Reverse mathematicsFrom 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 Equiconsistency 126.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 First-order logic 147.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 217.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 227.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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7.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8 Harvey Friedman 358.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Mathematical analysis 379.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3.4 Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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9.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Primitive recursive arithmetic 4510.1 Language and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.2 Logic-free calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

11 Proof theory 4811.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.2 Formal and informal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.3 Kinds of proof calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.4 Consistency proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.5 Structural proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.6 Proof-theoretic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.7 Tableau systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.8 Ordinal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.9 Logics from proof analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

12 Quantier (logic) 5212.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

13 Reverse mathematics 6013.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

13.1.1 Use of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.2 The big ve subsystems of second order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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13.2.1 The base system RCA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.2.2 Weak Knigs lemma WKL0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.2.3 Arithmetical comprehension ACA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.2.4 Arithmetical transnite recursion ATR0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2.5 11 comprehension 11-CA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.3 Additional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.4 -models and -models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

14 Second-order arithmetic 6614.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

14.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6614.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.1.4 The full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

14.2 Models of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6814.3 Denable functions of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.4 Subsystems of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14.4.1 Arithmetical comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.4.2 The arithmetical hierarchy for formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.4.3 Recursive comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.4.4 Weaker systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.4.5 Stronger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

14.5 Projective Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.6 Coding mathematics in second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

15 ZermeloFraenkel set theory 7315.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

15.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 7415.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7615.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7715.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7715.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vi CONTENTS

15.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

15.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 82

15.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8315.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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.

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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

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.

6.1 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.

6.2 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

12

6.3. SEE ALSO 13

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]

6.3 See also Large cardinal property

6.4 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 7

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).

7.1 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.

14

7.2. SYNTAX 15

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.

7.2 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.

7.2.1 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.

16 CHAPTER 7. FIRST-ORDER LOGIC

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 additional logicalsymbols:

Sometimes the truth constants T, Vpq, or , for true and F, Opq, or , for false are included. Without anysuch logical operators of valence 0, these two constants can only be expressed using quantiers.

Sometimes additional logical connectives are included, such as the Sheer stroke, Dpq (NAND), and exclusiveor, Jpq.

Non-logical symbols

The non-logical symbols represent predicates (relations), functions and constants on the domain of discourse. It usedto be standard practice to use a xed, innite set of non-logical symbols for all purposes. A more recent practice isto use dierent non-logical symbols according to the application one has in mind. Therefore it has become necessaryto name the set of all non-logical symbols used in a particular application. This choice is made via a signature.[2]

The traditional approach is to have only one, innite, set of non-logical symbols (one signature) for all applications.Consequently, under the traditional approach there is only one language of rst-order logic.[3] This approach is stillcommon, especially in philosophically oriented books.

1. For every integer n 0 there is a collection of n-ary, or n-place, predicate symbols. Because they representrelations between n elements, they are also called relation symbols. For each arity n we have an innite supplyof them:

Pn0, Pn1, Pn2, Pn3,

2. For every integer n 0 there are innitely many n-ary function symbols:

f n0, f n1, f n2, f n3,

In contemporary mathematical logic, the signature varies by application. Typical signatures in mathematics are {1,} or just {} for groups, or {0, 1, +, ,

7.2. SYNTAX 17

7.2.2 Formation rules

The formation rules dene the terms and formulas of rst order logic. When terms and formulas are representedas strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules aregenerally context-free (each production has a single symbol on the left side), except that the set of symbols may beallowed to be innite and there may be many start symbols, for example the variables in the case of terms.

Terms

The set of terms is inductively dened by the following rules:

1. Variables. Any variable is a term.

2. Functions. Any expression f(t1,...,tn) of n arguments (where each argument ti is a term and f is a functionsymbol of valence n) is a term. In particular, symbols denoting individual constants are 0-ary function symbols,and are thus terms.

Only expressions which can be obtained by nitely many applications of rules 1 and 2 are terms. For example, noexpression involving a predicate symbol is a term.

Formulas

The set of formulas (also called well-formed formulas [4] or ws) is inductively dened by the following rules:

1. Predicate symbols. If P is an n-ary predicate symbol and t1, ..., tn are terms then P(t1,...,t) is a formula.

2. Equality. If the equality symbol is considered part of logic, and t1 and t2 are terms, then t1 = t2 is a formula.

3. Negation. If is a formula, then : is a formula.

4. Binary connectives. If and are formulas, then (! ) is a formula. Similar rules apply to other binarylogical connectives.

