Bounded quantierFrom Wikipedia, the free encyclopedia

Contents

1 (, )-denition of limit 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Informal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Precise statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Comparison with innitesimal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Arithmetical hierarchy 52.1 The arithmetical hierarchy of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The arithmetical hierarchy of sets of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Relativized arithmetical hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Arithmetic reducibility and degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The arithmetical hierarchy of subsets of Cantor and Baire space . . . . . . . . . . . . . . . . . . . 72.6 Extensions and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Meaning of the notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.9 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.10 Relation to Turing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Axiom schema of predicative separation 10

4 Boolean satisability problem 114.1 Basic denitions and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Complexity and restricted versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Unrestricted satisability (SAT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2 3-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.3 Exactly-1 3-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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4.2.4 2-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.5 Horn-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.6 XOR-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.7 Schaefers dichotomy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Extensions of SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Self-reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Algorithms for solving SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.9.1 SAT problem format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9.2 Online SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9.3 Oine SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9.4 SAT applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.9.5 Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.9.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.9.7 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.9.8 Evaluation of SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Bounded quantier 215.1 Bounded quantiers in arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Bounded quantiers in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Branching quantier 236.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Relation to natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Computability theory 267.1 Computable and uncomputable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Turing computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Areas of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.3.1 Relative computability and the Turing degrees . . . . . . . . . . . . . . . . . . . . . . . . 287.3.2 Other reducibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3.3 Rices theorem and the arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 297.3.4 Reverse mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3.5 Numberings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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7.3.6 The priority method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3.7 The lattice of recursively enumerable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3.8 Automorphism problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3.9 Kolmogorov complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3.10 Frequency computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3.11 Inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3.12 Generalizations of Turing computability . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3.13 Continuous computability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.4 Relationships between denability, proof and computability . . . . . . . . . . . . . . . . . . . . . 327.5 Name of the subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.6 Professional organizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Conditional quantier 368.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Conjunctive normal form 379.1 Examples and Non-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.2 Conversion into CNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.3 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.4 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.5 Converting from rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Constructive set theory 4110.1 Intuitionistic ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

10.1.1 Predicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2 Myhills constructive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3 Aczels constructive ZermeloFraenkel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.4 Interpretability in type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.5 Interpretability in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

11 Context-sensitive grammar 4411.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.3 Kuroda normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4 Properties and uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11.4.1 Equivalence to linear bounded automaton . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4.2 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4.3 Computational problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4.4 As model of natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12 Counting quantication 4912.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13 Donkey sentence 5013.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.2 Discourse representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

14 ELEMENTARY 5514.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.2 Lower elementary recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3 Basis for ELEMENTARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.4 Descriptive characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

15 Existential quantication 5715.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

15.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.2.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.2.3 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

15.3 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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15.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

16 First-order logic 6116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

16.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6616.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6716.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6716.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 6816.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6816.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 6916.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

16.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

16.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

16.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

16.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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16.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17 Game semantics 8217.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.2 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.3 Intuitionistic logic, denotational semantics, linear logic, logical pluralism . . . . . . . . . . . . . . 8317.4 Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.6.1 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.6.2 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

18 Generalized quantier 8618.1 Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.2 Typed lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.3.2 Conservativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

18.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

19 KripkePlatek set theory 9019.1 The axioms of KP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.2 Proof that Cartesian products exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.3 Admissible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

20 Lindstrm quantier 9220.1 Generalization of rst-order quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9220.2 Expressiveness hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.3 As precursors to Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.4 Algorithmic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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21 Logic 9521.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

21.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9521.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 9621.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 9721.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

21.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9721.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

21.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10121.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10121.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10121.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 10221.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10321.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10421.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10521.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10721.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

22 Lvy hierarchy 10922.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10922.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

22.2.1 0=0=0 formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10922.2.2 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.3 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.4 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.5 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.6 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.7 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.2.8 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11122.2.9 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11122.2.10 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11122.2.11 4-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

22.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11122.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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22.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

23 Mathematical logic 11223.1 Subelds and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11223.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

23.2.1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11323.2.2 19th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11323.2.3 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

23.3 Formal logical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11523.3.1 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.3.2 Other classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.3.3 Nonclassical and modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11723.3.4 Algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

