UrelementFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of determinacy 11.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 11.3 Innite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Classical mathematics 42.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Computable analysis 53.1 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Computable real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.2 Computable real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Constructive analysis 74.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 The intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.2 The least upper bound principle and compact sets . . . . . . . . . . . . . . . . . . . . . . 84.1.3 Uncountability of the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Constructive proof 95.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.1 Non-constructive proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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5.1.2 Constructive proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Brouwerian counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Domain of discourse 126.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 Universe of discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.4 Booles 1854 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Element (mathematics) 147.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Empty set 178.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Equiconsistency 239.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 First-order logic 2510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 3210.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 3310.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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10.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

11 G space 4611.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12 Harvey Friedman 4712.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13 Hereditary set 4913.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2 In formulations of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

14 Isolated point 5014.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15 Lebesgue measure 5215.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16 Mathematical analysis 5716.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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16.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

16.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3.4 Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

16.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

17 Peano axioms 6517.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

17.2.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.2.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

17.3 First-order theory of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.3.1 Equivalent axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

17.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7017.4.1 Nonstandard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7017.4.2 Set-theoretic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7017.4.3 Interpretation in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

17.5 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7217.7 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7217.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7317.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

18 Perfect set 7518.1 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.2 Imperfection of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.4 Connection with other topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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18.5 Perfect spaces in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

19 Perfect set property 7719.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

20 Primitive recursive arithmetic 7820.1 Language and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7820.2 Logic-free calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7920.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7920.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

21 Proof theory 8121.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.2 Formal and informal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.3 Kinds of proof calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.4 Consistency proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.5 Structural proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.6 Proof-theoretic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.7 Tableau systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.8 Ordinal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.9 Logics from proof analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

22 Property of Baire 8522.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8522.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8522.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

23 Quantier (logic) 8623.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8923.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8923.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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23.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9223.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9223.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9223.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

24 Reverse mathematics 9424.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

24.1.1 Use of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.2 The big ve subsystems of second order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 95

24.2.1 The base system RCA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9524.2.2 Weak Knigs lemma WKL0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.2.3 Arithmetical comprehension ACA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9724.2.4 Arithmetical transnite recursion ATR0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.2.5 11 comprehension 11-CA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

24.3 Additional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.4 -models and -models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

25 Second-order arithmetic 10025.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

25.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.1.4 The full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

25.2 Models of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225.3 Denable functions of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.4 Subsystems of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

25.4.1 Arithmetical comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.4.2 The arithmetical hierarchy for formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.4.3 Recursive comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.4.4 Weaker systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.4.5 Stronger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

25.5 Projective Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.6 Coding mathematics in second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

26 Topology 10726.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10826.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10926.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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26.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 11226.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

26.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.4.3 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

26.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

26.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

27 Urelement 11727.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.2 Urelements in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.3 Quine atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

28 Well-founded relation 11928.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11928.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12028.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12028.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12128.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

29 ZermeloFraenkel set theory 12229.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12229.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

29.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 12329.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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29.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12629.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12629.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

29.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

29.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13029.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 131

29.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13129.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13429.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 1

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.

1.1 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).

1.2 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.

1

2 CHAPTER 1. 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.

1.3 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) .

1.4 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.

1.5 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

1.6. REFERENCES 3

1.6 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 AmericanMathematical 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. Bulletinde 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.

1.7 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 2

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.

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

2.2 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.

4

Chapter 3

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.

3.1 Basic constructions

3.1.1 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.

3.1.2 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.

3.2 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.

3.3 See also Specker sequence

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

5

6 CHAPTER 3. 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.

3.5 External links Computability and Complexity in Analysis Network

Chapter 4

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.

4.1 Examples

4.1.1 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.

7

8 CHAPTER 4. CONSTRUCTIVE ANALYSIS

4.1.2 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.

4.1.3 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.

4.2 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

4.3 See also Computable analysis Indecomposability

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

Chapter 5

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.

5.1 Examples

5.1.1 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:

9

10 CHAPTER 5. 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.)

5.1.2 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.

5.2 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,

5.3. SEE ALSO 11

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.

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

5.4 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

5.5 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

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

Chapter 6

Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simplyuniverse, is the set of entities over which certain variables of interest in some formal treatment may range.