5. Quantiers. If is a formula and x is a variable, then 8x' (for all x, ' holds) and 9x' (there exists x suchthat ' ) are formulas.

Only expressions which can be obtained by nitely many applications of rules 15 are formulas. The formulas ob-tained from the rst two rules are said to be atomic formulas.For example,

8x8y(P (f(x))! :(P (x)! Q(f(y); x; z)))

is a formula, if f is a unary function symbol, P a unary predicate symbol, and Q a ternary predicate symbol. On theother hand, 8xx! is not a formula, although it is a string of symbols from the alphabet.The role of the parentheses in the denition is to ensure that any formula can only be obtained in one way by followingthe inductive denition (in other words, there is a unique parse tree for each formula). This property is known asunique readability of formulas. There are many conventions for where parentheses are used in formulas. Forexample, some authors use colons or full stops instead of parentheses, or change the places in which parentheses areinserted. Each authors particular denition must be accompanied by a proof of unique readability.This denition of a formula does not support dening an if-then-else function ite(c, a, b), where c is a conditionexpressed as a formula, that would return a if c is true, and b if it is false. This is because both predicates andfunctions can only accept terms as parameters, but the rst parameter is a formula. Some languages built on rst-orderlogic, such as SMT-LIB 2.0, add this.[5]

18 CHAPTER 7. FIRST-ORDER LOGIC

Notational conventions

For convenience, conventions have been developed about the precedence of the logical operators, to avoid the needto write parentheses in some cases. These rules are similar to the order of operations in arithmetic. A commonconvention is:

: is evaluated rst

^ and _ are evaluated next

Quantiers are evaluated next

! is evaluated last.

Moreover, extra punctuation not required by the denition may be inserted to make formulas easier to read. Thus theformula

(:8xP (x)! 9x:P (x))

might be written as

(:[8xP (x)])! 9x[:P (x)]:

In some elds, it is common to use inx notation for binary relations and functions, instead of the prex notationdened above. For example, in arithmetic, one typically writes 2 + 2 = 4 instead of "=(+(2,2),4)". It is common toregard formulas in inx notation as abbreviations for the corresponding formulas in prex notation.The denitions above use inx notation for binary connectives such as ! . A less common convention is Polishnotation, in which one writes! , ^ , and so on in front of their arguments rather than between them. This conventionallows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practicebecause it is hard for humans to read it. In Polish notation, the formula

8x8y(P (f(x))! :(P (x)! Q(f(y); x; z)))

becomes "xyPfx PxQfyxz.

7.2.3 Free and bound variables

Main article: Free variables and bound variables

In a formula, a variable may occur free or bound. Intuitively, a variable is free in a formula if it is not quantied: in8y P (x; y) , variable x is free while y is bound. The free and bound variables of a formula are dened inductively asfollows.

1. Atomic formulas. If is an atomic formula then x is free in if and only if x occurs in . Moreover, thereare no bound variables in any atomic formula.

2. Negation. x is free in : if and only if x is free in . x is bound in : if and only if x is bound in .

3. Binary connectives. x is free in (! ) if and only if x is free in either or . x is bound in (! ) if andonly if x is bound in either or . The same rule applies to any other binary connective in place of! .

4. Quantiers. x is free in 8 y if and only if x is free in and x is a dierent symbol from y. Also, x is boundin 8 y if and only if x is y or x is bound in . The same rule holds with 9 in place of 8 .

7.3. SEMANTICS 19

For example, in 8 x 8 y (P(x)! Q(x,f(x),z)), x and y are bound variables, z is a free variable, andw is neither becauseit does not occur in the formula.Free and bound variables of a formula need not be disjoint sets: x is both free and bound in P (x)! 8xQ(x) .Freeness and boundness can be also specialized to specic occurrences of variables in a formula. For example, inP (x) ! 8xQ(x) , the rst occurrence of x is free while the second is bound. In other words, the x in P (x) is freewhile the x in 8xQ(x) is bound.A formula in rst-order logic with no free variables is called a rst-order sentence. These are the formulas that willhave well-dened truth values under an interpretation. For example, whether a formula such as Phil(x) is true mustdepend on what x represents. But the sentence 9xPhil(x) will be either true or false in a given interpretation.