23.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11723.5 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11823.6 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

23.6.1 Algorithmically unsolvable problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.7 Proof theory and constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.8 Connections with computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.9 Foundations of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12023.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12023.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

23.12.1 Undergraduate texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.12.2 Graduate texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12223.12.3 Research papers, monographs, texts, and surveys . . . . . . . . . . . . . . . . . . . . . . 12223.12.4 Classical papers, texts, and collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

23.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

24 Mathematics 12624.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

24.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12724.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

24.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13024.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

24.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 13324.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

24.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13524.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13624.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

24.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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24.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13924.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13924.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14124.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14224.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

25 Maxima and minima 14425.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14525.2 Finding functional maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14525.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14525.4 Functions of more than one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14625.5 Maxima or minima of a functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14725.6 In relation to sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14725.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14825.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14825.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

26 Peano axioms 14926.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14926.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

26.2.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15126.2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15126.2.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

26.3 First-order theory of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15226.3.1 Equivalent axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

26.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.4.1 Nonstandard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.4.2 Set-theoretic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.4.3 Interpretation in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

26.5 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15526.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15626.7 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15626.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15726.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

27 Plural quantication 15927.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15927.2 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

27.2.1 Multigrade (variably polyadic) predicates and relations . . . . . . . . . . . . . . . . . . . 15927.2.2 Nominalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

27.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16027.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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27.3.2 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16127.4 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16227.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16227.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16227.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

28 Polynomial hierarchy 16428.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16428.2 Relations between classes in the polynomial hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 16528.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16528.4 Problems in the polynomial hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16728.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16728.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

29 Predicate logic 16829.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16829.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16829.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

30 Primitive recursive function 17030.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

30.1.1 Role of the projection functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17130.1.2 Converting predicates to numeric functions . . . . . . . . . . . . . . . . . . . . . . . . . 17130.1.3 Computer language denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

30.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17130.2.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17130.2.2 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17230.2.3 Operations on integers and rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . 172

30.3 Relationship to recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17230.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17330.5 Some common primitive recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.6 Additional primitive recursive forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.7 Finitism and consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17630.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17630.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

31 Propositional formula 17831.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

31.1.1 Relationship between propositional and predicate formulas . . . . . . . . . . . . . . . . . 17931.1.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

31.2 An algebra of propositions, the propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 17931.2.1 Usefulness of propositional formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

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31.2.2 Propositional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18031.2.3 Truth-value assignments, formula evaluations . . . . . . . . . . . . . . . . . . . . . . . . 180

31.3 Propositional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18131.3.1 Connectives of rhetoric, philosophy and mathematics . . . . . . . . . . . . . . . . . . . . 18131.3.2 Engineering connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18131.3.3 CASE connective: IF ... THEN ... ELSE ... . . . . . . . . . . . . . . . . . . . . . . . . . 18131.3.4 IDENTITY and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

31.4 More complex formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18331.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18331.4.2 Axiom and denition schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18431.4.3 Substitution versus replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

31.5 Inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18431.6 Parsing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

31.6.1 Connective seniority (symbol rank) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18531.6.2 Commutative and associative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18631.6.3 Distributive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18631.6.4 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18631.6.5 Laws of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18731.6.6 Laws of evaluation: Identity, nullity, and complement . . . . . . . . . . . . . . . . . . . . 18731.6.7 Double negative (Involution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

31.7 Well-formed formulas (ws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18731.7.1 Ws versus valid formulas in inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

31.8 Reduced sets of connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18831.8.1 The stroke (NAND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18831.8.2 IF ... THEN ... ELSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

31.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19031.9.1 Reduction to normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19031.9.2 Reduction by use of the map method (Veitch, Karnaugh) . . . . . . . . . . . . . . . . . . 191

31.10Impredicative propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19231.11Propositional formula with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

31.11.1 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19331.11.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

31.12Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19431.13Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19631.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

32 Quanticational variability eect 20432.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20432.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20432.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20432.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

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33 Quantier (linguistics) 20633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20633.2 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20633.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20733.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20733.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

34 Quantier (logic) 20834.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20834.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20834.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20934.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21034.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21034.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21134.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21134.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21334.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21334.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21434.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21434.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21434.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

