view this example, notice that according to classical logic, if the locally non-zero condition fails,then it must fail at some specic point x; and then f(x) will equal 0, so that IVT is valid automatically. Thus inclassical analysis, which uses classical logic, in order to prove the full IVT, it is sucient to prove the constructiveversion. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does notaccept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is theconstructive version involving the locally non-zero condition, with the full IVT following by pure logic afterwards.Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach givesa better insight into the true meaning of theorems, in much this way.

9

10 CHAPTER 5. CONSTRUCTIVE ANALYSIS

5.1.2 The least upper bound principle and compact setsAnother dierence between classical and constructive analysis is that constructive analysis does not accept the leastupper bound principle, that any subset of the real line R has a least upper bound (or supremum), possibly innite.However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any locatedsubset of the real line has a supremum. (Here a subset S of R is located if, whenever x < y are real numbers, eitherthere exists an element s of S such that x < s, or y is an upper bound of S.) Again, this is classically equivalent to thefull least upper bound principle, since every set is located in classical mathematics. And again, while the denitionof located set is complicated, nevertheless it is satised by several commonly studied sets, including all intervals andcompact sets.Closely related to this, in constructive mathematics, fewer characterisations of compact spaces are constructivelyvalidor from another point of view, there are several dierent concepts which are classically equivalent but notconstructively equivalent. Indeed, if the interval [a,b] were sequentially compact in constructive analysis, then theclassical IVT would follow from the rst constructive version in the example; one could nd c as a cluster point ofthe innite sequence (cn)n.

5.1.3 Uncountability of the real numbersA constructive version of the famous theorem of Cantor, that the real numbers are uncountable is: Let {an} bea sequence of real numbers. Let x0 and y0 be real numbers, x0 < y0. Then there exists a real number x with x0 x y0 and x an (n Z+) . . . The proof is essentially Cantors 'diagonal' proof. (Theorem 1 in Errett Bishop,Foundations of Constructive Analysis, 1967, page 25.) It should be stressed that the constructive component of thediagonal argument already appeared in Cantors work.[1] According to Kanamori, a historical misrepresentation hasbeen perpetuated that associates diagonalization with non-constructivity.

5.2 References[1] Akihiro Kanamori, The Mathematical Development of Set Theory from Cantor to Cohen, Bulletin of Symbolic Logic /

Volume 2 / Issue 01 / March 1996, pp 1-71

5.3 See also Computable analysis Indecomposability

5.4 Further reading Bridger, Mark (2007). Real Analysis: A Constructive Approach. Hoboken: Wiley. ISBN 0-471-79230-6.

Chapter 6

Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical objectby creating or providing a method for creating the object. This is in contrast to a non-constructive proof (alsoknown as an existence proof or pure existence theorem) which proves the existence of a particular kind of objectwithout providing an example.Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently theproposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has beenaccepted in some varieties of constructive mathematics, including intuitionism.Constructivism is a mathematical philosophy that rejects all but constructive proofs in mathematics. This leads toa restriction on the proof methods allowed (prototypically, the law of the excluded middle is not accepted) and adierent meaning of terminology (for example, the term or has a stronger meaning in constructive mathematicsthan in classical).Constructive proofs can be seen as dening certied mathematical algorithms: this idea is explored in the BrouwerHeytingKolmogorov interpretation of constructive logic, the CurryHoward correspondence between proofs andprograms, and such logical systems as Per Martin-Lf's Intuitionistic Type Theory, and Thierry Coquand and GrardHuet's Calculus of Constructions.

6.1 Examples

6.1.1 Non-constructive proofs

First consider the theorem that there are an innitude of prime numbers. Euclid's proof is constructive. But acommon way of simplifying Euclids proof postulates that, contrary to the assertion in the theorem, there are onlya nite number of them, in which case there is a largest one, denoted n. Then consider the number n! + 1 (1 + theproduct of the rst n numbers). Either this number is prime, or all of its prime factors are greater than n. Withoutestablishing a specic prime number, this proves that one exists that is greater than n, contrary to the original postulate.Now consider the theorem There exist irrational numbers a and b such that ab is rational. This theorem can beproven using a constructive proof, or using a non-constructive proof.The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since atleast 1970:[1][2]

CURIOSA339. A Simple Proof That a Power of an Irrational Number to an Irrational Exponent May Be Rational.p2p2 is either rational or irrational. If it is rational, our statement is proved. If it is irrational, (

p2p2)p2 =

2 proves our statement.Dov Jarden Jerusalem

In a bit more detail:

11

12 CHAPTER 6. CONSTRUCTIVE PROOF

Recall that p2 is irrational, and 2 is rational. Consider the number q = p2p2 . Either it is rational or it is

irrational.

If q is rational, then the theorem is true, with a and b both beingp2 .

If q is irrational, then the theorem is true, with a beingp2p2 and b being

p2 , since

p2

p2p2

=p2(p2p2)

=p22= 2:

This proof is non-constructive because it relies on the statement Either q is rational or it is irrationalan instanceof the law of excluded middle, which is not valid within a constructive proof. The non-constructive proof doesnot construct an example a and b; it merely gives a number of possibilities (in this case, two mutually exclusivepossibilities) and shows that one of thembut does not show which onemust yield the desired example.