6.1 OverviewThe domain of discourse is usually identied in the preliminaries, so that there is no need in the further treatment tospecify each time the range of the relevant variables.[1] Many logicians distinguish, sometimes only tacitly, betweenthe domain of a science and the universe of discourse of a formalization of the science.[2] Giuseppe Peanoformalized number theory (arithmetic of positive integers) taking its domain to be the positive integers and theuniverse of discourse to include all individuals not just integers.

6.2 ExamplesFor example, in an interpretation of rst-order logic, the domain of discourse is the set of individuals that thequantiers range over. In one interpretation, the domain of discourse could be the set of real numbers; in anotherinterpretation, it could be the set of natural numbers. If no domain of discourse has been identied, a propositionsuch as x (x2 2) is ambiguous. If the domain of discourse is the set of real numbers, the proposition is false, withx = 2 as counterexample; if the domain is the set of naturals, the proposition is true, since 2 is not the square of anynatural number.

6.3 Universe of discourseThe term universe of discourse generally refers to the collection of objects being discussed in a specic discourse.In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The conceptuniverse of discourse is generally attributed to Augustus De Morgan (1846) but the name was used for the rst timein history by George Boole (1854) on page 42 of his Laws of Thought in a long and incisive passage well worthstudy. Booles denition is quoted below. The concept, probably discovered independently by Boole in 1847, playeda crucial role in his philosophy of logic especially in his stunning principle of wholistic reference.A database is a model of some aspect of the reality of an organisation. It is conventional to call this reality theuniverse of discourse or domain of discourse.

6.4 Booles 1854 denitionIn every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercoursewith others, there is an assumed or expressed limit within which the subjects of its operation are conned. The mostunfettered discourse is that in which the words we use are understood in the widest possible application, and for them

12

6.5. SEE ALSO 13

the limits of discourse are co-extensive with those of the universe itself. But more usually we conne ourselves to aless spacious eld. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of menonly under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life,or of men under some other condition or relation. Now, whatever may be the extent of the eld within which all theobjects of our discourse are found, that eld may properly be termed the universe of discourse. Furthermore, thisuniverse of discourse is in the strictest sense the ultimate subject of the discourse.[3]

6.5 See also Domain of a function Domain theory Interpretation (logic) Term algebra Universe (mathematics)

6.6 References[1] Corcoran, John. Universe of discourse. Cambridge Dictionary of Philosophy, Cambridge University Press, 1995, p. 941.

[2] Jos Miguel Sagillo, Domains of sciences, universe of discourse, and omega arguments, History and philosophy of logic,vol. 20 (1999), pp. 267280.

[3] George Boole. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Bualo:Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review.24 (2004) 167169.

Chapter 7

Element (mathematics)

In mathematics, an element, ormember, of a set is any one of the distinct objects that make up that set.

7.1 SetsWriting A = {1, 2, 3, 4} means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A,for example {1, 2}, are subsets of A.Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3,and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colorsred, green and blue.

7.2 Notation and terminology

First usage of the symbol in the work Arithmetices principia nova methodo exposita by Giuseppe Peano.

The relation is an element of, also called set membership, is denoted by the symbol "". Writing

x 2 A

means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A"and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, howeversome authors use them to mean instead "x is a subset of A".[1] Logician George Boolos strongly urged that containsbe used for membership only and includes for the subset relation only.[2]

Another possible notation for the same relation is

14

7.3. CARDINALITY OF SETS 15

A 3 x;meaning "A contains x", though it is used less often.The negation of set membership is denoted by the symbol "". Writing

x /2 Ameans that "x is not an element of A".The symbol was rst used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita. Herehe wrote on page X:

Signum signicat est. Ita a b legitur a est quoddam b; ...

which means

The symbol means is. So a b has to be read as a is a b; ...

Thereby is a derivation from the lowercase Greek letter epsilon ("") and shall be the rst letter of the word ,which means is.The Unicode characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209('not an element of'). The equivalent LaTeX commands are "\in, "\ni and "\notin. Mathematica has commands"\[Element]" and "\[NotElement]".

7.3 Cardinality of setsMain article: Cardinality

The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set. Inthe above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3. An inniteset is a set with an innite number of elements, while a nite set is a set with a nite number of elements. The aboveexamples are examples of nite sets. An example of an innite set is the set of positive integers = { 1, 2, 3, 4, ... }.