35 Quantier rank 21635.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21635.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21635.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21735.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21735.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

36 Quantier variance 21836.1 Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21836.2 Usage, not 'existence'? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21936.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21936.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

37 Second-order arithmetic 22137.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

37.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22137.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22237.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22237.1.4 The full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

37.2 Models of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22337.3 Denable functions of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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37.4 Subsystems of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22437.4.1 Arithmetical comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22437.4.2 The arithmetical hierarchy for formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22437.4.3 Recursive comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22537.4.4 Weaker systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22537.4.5 Stronger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

37.5 Projective Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22637.6 Coding mathematics in second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 22637.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22637.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

38 Sentence (logic) 22838.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22838.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22838.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

39 Subtyping 23039.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23039.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23139.3 Subtyping schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23139.4 Record types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23239.5 Function types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23239.6 Coercions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23239.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23339.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23339.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23339.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

40 System F 23540.1 Logic and predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23540.2 System F Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23640.3 Use in programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23740.4 System F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23740.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23740.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23740.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23840.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23840.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

41 System F-sub 23941.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23941.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

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42 Two-variable logic 24042.1 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24042.2 Counting quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24042.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

43 Type theory 24143.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24143.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24143.3 Dierence from set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24243.4 Optional features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

43.4.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24243.4.2 Dependent types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24243.4.3 Equality types (or identity types) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24343.4.4 Inductive types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24343.4.5 Universe types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24343.4.6 Computational component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

43.5 Systems of type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.5.1 Major . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.5.2 Minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.5.3 Active . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

43.6 Practical impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.6.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.6.2 Mathematical foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.6.3 Proof assistants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24543.6.4 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24543.6.5 Social sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

43.7 Relation to category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24543.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24643.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24643.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24643.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

44 Typed lambda calculus 24844.1 Kinds of typed lambda calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24844.2 Applications to programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

45 Uniqueness quantication 25045.1 Proving uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

CONTENTS xv

45.2 Reduction to ordinary existential and universal quantication . . . . . . . . . . . . . . . . . . . . 25045.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25145.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25145.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

46 Universal quantication 25246.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

46.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25346.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

46.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25346.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25446.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25546.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

46.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25546.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25646.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25646.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25646.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

47 Unsatisable core 25747.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

48 Witness (mathematics) 25848.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25848.2 Henkin witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25848.3 Relation to game semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25848.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25848.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

49 ZermeloFraenkel set theory 26049.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26049.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

49.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26149.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 26149.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26149.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26249.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26249.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26349.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26449.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26449.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

49.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xvi CONTENTS

49.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26549.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

49.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26649.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26749.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26749.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26849.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 269

49.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26949.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27649.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Chapter 1

(, )-denition of limit

x

y

c

L

Whenever a point x is within units of c, f(x) is within units of L

In calculus, the (, )-denition of limit ("epsilon-delta denition of limit) is a formalization of the notion of limit.It was rst given by Bernard Bolzano in 1817. Augustin-Louis Cauchy never gave an ( "; ) denition of limit inhis Cours d'Analyse, but occasionally used "; arguments in proofs. The denitive modern statement was ultimately

1

2 CHAPTER 1. (, )-DEFINITION OF LIMIT

provided by Karl Weierstrass.[1][2]

1.1 HistoryIsaac Newton was aware, in the context of the derivative concept, that the limit of the ratio of evanescent quantitieswas not itself a ratio, as when he wrote:

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they canapproach so closely that their dierence is less than any given quantity...

Occasionally Newton explained limits in terms similar to the epsilon-delta denition.[3] Augustin-Louis Cauchy gavea denition of limit in terms of a more primitive notion he called a variable quantity. He never gave an epsilon-delta denition of limit (Grabiner 1981). Some of Cauchys proofs contain indications of the epsilon, delta method.Whether or not his foundational approach can be considered a harbinger of Weierstrasss is a subject of scholarlydispute. Grabiner feels that it is, while Schubring (2005) disagrees.[1] Nakane concludes that Cauchy and Weierstrassgave the same name to dierent notions of limit.[4]