7.2.4 ExamplesOrdered abelian groups

In mathematics the language of ordered abelian groups has one constant symbol 0, one unary function symbol , onebinary function symbol +, and one binary relation symbol . Then:

The expressions +(x, y) and +(x, +(y, (z))) are terms. These are usually written as x + y and x + y z. The expressions +(x, y) = 0 and (+(x, +(y, (z))), +(x, y)) are atomic formulas.

These are usually written as x + y = 0 and x + y z x + y.

The expression (8x8y(+(x; y); z) ! 8x8y+(x; y) = 0) is a formula, which is usually written as8x8y(x+ y z)! 8x8y(x+ y = 0):

Loving relation

English sentences like everyone loves someone can be formalized by rst-order logic formulas like xy L(x,y).This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantiers and andthe loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with8 dierent meanings can be built. The following diagrams show models for each of them, assuming that there areexactly ve individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x andcolumn y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satised byseveral models.Each model, represented by a logical matrix, satises the formulas in its caption in a minimal way, i.e. whiteningany red cell in any matrix would make it non-satisfying the corresponding formula. For example, formula 1 is alsosatised by the matrices at 3, 6, and 10, but not by those at 2, 4, 5, and 7. Conversely, the matrix shown at 6 satises1, 2, 5, 6, 7, and 8, but not 3, 4, 9, and 10.Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) ofthe implication e.g. formula 3 implies formula 1, i.e.: each matrix fullling formula 3 also fullls formula 1, butnot vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices,[6] which satisfyformula 2, happen to satisfy also formula 5, whereas others,[7] also satisfying formula 2, do not; therefore formula 5is not a logical consequence of formula 2.The sequence of the quantiers is important! So it is instructive to distinguish formulas 1: x y L(y,x), and 3: xy L(x,y). In both cases everyone is loved; but in the rst case everyone (x) is loved by someone (y), in the secondcase everyone (y) is loved by just exactly one person (x).

7.3 SemanticsAn interpretation of a rst-order language assigns a denotation to all non-logical constants in that language. It alsodetermines a domain of discourse that species the range of the quantiers. The result is that each term is assigned anobject that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic

20 CHAPTER 7. FIRST-ORDER LOGIC

meaning to the terms and formulas of the language. The study of the interpretations of formal languages is calledformal semantics. What follows is a description of the standard or Tarskian semantics for rst-order logic. (It is alsopossible to dene game semantics for rst-order logic, but aside from requiring the axiom of choice, game semanticsagree with Tarskian semantics for rst-order logic, so game semantics will not be elaborated herein.)The domain of discourseD is a nonempty set of objects of some kind. Intuitively, a rst-order formula is a statementabout these objects; for example, 9xP (x) states the existence of an object x such that the predicate P is true wherereferred to it. The domain of discourse is the set of considered objects. For example, one can takeD to be the set ofinteger numbers.The interpretation of a function symbol is a function. For example, if the domain of discourse consists of integers, afunction symbol f of arity 2 can be interpreted as the function that gives the sum of its arguments. In other words,the symbol f is associated with the function I(f) which, in this interpretation, is addition.The interpretation of a constant symbol is a function from the one-element setD0 toD, which can be simply identiedwith an object in D. For example, an interpretation may assign the value I(c) = 10 to the constant symbol c .The interpretation of an n-ary predicate symbol is a set of n-tuples of elements of the domain of discourse. Thismeans that, given an interpretation, a predicate symbol, and n elements of the domain of discourse, one can tellwhether the predicate is true of those elements according to the given interpretation. For example, an interpretationI(P) of a binary predicate symbol P may be the set of pairs of integers such that the rst one is less than the second.According to this interpretation, the predicate P would be true if its rst argument is less than the second.

7.3.1 First-order structures

Main article: Structure (mathematical logic)

The most common way of specifying an interpretation (especially in mathematics) is to specify a structure (alsocalled a model; see below). The structure consists of a nonempty set D that forms the domain of discourse and aninterpretation I of the non-logical terms of the signature. This interpretation is itself a function:

Each function symbol f of arity n is assigned a function I(f) fromDn toD . In particular, each constant symbolof the signature is assigned an individual in the domain of discourse.

Each predicate symbol P of arity n is assigned a relation I(P) overDn or, equivalently, a function fromDn toftrue; falseg . Thus each predicate symbol is interpreted by a Boolean-valued function on D.