7.4 ExamplesUsing the sets dened above, namely A = {1, 2, 3, 4 }, B = {1, 2, {3, 4}} and C = { red, green, blue }:

2 A {3,4} B {3,4} is a member of B Yellow C The cardinality of D = { 2, 4, 8, 10, 12 } is nite and equal to 5. The cardinality of P = { 2, 3, 5, 7, 11, 13, ...} (the prime numbers) is innite (this was proven by Euclid).

7.5 References[1] Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12[2] George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture). (Speech). Massachusetts Institute of Technology,

Cambridge, MA.

16 CHAPTER 7. ELEMENT (MATHEMATICS)

7.6 Further reading Halmos, Paul R. (1974) [1960], Naive Set Theory, Undergraduate Texts in Mathematics (Hardcover ed.), NY:Springer-Verlag, ISBN 0-387-90092-6 - Naive means that it is not fully axiomatized, not that it is silly oreasy (Halmoss treatment is neither).

Jech, Thomas (2002), Set Theory, Stanford Encyclopedia of Philosophy Suppes, Patrick (1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., ISBN 0-486-61630-4 -Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, theaxiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thoroughunderstanding of set element.

7.7 External links Weisstein, Eric W., Element, MathWorld.

Chapter 8

Empty set

"" redirects here. For similar symbols, see (disambiguation).In mathematics, and more specically set theory, the empty set is the unique set having no elements; its size or

cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for empty set, but is now a technical term in measure theory. The empty setmay also be called the void set.

8.1 NotationCommon notations for the empty set include "{}", "", and " ; ". The latter two symbols were introduced by theBourbaki group (specically Andr Weil) in 1939, inspired by the letter in the Norwegian and Danish alphabets(and not related in any way to the Greek letter ).[1]

The empty-set symbol is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

8.2 PropertiesIn standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of the emptyset rather than an empty set.The mathematical symbols employed below are explained here.For any set A:

The empty set is a subset of A:8A : ; A

The union of A with the empty set is A:8A : A [ ; = A

The intersection of A with the empty set is the empty set:8A : A \ ; = ;

The Cartesian product of A and the empty set is the empty set:8A : A ; = ;

17

18 CHAPTER 8. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

Its only subset is the empty set itself:8A : A ; ) A = ;

The power set of the empty set is the set containing only the empty set:2; = f;g

8.2. PROPERTIES 19

A symbol for the empty set

Its number of elements (that is, its cardinality) is zero:card(;) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic denition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

For every element of ; the property holds (vacuous truth); There is no element of ; for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

For every element of V the property holds; There is no element of V for which the property holds,

V = ;

20 CHAPTER 8. EMPTY SET

By the denition of subset, the empty set is a subset of any set A, as every element x of ; belongs to A. If it is nottrue that every element of ; is in A, there must be at least one element of ; that is not present in A. Since there areno elements of ; at all, there is no element of ; that is not in A. Hence every element of ; is in A, and ; is a subsetof A. Any statement that begins for every element of ; " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as everything is true of the elements of the empty set.

8.2.1 Operations on the empty setOperations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

8.3 In other areas of mathematics

8.3.1 Extended real numbersSince the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two numbersor points to the real numbers, namely negative innity, denoted 1; which is dened to be less than every otherextended real number, and positive innity, denoted +1; which is dened to be greater than every other extendedreal number, then:

sup ; = min(f1;+1g [ R) = 1;

and

inf ; = max(f1;+1g [ R) = +1:

That is, the least upper bound (sup or supremum) of the empty set is negative innity, while the greatest lower bound(inf or inmum) is positive innity. By analogy with the above, in the domain of the extended reals, negative innityis the identity element for the maximum and supremum operators, while positive innity is the identity element forminimum and inmum.

8.3.2 TopologyConsidered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a clopen set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every nite set is compact.The closure of the empty set is empty. This is known as preservation of nullary unions.

8.3.3 Category theoryIf A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

8.4. QUESTIONED EXISTENCE 21

The empty set can be turned into a topological space, called the empty space, in just one way: by dening the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

8.4 Questioned existence

8.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

8.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bagan empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements Nothing is better than eternal happiness and "[A]ham sandwich is better than nothing in a mathematical tone. According to Darling, the former is equivalent to Theset of all things that are better than eternal happiness is ; " and the latter to The set {ham sandwich} is better thanthe set ; ". It is noted that the rst compares elements of sets, while the second compares the sets themselves.[4]Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, we should not assume thatits utility in calculation is dependent upon its actually denoting some object.

it is also the case that:

All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers, in the set-theoretical sensenamely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantication over individuals, without reifying sets as singular entities having other entities as members.[6]

22 CHAPTER 8. EMPTY SET

8.5 See also Inhabited set Nothing

8.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] George Boolos, 1984, To be is to be the value of a variable, The Journal of Philosophy 91: 43049. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jerey, and Burgess, J., eds.) Harvard Univ. Press: 5472.