1.2 Informal statementLet f be a function. To say that

limx!c f(x) = L

means that f(x) can be made as close as desired to L by making the independent variable x close enough, but notequal, to the value c.How close is close enough to c" depends on how close one wants to make f(x) to L. It also of course depends onwhich function f is and on which number c is. Therefore let the positive number (epsilon) be how close one wishesto make f(x) to L; strictly one wants the distance to be less than . Further, if the positive number is how close onewill make x to c, and if the distance from x to c is less than (but not zero), then the distance from f(x) to L will beless than . Therefore depends on . The limit statement means that no matter how small is made, can be madesmall enough.The letters and can be understood as error and distance, and in fact Cauchy used as an abbreviation forerror in some of his work.[1] In these terms, the error () in the measurement of the value at the limit can be madeas small as desired by reducing the distance () to the limit point.This denition also works for functions with more than one argument. For such functions, can be understood as theradius of a circle or a sphere or some higher-dimensional analogy centered at the point where the existence of a limitis being proven, in the domain of the function and, for which, every point inside maps to a function value less than away from the value of the function at the limit point.

1.3 Precise statementThe ("; ) denition of the limit of a function is as follows:[5]

Let f : D ! R be a function dened on a subset D R , let c be a limit point of D , and let L be a real number.Then

the function f has a limit L at c

is dened to mean

for all " > 0 , there exists a > 0 such that for all x in D that satisfy 0 < jx cj < , the inequalityjf(x) Lj < " holds.

1.4. WORKED EXAMPLE 3

Symbolically:

limx!c f(x) = L () (8" > 0)(9 > 0)(8x 2 D)(0 < jx cj < ) jf(x) Lj < ")

1.4 Worked exampleLet us prove the statement that

limx!5

(3x 3) = 12:

This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introductionto proof. According to the formal denition above, a limit statement is correct if and only if conning x to units ofc will inevitably conne f(x) to " units of L . In this specic case, this means that the statement is true if and onlyif conning x to units of 5 will inevitably conne

3x 3

to " units of 12. The overall key to showing this implication is to demonstrate how and " must be related to eachother such that the implication holds. Mathematically, we want to show that

0 < jx 5j < ) j(3x 3) 12j < ":

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

jx 5j < "/3;which immediately gives the required result if we choose

= "/3:

Thus the proof is completed. The key to the proof lies in the ability of one to choose boundaries in x , and thenconclude corresponding boundaries in f(x) , which in this case were related by a factor of 3, which is entirely due tothe slope of 3 in the line

y = 3x 3:

1.5 ContinuityA function f is said to be continuous at c if it is both dened at c and its value at c equals the limit of f as x approachesc:

limx!c f(x) = f(c):

If the condition 0 < |x c| is left out of the denition of limit, then requiring f(x) to have a limit at c would be thesame as requiring f(x) to be continuous at c.f is said to be continuous on an interval I if it is continuous at every point c of I.

4 CHAPTER 1. (, )-DEFINITION OF LIMIT

1.6 Comparison with innitesimal denitionKeisler proved that a hyperreal denition of limit reduces the quantier complexity by two quantiers.[6] Namely,f(x) converges to a limit L as x tends to a if and only if for every innitesimal e, the value f(x + e) is innitelyclose to L; see microcontinuity for a related denition of continuity, essentially due to Cauchy. Innitesimal cal-culus textbooks based on Robinson's approach provide denitions of continuity, derivative, and integral at standardpoints in terms of innitesimals. Once notions such as continuity have been thoroughly explained via the approachusing microcontinuity, the epsilon-delta approach is presented as well. Karel Hrbacek argues that the denitions ofcontinuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the - methodin order to cover also non-standard values of the input[7] Baszczyk et al. argue that microcontinuity is useful indeveloping a transparent denition of uniform continuity, and characterize the criticism by Hrbacek as a dubiouslament.[8] Hrbacek proposes an alternative non-standard analysis, which is unlike Robinsons having many levelsof innitesimals, so that limits at one level can be dened in terms of innitesimals at the next level.[9]

1.7 See also Continuous function Limit of a sequence List of calculus topics

1.8 Notes[1] Grabiner, Judith V. (March 1983), Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus (PDF),

The American Mathematical Monthly (Mathematical Association of America) 90 (3): 185194, doi:10.2307/2975545,JSTOR 2975545, archived from the original on 2009-05-03, retrieved 2009-05-01

[2] Cauchy, A.-L. (1823), Septime Leon - Valeurs de quelques expressions qui se prsentent sous les formes indtermines11 ;10; : : : Relation qui existe entre le rapport aux dirences nies et la fonction drive, Rsum des leons donnes lcole royale polytechnique sur le calcul innitsimal, Paris, archived from the original on 2009-05-03, retrieved 2009-05-01, p. 44.. Accessed 2009-05-01.