7.3.2 Evaluation of truth values

A formula evaluates to true or false given an interpretation, and a variable assignment that associates an elementof the domain of discourse with each variable. The reason that a variable assignment is required is to give meaningsto formulas with free variables, such as y = x . The truth value of this formula changes depending on whether x andy denote the same individual.First, the variable assignment can be extended to all terms of the language, with the result that each term maps toa single element of the domain of discourse. The following rules are used to make this assignment:

1. Variables. Each variable x evaluates to (x)

2. Functions. Given terms t1; : : : ; tn that have been evaluated to elements d1; : : : ; dn of the domain of discourse,and a n-ary function symbol f, the term f(t1; : : : ; tn) evaluates to (I(f))(d1; : : : ; dn) .

Next, each formula is assigned a truth value. The inductive denition used to make this assignment is called theT-schema.

1. Atomic formulas (1). A formula P (t1; : : : ; tn) is associated the value true or false depending on whetherhv1; : : : ; vni 2 I(P ) , where v1; : : : ; vn are the evaluation of the terms t1; : : : ; tn and I(P ) is the interpreta-tion of P , which by assumption is a subset of Dn .

7.3. SEMANTICS 21

2. Atomic formulas (2). A formula t1 = t2 is assigned true if t1 and t2 evaluate to the same object of the domainof discourse (see the section on equality below).

3. Logical connectives. A formula in the form : , ! , etc. is evaluated according to the truth table forthe connective in question, as in propositional logic.

4. Existential quantiers. A formula 9x(x) is true according to M and if there exists an evaluation 0 ofthe variables that only diers from regarding the evaluation of x and such that is true according to theinterpretation M and the variable assignment 0 . This formal denition captures the idea that 9x(x) is trueif and only if there is a way to choose a value for x such that (x) is satised.

5. Universal quantiers. A formula 8x(x) is true according toM and if (x) is true for every pair composedby the interpretationM and some variable assignment0 that diers from only on the value of x. This capturesthe idea that 8x(x) is true if every possible choice of a value for x causes (x) to be true.

If a formula does not contain free variables, and so is a sentence, then the initial variable assignment does not aectits truth value. In other words, a sentence is true according to M and if and only if it is true according to M andevery other variable assignment 0 .There is a second common approach to dening truth values that does not rely on variable assignment functions.Instead, given an interpretation M, one rst adds to the signature a collection of constant symbols, one for eachelement of the domain of discourse in M; say that for each d in the domain the constant symbol cd is xed. Theinterpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain.One now denes truth for quantied formulas syntactically, as follows:

1. Existential quantiers (alternate). A formula 9x(x) is true according toM if there is some d in the domainof discourse such that (cd) holds. Here (cd) is the result of substituting cd for every free occurrence of x in.

2. Universal quantiers (alternate). A formula 8x(x) is true according toM if, for every d in the domain ofdiscourse, (cd) is true according to M.

This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments.

7.3.3 Validity, satisability, and logical consequenceSee also: Satisability

If a sentence evaluates to True under a given interpretationM, one says thatM satises ; this is denotedM '. A sentence is satisable if there is some interpretation under which it is true.Satisability of formulas with free variables is more complicated, because an interpretation on its own does notdetermine the truth value of such a formula. The most common convention is that a formula with free variables issaid to be satised by an interpretation if the formula remains true regardless which individuals from the domain ofdiscourse are assigned to its free variables. This has the same eect as saying that a formula is satised if and only ifits universal closure is satised.A formula is logically valid (or simply valid) if it is true in every interpretation. These formulas play a role similarto tautologies in propositional logic.A formula is a logical consequence of a formula if every interpretation that makes true also makes true. Inthis case one says that is logically implied by .

7.3.4 AlgebraizationsAn alternate approach to the semantics of rst-order logic proceeds via abstract algebra. This approach generalizesthe LindenbaumTarski algebras of propositional logic. There are three ways of eliminating quantied variables fromrst-order logic that do not involve replacing quantiers with other variable binding term operators:

Cylindric algebra, by Alfred Tarski and his coworkers;

22 CHAPTER 7. FIRST-ORDER LOGIC

Polyadic algebra, by Paul Halmos; Predicate functor logic, mainly due to Willard Quine.

These algebras are all lattices that properly extend the two-element Boolean algebra.Tarski and Givant (1987) showed that the fragment of rst-order logic that has no atomic sentence lying in the scopeof more than three quantiers has the same expressive power as relation algebra. This fragment is of great interestbecause it suces for Peano arithmetic and most axiomatic set theory, including the canonical ZFC. They also provethat rst-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projectionfunctions.