8.7 References 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). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

8.8 External links Weisstein, Eric W., Empty Set, MathWorld.

Chapter 9

Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of theother theory, and vice versa. In this case, they are, roughly speaking, as consistent as each other.In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believedto be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistentif wecan do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T areequiconsistent.

9.1 ConsistencyIn mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enoughto model dierent mathematical objects, it is natural to wonder about their own consistency.Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematicalmethods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, theprogram quickly became the establishment of the consistency of arithmetic bymethods formalizable within arithmeticitself.Gdel's incompleteness theorems show that Hilberts program cannot be realized: If a consistent recursively enumer-able theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strongenough to model a weak fragment of arithmetic (Robinson arithmetic suces), then the theory cannot prove its ownconsistency. There are some technical caveats as to what requirements the formal statement representing the meta-mathematical statement The theory is consistent needs to satisfy, but the outcome is that if a (suciently strong)theory can prove its own consistency then either there is no computable way of identifying whether a statement is evenan axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, includingfalse statements such as its own consistency).Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories.Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Twotheories are equiconsistent if each one is consistent relative to the other.

9.2 Consistency strengthIf T is consistent relative to S, but S is not known to be consistent relative to T, then we say that S has greaterconsistency strength than T. When discussing these issues of consistency strength the metatheory in which thediscussion takes places needs to be carefully addressed. For theories at the level of second-order arithmetic, thereverse mathematics program has much to say. Consistency strength issues are a usual part of set theory, since thisis a recursive theory that can certainly model most of mathematics. The usual set of axioms of set theory is calledZFC. When a set theoretic statement A is said to be equiconsistent to another B, what is being claimed is that in themetatheory (Peano Arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent.Usually, primitive recursive arithmetic can be adopted as the metatheory in question, but even if the metatheory is

23

24 CHAPTER 9. EQUICONSISTENCY

ZFC (for Ernst Zermelo and Abraham Fraenkel with Zermelos axiom of choice) or an extension of it, the notion ismeaningful. Thus, the method of forcing allows one to show that the theories ZFC, ZFC+CH and ZFC+CH are allequiconsistent.When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, orZF+AD, set theory with the axiom of determinacy), the notions described above are adapted accordingly. Thus, ZFis equiconsistent with ZFC, as shown by Gdel.The consistency strength of numerous combinatorial statements can be calibrated by large cardinals. For example,the negation of Kurepas hypothesis is equiconsistent with an inaccessible cardinal, the non-existence of special !2 -Aronszajn trees is equiconsistent with aMahlo cardinal, and the non-existence of!2 -Aronszajn trees is equiconsistentwith a weakly compact cardinal.[1]

9.3 See also Large cardinal property

9.4 References[1] Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, p. 225, ISBN 978-1-84890-

050-9, Zbl 1262.03001

Akihiro Kanamori (2003). The Higher Innite. Springer. ISBN 3-540-00384-3

Chapter 10

First-order logic

First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is alsoknown as rst-order predicate calculus, the lower predicate calculus, quantication theory, and predicate logic.First-order logic uses quantied variables over (non-logical) objects. This distinguishes it from propositional logicwhich does not use quantiers.A theory about some topic is usually rst-order logic together with a specied domain of discourse over which thequantied variables range, nitelymany functions whichmap from that domain into it, nitelymany predicates denedon that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes theory isunderstood in a more formal sense, which is just a set of sentences in rst-order logic.The adjective rst-order distinguishes rst-order logic from higher-order logic in which there are predicates havingpredicates or functions as arguments, or in which one or both of predicate quantiers or function quantiers arepermitted.[1] In rst-order theories, predicates are often associated with sets. In interpreted higher-order theories,predicates may be interpreted as sets of sets.There are many deductive systems for rst-order logic that are sound (all provable statements are true in all models)and complete (all statements which are true in all models are provable). Although the logical consequence relation isonly semidecidable, much progress has been made in automated theorem proving in rst-order logic. First-order logicalso satises several metalogical theorems that make it amenable to analysis in proof theory, such as the LwenheimSkolem theorem and the compactness theorem.First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundationsof mathematics. Mathematical theories, such as number theory and set theory, have been formalized into rst-orderaxiom schemas such as Peano arithmetic and ZermeloFraenkel set theory (ZF) respectively.No rst-order theory, however, has the strength to describe uniquely a structure with an innite domain, such as thenatural numbers or the real line. A uniquely describing, i.e. categorical, axiom system for such a structure can beobtained in stronger logics such as second-order logic.For a history of rst-order logic and how it came to dominate formal logic, see Jos Ferreirs (2001).