[3] Pourciau, B. (2001), Newton and the Notion of Limit, Historia Mathematica 28 (1), doi:10.1006/hmat.2000.2301

[4] Nakane, Michiyo. Did Weierstrasss dierential calculus have a limit-avoiding character? His denition of a limit in style. BSHM Bull. 29 (2014), no. 1, 5159.

[5] Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math. p. 83. ISBN 978-0070542358.

[6] Keisler, H. Jerome (2008), Quantiers in limits (PDF), Andrzej Mostowski and foundational studies, IOS, Amsterdam,pp. 151170

[7] Hrbacek, K. (2007), Stratied Analysis?", in Van Den Berg, I.; Neves, V., The Strength of Nonstandard Analysis, Springer

[8] Baszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), Ten misconceptions from the history of analysis and their debunk-ing, Foundations of Science, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8

[9] Hrbacek, K. (2009). Relative set theory: Internal view. Journal of Logic and Analysis 1.

1.9 Bibliography Grabiner, Judith V. The origins of Cauchys rigorous calculus. MIT Press, Cambridge, Mass.-London, 1981. Schubring, Gert (2005), Conicts Between Generalization, Rigor, and Intuition: Number Concepts Underlyingthe Development of Analysis in 17th19th Century France and Germany, Springer, ISBN 0-387-22836-5

Chapter 2

Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy orKleene-Mostowski hierarchy classiescertain sets based on the complexity of formulas that dene them. Any set that receives a classication is calledarithmetical.The arithmetical hierarchy is important in recursion theory, eective descriptive set theory, and the study of formaltheories such as Peano arithmetic.The Tarski-Kuratowski algorithm provides an easy way to get an upper bound on the classications assigned to aformula and the set it denes.The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additionalformulas and sets.

2.1 The arithmetical hierarchy of formulas

The arithmetical hierarchy assigns classications to the formulas in the language of rst-order arithmetic. The clas-sications are denoted 0n and 0n for natural numbers n (including 0). The Greek letters here are lightface symbols,which indicates that the formulas do not contain set parameters.If a formula is logically equivalent to a formula with only bounded quantiers then is assigned the classications00 and 00 .The classications 0n and 0n are dened inductively for every natural number n using the following rules:

If is logically equivalent to a formula of the form 9n19n2 9nk , where is 0n , then is assigned theclassication 0n+1 .

If is logically equivalent to a formula of the form 8n18n2 8nk , where is 0n , then is assigned theclassication 0n+1 .

Also, a 0n formula is equivalent to a formula that begins with some existential quantiers and alternates n 1 timesbetween series of existential and universal quantiers; while a 0n formula is equivalent to a formula that begins withsome universal quantiers and alternates similarly.Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantiers isassigned at least one classication. Because redundant quantiers can be added to any formula, once a formula isassigned the classication 0n or 0n it will be assigned the classications 0m and 0m for every m greater than n.The most important classication assigned to a formula is thus the one with the least n, because this is enough todetermine all the other classications.

5

6 CHAPTER 2. ARITHMETICAL HIERARCHY

2.2 The arithmetical hierarchy of sets of natural numbers

A set X of natural numbers is dened by formula in the language of Peano arithmetic (the rst-order language withsymbols 0 for zero, S for the successor function, "+" for addition, "" for multiplication, and "=" for equality), ifthe elements of X are exactly the numbers that satisfy . That is, for all natural numbers n,

n 2 X , N j= (n);

where n is the numeral in the language of arithmetic corresponding to n . A set is denable in rst order arithmeticif it is dened by some formula in the language of Peano arithmetic.Each set X of natural numbers that is denable in rst order arithmetic is assigned classications of the form 0n ,0n , and 0n , where n is a natural number, as follows. If X is denable by a 0n formula then X is assigned theclassication 0n . If X is denable by a 0n formula then X is assigned the classication 0n . If X is both 0n and0n then X is assigned the additional classication 0n .Note that it rarely makes sense to speak of 0n formulas; the rst quantier of a formula is either existential oruniversal. So a 0n set is not dened by a 0n formula; rather, there are both 0n and 0n formulas that dene the set.A parallel denition is used to dene the arithmetical hierarchy on nite Cartesian powers of the natural numbers.Instead of formulas with one free variable, formulas with k free number variables are used to dene the arithmeticalhierarchy on sets of k-tuples of natural numbers.