(It turns out thatp2p2 is irrational because of the GelfondSchneider theorem, but this fact is irrelevant to the

correctness of the non-constructive proof.)

6.1.2 Constructive proofsA constructive proof of the above theorem on irrational powers of irrationals would give an actual example, such as:

a =p2 ; b = log2 9 ; ab = 3 :

The square root of 2 is irrational, and 3 is rational. log2 9 is also irrational: if it were equal to mn , then, by theproperties of logarithms, 9n would be equal to 2m, but the former is odd, and the latter is even.A more substantial example is the graph minor theorem. A consequence of this theorem is that a graph can be drawnon the torus if, and only if, none of its minors belong to a certain nite set of "forbidden minors". However, the proofof the existence of this nite set is not constructive, and the forbidden minors are not actually specied. They are stillunknown.

6.2 Brouwerian counterexamplesIn constructive mathematics, a statement may be disproved by giving a counterexample, as in classical mathematics.However, it is also possible to give a Brouwerian counterexample to show that the statement is non-constructive.This sort of counterexample shows that the statement implies some principle that is known to be non-constructive.If it can be proved constructively that a statement implies some principle that is not constructively provable, thenthe statement itself cannot be constructively provable. For example, a particular statement may be shown to implythe law of the excluded middle. An example of a Brouwerian counterexample of this type is Diaconescus theorem,which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiomof choice implies the law of excluded middle in such systems. The eld of constructive reverse mathematics developsthis idea further by classifying various principles in terms of how nonconstructive they are, by showing they areequivalent to various fragments of the law of the excluded middle.Brouwer also provided weak counterexamples.[3] Such counterexamples do not disprove a statement, however; theyonly show that, at present, no constructive proof of the statement is known. One weak counterexample begins bytaking some unsolved problem of mathematics, such as Goldbachs conjecture. Dene a function f of a naturalnumber x as follows:

f(x) =

(0 if Goldbach's conjecture is false1 if Goldbach's conjecture is true

Although this is a denition by cases, it is still an admissible denition in constructivemathematics. Several facts aboutf can be proved constructively. However, based on the dierent meaning of the words in constructive mathematics,

6.3. SEE ALSO 13

if there is a constructive proof that "f(0) = 1 or f(0) 1 then this would mean that there is a constructive proof ofGoldbachs conjecture (in the former case) or a constructive proof that Goldbachs conjecture is false (in the lattercase). Because no such proof is known, the quoted statementmust also not have a known constructive proof. However,it is entirely possible that Goldbachs conjecture may have a constructive proof (as we do not know at present whetherit does), in which case the quoted statement would have a constructive proof as well, albeit one that is unknown atpresent. The main practical use of weak counterexamples is to identify the hardness of a problem. For example, thecounterexample just shown shows that the quoted statement is at least as hard to prove as Goldbachs conjecture.Weak counterexamples of this sort are often related to the limited principle of omniscience.

6.3 See also Existence theorem#'Pure' existence results Non-constructive algorithm existence proofs Errett Bishop - author of the book Foundations of Constructive Analysis.

6.4 References[1] J. Roger Hindley, The Root-2 Proof as an Example of Non-constructivity, unpublished paper, September 2014, full text

[2] Dov Jarden, A simple proof that a power of an irrational number to an irrational exponent may be rational, Curiosa No.339 in Scripta Mathematica 19:229 (1953)

[3] A. S. Troelstra, Principles of Intuitionism, Lecture Notes in Mathematics 95, 1969, p. 102

6.5 Further reading J. Franklin and A. Daoud (2011) Proof in Mathematics: An Introduction. Kew Books, ISBN 0-646-54509-4,ch. 4

Hardy, G.H. & Wright, E.M. (1979) An Introduction to the Theory of Numbers (Fifth Edition). Oxford Uni-versity Press. ISBN 0-19-853171-0

Anne Sjerp Troelstra and Dirk van Dalen (1988) Constructivism inMathematics: Volume 1 Elsevier Science.ISBN 978-0-444-70506-8

6.6 External links Weak counterexamples by Mark van Atten, Stanford Encyclopedia of Philosophy

Chapter 7

Counting

Counting is the action of nding the number of elements of a nite set of objects. The traditional way of countingconsists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order,while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarkedelements are left; if the counter was set to one after the rst object, the value after visiting the nal object gives thedesired number of elements. The related term enumeration refers to uniquely identifying the elements of a nite(combinatorial) set or innite set by assigning a number to each element.Counting sometimes involves numbers other than one; for example, when counting money, counting out change,counting by twos (2, 4, 6, 8, 10, 12, ...), or counting by ves (5, 10, 15, 20, 25, ...).There is archeological evidence suggesting that humans have been counting for at least 50,000 years.[1] Counting wasprimarily used by ancient cultures to keep track of social and economic data such as number of group members, preyanimals, property, or debts (i.e., accountancy). The development of counting led to the development of mathematicalnotation, numeral systems, and writing.