7.3.5 First-order theories, models, and elementary classesA rst-order theory of a particular signature is a set of axioms, which are sentences consisting of symbols from thatsignature. The set of axioms is often nite or recursively enumerable, in which case the theory is called eective.Some authors require theories to also include all logical consequences of the axioms. The axioms are considered tohold within the theory and from them other sentences that hold within the theory can be derived.A rst-order structure that satises all sentences in a given theory is said to be amodel of the theory. An elementaryclass is the set of all structures satisfying a particular theory. These classes are a main subject of study in modeltheory.Many theories have an intended interpretation, a certain model that is kept in mind when studying the theory.For example, the intended interpretation of Peano arithmetic consists of the usual natural numbers with their usualoperations. However, the LwenheimSkolem theorem shows that most rst-order theories will also have other,nonstandard models.A theory is consistent if it is not possible to prove a contradiction from the axioms of the theory. A theory is completeif, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of thetheory. Gdels incompleteness theorem shows that eective rst-order theories that include a sucient portion ofthe theory of the natural numbers can never be both consistent and complete.For more information on this subject see List of rst-order theories and Theory (mathematical logic)

7.3.6 Empty domainsMain article: Empty domain

The denition above requires that the domain of discourse of any interpretation must be a nonempty set. There aresettings, such as inclusive logic, where empty domains are permitted. Moreover, if a class of algebraic structuresincludes an empty structure (for example, there is an empty poset), that class can only be an elementary class inrst-order logic if empty domains are permitted or the empty structure is removed from the class.There are several diculties with empty domains, however:

Many common rules of inference are only valid when the domain of discourse is required to be nonempty. Oneexample is the rule stating that _9x implies 9x(_ ) when x is not a free variable in . This rule, whichis used to put formulas into prenex normal form, is sound in nonempty domains, but unsound if the emptydomain is permitted.

The denition of truth in an interpretation that uses a variable assignment function cannot work with emptydomains, because there are no variable assignment functions whose range is empty. (Similarly, one cannotassign interpretations to constant symbols.) This truth denition requires that one must select a variable as-signment function ( above) before truth values for even atomic formulas can be dened. Then the truth valueof a sentence is dened to be its truth value under any variable assignment, and it is proved that this truthvalue does not depend on which assignment is chosen. This technique does not work if there are no assignmentfunctions at all; it must be changed to accommodate empty domains.

Thus, when the empty domain is permitted, it must often be treated as a special case. Most authors, however, simplyexclude the empty domain by denition.

7.4. DEDUCTIVE SYSTEMS 23

7.4 Deductive systemsA deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequenceof another formula. There are many such systems for rst-order logic, including Hilbert-style deductive systems,natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property thata deduction is a nite syntactic object; the format of this object, and the way it is constructed, vary widely. Thesenite deductions themselves are often called derivations in proof theory. They are also often called proofs, but arecompletely formalized unlike natural-language mathematical proofs.A deductive system is sound if any formula that can be derived in the system is logically valid. Conversely, a deductivesystem is complete if every logically valid formula is derivable. All of the systems discussed in this article are bothsound and complete. They also share the property that it is possible to eectively verify that a purportedly validdeduction is actually a deduction; such deduction systems are called eective.A key property of deductive systems is that they are purely syntactic, so that derivations can be veried withoutconsidering any interpretation. Thus a sound argument is correct in every possible interpretation of the language,regardless whether that interpretation is about mathematics, economics, or some other area.In general, logical consequence in rst-order logic is only semidecidable: if a sentence A logically implies a sentenceB then this can be discovered (for example, by searching for a proof until one is found, using some eective, sound,complete proof system). However, if A does not logically imply B, this does not mean that A logically implies thenegation of B. There is no eective procedure that, given formulas A and B, always correctly decides whether Alogically implies B.