2.3 Relativized arithmetical hierarchies

Just as we can dene what it means for a set X to be recursive relative to another set Y by allowing the computationdening X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and dene what itmeans for X to be 0n , 0n or 0n in Y, denoted respectively 0;Yn 0;Yn and 0;Yn . To do so, x a set of integersY and add a predicate for membership in Y to the language of Peano arithmetic. We then say that X is in 0;Yn ifit is dened by a 0n formula in this expanded language. In other words X is 0;Yn if it is dened by a 0n formulaallowed to ask questions about membership in Y. Alternatively one can view the 0;Yn sets as those sets that can bebuilt starting with sets recursive in Y and alternately taking unions and intersections of these sets up to n times.For example let Y be a set of integers. Let X be the set of numbers divisible by an element of Y. Then X is denedby the formula (n) = 9m9t(Y (m) ^m t = n) so X is in 0;Y1 (actually it is in 0;Y0 as well since we couldbound both quantiers by n).

2.4 Arithmetic reducibility and degrees

Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility.A set is arithmetical (also arithmetic and arithmetically denable) if it is dened by some formula in the languageof Peano arithmetic. Equivalently X is arithmetical if X is 0n or 0n for some integer n. A set X is arithmeticalin a set Y, denoted X A Y , if X is denable a some formula in the language of Peano arithmetic extended by apredicate for membership in Y. Equivalently, X is arithmetical in Y if X is in 0;Yn or 0;Yn for some integer n. Asynonym for X A Y is: X is arithmetically reducible to Y.The relation X A Y is reexive and transitive, and thus the relation A dened by the rule

X A Y , X A Y ^ Y A X

is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partiallyordered under A .

2.5. THE ARITHMETICAL HIERARCHY OF SUBSETS OF CANTOR AND BAIRE SPACE 7

2.5 The arithmetical hierarchy of subsets of Cantor and Baire spaceThe Cantor space, denoted 2! , is the set of all innite sequences of 0s and 1s; the Baire space, denoted !! or N ,is the set of all innite sequences of natural numbers. Note that elements of the Cantor space can be identied withsets of integers and elements of the Baire space with functions from integers to integers.The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantiers cannaturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classication 0n if it isdenable by a 0n formula. The set is assigned the classication 0n if it is denable by a 0n formula. If the set is both0n and 0n then it is given the additional classication 0n . For example let O 2! be the set of all innite binarystrings which aren't all 0 (or equivalently the set of all non-empty sets of integers). AsO = fX 2 2!j9n(X(n) = 1)gwe see that O is dened by a 01 formula and hence is a 01 set.Note that while both the elements of the Cantor space (regarded as sets of integers) and subsets of the Cantor spaceare classied in arithmetic hierarchies, these are not the same hierarchy. In fact the relationship between the twohierarchies is interesting and non-trivial. For instance the 0n elements of the Cantor space are not (in general)the same as the elements X of the Cantor space so that fXg is a 0n subset of the Cantor space. However, manyinteresting results relate the two hierarchies.There are two ways that a subset of Baire space can be classied in the arithmetical hierarchy.

A subset of Baire space has a corresponding subset of Cantor space under the map that takes each functionfrom ! to ! to the characteristic function of its graph. A subset of Baire space is given the classication 1n ,1n , or 1n if and only if the corresponding subset of Cantor space has the same classication.

An equivalent denition of the analytical hierarchy on Baire space is given by dening the analytical hierarchyof formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets ofCantor space can be dened from the hierarchy on Baire space. This alternate denition gives exactly the sameclassications as the rst denition.