7.1 Forms of countingFurther information: Prehistoric numerals and Numerical digit

Counting can occur in a variety of forms.Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is oftenused to count objects that are present already, instead of counting a variety of things over time.Counting can also be in the form of tally marks, making a mark for each number and then counting all of the markswhen done tallying. This is useful when counting objects over time, such as the number of times something occursduring the course of a day. Tallying is base 1 counting; normal counting is done in base 10. Computers use base 2counting (0s and 1s).Counting can also be in the form of nger counting, especially when counting small numbers. This is often used bychildren to facilitate counting and simple mathematical operations. Finger-counting uses unary notation (one nger= one unit), and is thus limited to counting 10 (unless you start in with your toes). Older nger counting used the fourngers and the three bones in each nger (phalanges) to count to the number twelve.[2] Other hand-gesture systemsare also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By usingnger binary (base 2 counting), it is possible to keep a nger count up to 1023 = 210 1.Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.

7.2 Inclusive countingInclusive counting is usually encountered when dealing with time in the Romance languages.[3] Normally when count-ing 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day.

14

7.3. EDUCATION AND DEVELOPMENT 15

When counting inclusively, the Sunday (the start day) will be day 1 and therefore the following Sunday will be theeighth day. For example, the French phrase for fortnight is quinzaine (15 [days]), and similar words are presentin Greek (, dekapenthmero), Spanish (quincena) and Portuguese (quinzena). In contrast, the En-glish word fortnight itself derives from a fourteen-night, as the archaic "sennight" does from a seven-night"; theEnglish words are not examples of inclusive counting.Names based on inclusive counting appear in other calendars as well: in the Roman calendar the nones (meaningnine) is 8 days before the ides; and in the Christian calendar Quinquagesima (meaning 50) is 49 days before EasterSunday.Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one noteis a second interval, going up two notes is a third interval, etc., and going up seven notes is an octave.

7.3 Education and developmentMain article: Pre-math skills

Learning to count is an important educational/developmental milestone in most cultures of the world. Learningto count is a childs very rst step into mathematics, and constitutes the most fundamental idea of that discipline.However, some cultures in Amazonia and the Australian Outback do not count,[4][5] and their languages do not havenumber words.Many children at just 2 years of age have some skill in reciting the count list (i.e., saying one, two, three, ...). Theycan also answer questions of ordinality for small numbers, e.g., What comes after three?". They can even be skilledat pointing to each object in a set and reciting the words one after another. This leads many parents and educatorsto the conclusion that the child knows how to use counting to determine the size of a set.[6] Research suggests thatit takes about a year after learning these skills for a child to understand what they mean and why the procedures areperformed.[7][8] In the mean time, children learn how to name cardinalities that they can subitize.

7.4 Counting in mathematicsInmathematics, the essence of counting a set and nding a result n, is that it establishes a one to one correspondence (orbijection) of the set with the set of numbers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematicalinduction, is that no bijection can exist between {1, 2, ..., n} and {1, 2, ..., m} unless n = m; this fact (together withthe fact that two bijections can be composed to give another bijection) ensures that counting the same set in dierentways can never result in dierent numbers (unless an error is made). This is the fundamental mathematical theoremthat gives counting its purpose; however you count a (nite) set, the answer is the same. In a broader context, thetheorem is an example of a theorem in the mathematical eld of (nite) combinatoricshence (nite) combinatoricsis sometimes referred to as the mathematics of counting.Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural numbern; these are called innite sets, while those sets for which such a bijection does exist (for some n) are called nite sets.Innite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usualsense for nite sets are false for innite sets. Furthermore, dierent denitions of the concepts in terms of whichthese theorems are stated, while equivalent for nite sets, are inequivalent in the context of innite sets.The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with somewell understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then itis called "countably innite. This kind of counting diers in a fundamental way from counting of nite sets, in thatadding new elements to a set does not necessarily increase its size, because the possibility of a bijection with theoriginal set is not excluded. For instance, the set of all integers (including negative numbers) can be brought intobijection with the set of natural numbers, and even seemingly much larger sets like that of all nite sequences ofrational numbers are still (only) countably innite. Nevertheless, there are sets, such as the set of real numbers, thatcan be shown to be too large to admit a bijection with the natural numbers, and these sets are called "uncountable.Sets for which there exists a bijection between them are said to have the same cardinality, and in the most generalsense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of thenatural numbers, there is an innite hierarchy of innite cardinalities, although only very few such cardinalities occurin ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities).

16 CHAPTER 7. COUNTING

Counting, mostly of nite sets, has various applications in mathematics. One important principle is that if two setsX and Y have the same nite number of elements, and a function f: X Y is known to be injective, then it isalso surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets Xand Y have nite numbers of elements n and m with n > m, then any map f: X Y is not injective (so there existtwo distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if fwere injective, then so would its restriction to a strict subset S of X with m elements, which restriction would thenbe surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similarcounting arguments can prove the existence of certain objects without explicitly providing an example. In the case ofinnite sets this can even apply in situations where it is impossible to give an example; for instance there must existsreal nu