10.1 IntroductionWhile propositional logic deals with simple declarative propositions, rst-order logic additionally covers predicatesand quantication.A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Considerthe two sentences Socrates is a philosopher and Plato is a philosopher. In propositional logic, these sentencesare viewed as being unrelated and are denoted, for example, by p and q. However, the predicate is a philosopheroccurs in both sentences which have a common structure of "a is a philosopher. The variable a is instantiated asSocrates in the rst sentence and is instantiated as Plato in the second sentence. The use of predicates, such asis a philosopher in this example, distinguishes rst-order logic from propositional logic.Predicates can be compared. Consider, for example, the rst-order formula if a is a philosopher, then a is a scholar.This formula is a conditional statement with "a is a philosopher as hypothesis and "a is a scholar as conclusion.

25

26 CHAPTER 10. FIRST-ORDER LOGIC

The truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates is aphilosopher and is a scholar.Variables can be quantied over. The variable a in the previous formula can be quantied over, for instance, in therst-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantier for every inthis sentence expresses the idea that the claim if a is a philosopher, then a is a scholar holds for all choices of a.The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to thesentence There exists a such that a is a philosopher and a is not a scholar. The existential quantier there existsexpresses the idea that the claim "a is a philosopher and a is not a scholar holds for some choice of a.The predicates is a philosopher and is a scholar each take a single variable. Predicates can take several variables.In the rst-order sentence Socrates is the teacher of Plato, the predicate is the teacher of takes two variables.To interpret a rst-order formula, one species what each predicate means and the entities that can instantiate thepredicated variables. These entities form the domain of discourse or universe, which is usually required to be anonempty set. Given that the interpretation with the domain of discourse as consisting of all human beings and thepredicate is a philosopher understood as have written the Republic, the sentence There exists a such that a is aphilosopher is seen as being true, as witnessed by Plato.

10.2 SyntaxThere are two key parts of rst-order logic. The syntax determines which collections of symbols are legal expressionsin rst-order logic, while the semantics determine the meanings behind these expressions.

10.2.1 AlphabetUnlike natural languages, such as English, the language of rst-order logic is completely formal, so that it can bemechanically determined whether a given expression is legal. There are two key types of legal expressions: terms,which intuitively represent objects, and formulas, which intuitively express predicates that can be true or false. Theterms and formulas of rst-order logic are strings of symbols which together form the alphabet of the language. Aswith all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are oftenregarded simply as letters and punctuation symbols.It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, andnon-logical symbols, whose meaning varies by interpretation. For example, the logical symbol ^ always representsand"; it is never interpreted as or. On the other hand, a non-logical predicate symbol such as Phil(x) could beinterpreted to mean "x is a philosopher, "x is a man named Philip, or any other unary predicate, depending on theinterpretation at hand.

Logical symbols

There are several logical symbols in the alphabet, which vary by author but usually include:

The quantier symbols and The logical connectives: for conjunction, for disjunction, for implication, for biconditional, fornegation. Occasionally other logical connective symbols are included. Some authors use Cpq, instead of ,and Epq, instead of , especially in contexts where is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may replace ; a tilde (~), Np, or Fpq, may replace ; ||, or Apq may replace; and &, Kpq, or the middle dot, , may replace , especially if these symbols are not available for technicalreasons. (Note: the aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation.)

Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context. An innite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, . Subscriptsare often used to distinguish variables: x0, x1, x2, .

An equality symbol (sometimes, identity symbol) =; see the section on equality below.

10.2. SYNTAX 27

It should be noted that not all of these symbols are required only one of the quantiers, negation and conjunc-tion, variables, brackets and equality suce. There are numerous minor variations that may dene