A parallel denition is used to dene the arithmetical hierarchy on nite Cartesian powers of Baire space or Cantorspace, using formulas with several free variables. The arithmetical hierarchy can be dened on any eective Polishspace; the denition is particularly simple for Cantor space and Baire space because they t with the language ofordinary second-order arithmetic.Note that we can also dene the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some setof integers. In fact boldface 0n is just the union of 0;Yn for all sets of integers Y. Note that the boldface hierarchyis just the standard hierarchy of Borel sets.

2.6 Extensions and variationsIt is possible to dene the arithmetical hierarchy of formulas using a language extended with a function symbol foreach primitive recursive function. This variation slightly changes the classication of some sets.A more semantic variation of the hierarchy can be dened on all nitary relations on the natural numbers; the followingdenition is used. Every computable relation is dened to be 00 and 00 . The classications 0n and 0n are denedinductively with the following rules.

If the relationR(n1; : : : ; nl;m1; : : : ;mk) is0n then the relationS(n1; : : : ; nl) = 8m1 8mkR(n1; : : : ; nl;m1; : : : ;mk)is dened to be 0n+1

If the relationR(n1; : : : ; nl;m1; : : : ;mk) is0n then the relationS(n1; : : : ; nl) = 9m1 9mkR(n1; : : : ; nl;m1; : : : ;mk)is dened to be 0n+1

This variation slightly changes the classication of some sets. It can be extended to cover nitary relations on thenatural numbers, Baire space, and Cantor space.

8 CHAPTER 2. ARITHMETICAL HIERARCHY

2.7 Meaning of the notationThe following meanings can be attached to the notation for the arithmetical hierarchy on formulas.The subscript n in the symbols 0n and 0n indicates the number of alternations of blocks of universal and existentialnumber quantiers that are used in a formula. Moreover, the outermost block is existential in 0n formulas anduniversal in 0n formulas.The superscript 0 in the symbols 0n , 0n , and 0n indicates the type of the objects being quantied over. Type0 objects are natural numbers, and objects of type i + 1 are functions that map the set of objects of type i to thenatural numbers. Quantication over higher type objects, such as functions from natural numbers to natural numbers,is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantiers overnumbers, the superscript 1 would indicate quantication over functions from numbers to numbers (type 1 objects),the superscript 2 would correspond to quantication over functions that take a type 1 object and return a number, andso on.

2.8 Examples

The 01 sets of numbers are those denable by a formula of the form 9n1 9nk (n1; : : : ; nk;m) where has only bounded quantiers. These are exactly the recursively enumerable sets.

The set of natural numbers that are indices for Turing machines that compute total functions is 02 . Intuitively,an index e falls into this set if and only if for every m there is an s such that the Turing machine with indexe halts on input m after s steps. A complete proof would show that the property displayed in quotes in theprevious sentence is denable in the language of Peano arithmetic by a 01 formula.

Every 01 subset of Baire space or Cantor space is an open set in the usual topology on the space. Moreover,for any such set there is a computable enumeration of Gdel numbers of basic open sets whose union is theoriginal set. For this reason, 01 sets are sometimes called eectively open. Similarly, every 01 set is closedand the 01 sets are sometimes called eectively closed.

Every arithmetical subset of Cantor space or Baire space is a Borel set. The lightface Borel hierarchy extendsthe arithmetical hierarchy to include additional Borel sets. For example, every 02 subset of Cantor or Bairespace is a G set (that is, a set which equals the intersection of countably many open sets). Moreover, eachof these open sets is 01 and the list of Gdel numbers of these open sets has a computable enumeration.If (X;n;m) is a 00 formula with a free set variable X and free number variables n;m then the 02 setfX j 8n9m(X;n;m)g is the intersection of the 01 sets of the form fX j 9m(X;n;m)g as n ranges overthe set of natural numbers.

2.9 PropertiesThe following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchyof subsets of Cantor or Baire space.

The collections 0n and 0n are closed under nite unions and nite intersections of their respective elements.

A set is 0n if and only if its complement is 0n . A set is 0n if and only if the set is both 0n and 0n , in whichcase its complement will also be 0n .

The inclusions 0n ( 0n and 0n ( 0n hold for n 1 .