7.4.1 Rules of inference

Further information: List of rules of inference

A rule of inference states that, given a particular formula (or set of formulas) with a certain property as a hypothesis,another specic formula (or set of formulas) can be derived as a conclusion. The rule is sound (or truth-preserving)if it preserves validity in the sense that whenever any interpretation satises the hypothesis, that interpretation alsosatises the conclusion.For example, one common rule of inference is the rule of substitution. If t is a term and is a formula possiblycontaining the variable x, then [t/x] (often denoted [x/t]) is the result of replacing all free instances of x by t in. The substitution rule states that for any and any term t, one can conclude [t/x] from provided that no freevariable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then tosubstitute t for x it is rst necessary to change the bound variables of to dier from the free variables of t.)To see why the restriction on bound variables is necessary, consider the logically valid formula given by 9x(x = y), in the signature of (0,1,+,,=) of arithmetic. If t is the term x + 1, the formula [t/y] is 9x(x = x+1) , which willbe false in many interpretations. The problem is that the free variable x of t became bound during the substitution.The intended replacement can be obtained by renaming the bound variable x of to something else, say z, so thatthe formula after substitution is 9z(z = x+ 1) , which is again logically valid.The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one cantell whether it was correctly applied without appeal to any interpretation. It has (syntactically dened) limitations onwhen it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is oftenthe case, these limitations are necessary because of interactions between free and bound variables that occur duringsyntactic manipulations of the formulas involved in the inference rule.

7.4.2 Hilbert-style systems and natural deduction

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesisthat has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. Thelogical axioms consist of several axiom schemas of logically valid formulas; these encompass a signicant amount ofpropositional logic. The rules of inference enable the manipulation of quantiers. Typical Hilbert-style systems havea small number of rules of inference, along with several innite schemas of logical axioms. It is common to have onlymodus ponens and universal generalization as rules of inference.

24 CHAPTER 7. FIRST-ORDER LOGIC

Natural deduction systems resemble Hilbert-style systems in that a deduction is a nite list of formulas. However,natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that canbe used to manipulate the logical connectives in formulas in the proof.

7.4.3 Sequent calculus

Further information: Sequent calculus

The sequent calculus was developed to study the properties of natural deduction systems. Instead of working withone formula at a time, it uses sequents, which are expressions of the form

A1; : : : ; An ` B1; : : : ; Bk;

where A1, ..., An, B1, ..., Bk are formulas and the turnstile symbol ` is used as punctuation to separate the two halves.Intuitively, a sequent expresses the idea that (A1 ^ ^An) implies (B1 _ _Bk) .

7.4.4 Tableaux method

Further information: Method of analytic tableaux

Unlike the methods just described, the derivations in the tableaux method are not lists of formulas. Instead, a deriva-tion is a tree of formulas. To show that a formula A is provable, the tableaux method attempts to demonstrate thatthe negation of A is unsatisable. The tree of the derivation has :A at its root; the tree branches in a way that reectsthe structure of the formula. For example, to show that C _ D is unsatisable requires showing that C and D areeach unsatisable; this corresponds to a branching point in the tree with parent C _D and children C and D.

7.4.5 Resolution

The resolution rule is a single rule of inference that, together with unication, is sound and complete for rst-orderlogic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisable.Resolution is commonly used in automated theorem proving.The resolutionmethod works only with formulas that are disjunctions of atomic formulas; arbitrary formulas must rstbe converted to this form through Skolemization. The resolution rule states that from the hypothesesA1_ _Ak_Cand B1 _ _Bl _ :C , the conclusion A1 _ _Ak _B1 _ _Bl can be obtained.

7.4.6 Provable identities

The following sentences can be called identities because the main connective in each is the biconditional.

:8xP (x), 9x:P (x):9xP (x), 8x:P (x)8x 8y P (x; y), 8y 8xP (x; y)9x 9y P (x; y), 9y 9xP (x; y)8xP (x) ^ 8xQ(x), 8x (P (x) ^Q(x))9xP (x) _ 9xQ(x), 9x (P (x) _Q(x))P ^ 9xQ(x), 9x (P ^Q(x)) (where x must not occur free in P )P _ 8xQ(x), 8x (P _Q(x)) (where x must not occur free in P )

7.5. EQUALITY AND ITS AXIOMS 25

7.5 Equality and its axiomsThere are several dierent conventions for using equality (or identity) in rst-order logic. The most common con-vention, known as rst-order logic with equality, includes the equality symbol as a primitive logical symbol whichis always interpreted as the real equality relation between members of the domain of discourse, such that the twogiven members are the same member. This approach also adds certain axioms about equality to the deductive systememployed. These equality axioms are:

1. Reexivity. For each variable x, x = x.2. Substitution for functions. For all variables x and y, and any function symbol f,

x = y f(...,x,...) = f(...,y,...).3. Substitution for formulas