The inclusions 0n ( 0n+1 and 0n ( 0n+1 hold for all n and the inclusion 0n [ 0n ( 0n+1 holds forn 1 . Thus the hierarchy does not collapse.

2.10. RELATION TO TURING MACHINES 9

2.10 Relation to Turing machinesThe Turing computable sets of natural numbers are exactly the sets at level 01 of the arithmetical hierarchy. Therecursively enumerable sets are exactly the sets at level 01 .No oracle machine is capable of solving its own halting problem (a variation of Turings proof applies). The haltingproblem for a 0;Yn oracle in fact sits in 0;Yn+1 .Posts theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and theTuring degrees. In particular, it establishes the following facts for all n 1:

The set ;(n) (the nth Turing jump of the empty set) is many-one complete in 0n . The set N n ;(n) is many-one complete in 0n . The set ;(n1) is Turing complete in 0n .

The polynomial hierarchy is a feasible resource-bounded version of the arithmetical hierarchy in which polyno-mial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on theTuring machines involved). It gives a ner classication of some sets of natural numbers that are at level 01 of thearithmetical hierarchy.

2.11 See also Interpretability logic Hierarchy (mathematics) Polynomial hierarchy

2.12 References Japaridze, Giorgie (1994), The logic of arithmetical hierarchy, Annals of Pure and Applied Logic 66 (2):

89112, doi:10.1016/0168-0072(94)90063-9, Zbl 0804.03045.

Moschovakis, Yiannis N. (1980), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics100, North Holland, ISBN 0-444-70199-0, Zbl 0433.03025.

Nies, Andr (2009), Computability and randomness, Oxford Logic Guides 51, Oxford: Oxford UniversityPress, ISBN 978-0-19-923076-1, Zbl 1169.03034.

Rogers, H., jr (1967), Theory of recursive functions and eective computability, Maidenhead: McGraw-Hill,Zbl 0183.01401.

Chapter 3

Axiom schema of predicative separation

In axiomatic set theory, the axiom schema of predicative separation, or of restricted, or 0 separation, is aschema of axioms which is a restriction of the usual axiom schema of separation in ZermeloFraenkel set theory. Itonly asserts the existence of a subset of a set if that subset can be dened without reference to the entire universeof sets. The axiom appears in the systems of constructive set theory CST and CZF, as well as in the system ofKripkePlatek set theory. The name 0 comes from the Levy hierarchy (in analogy with the arithmetic hierarchy).The formal statement of this is the same as full separation schema, but with a restriction on the formulas that may beused. For any formula :

8x 9y 8z (z 2 y $ z 2 x ^ (z))

provided, as usual, that the variable y is not free in ; but also provided that contains only bounded quantiers. Thatis, all quantiers in (if there are any) must appear in the form 9x 2 y (x) or 8x 2 y (x) for some sub-formula.The meaning of this is that, given any set x, and any predicate there is a set y whose elements are the elements ofx which satisfy , provided only quanties over existing sets, and never quanties over all sets. This restriction isnecessary from a predicative point of view, since the universe of all sets contains the set being dened. If it werereferenced in the denition of the set, the denition would be circular.Although the schema contains one axiom for each restricted formula , it is possible in CZF to replace this schemawith a nite number of axioms.

10

Chapter 4

Boolean satisability problem

3SAT redirects here. For the Central European television network, see 3sat.

In computer science, the Boolean Satisability Problem (sometimes called Propositional Satisability Problemand abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation thatsatises a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can beconsistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is thecase, the formula is called satisable. On the other hand, if no such assignment exists, the function expressed by theformula is identically FALSE for all possible variable assignments and the formula is unsatisable. For example, theformula "a AND NOT b" is satisable because one can nd the values a = TRUE and b = FALSE, which make (aAND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisable.SAT is one of the rst problems that was proven to be NP-complete. This means that all problems in the complexityclass NP, which includes a wide range of natural decision and optimization problems, are at most as dicult to solve asSAT. There is no known algorithm that eciently solves SAT, and it is generally believed that no such algorithm exists;yet this belief has not been proven mathematically, and resolving the question whether SAT has an ecient algorithmis equivalent to the P versus NP problem, which is the most famous open problem in the theory of computing.Despite the fact that no algorithms are known that solve SAT eciently, correctly, and for all possible input instances,many instances of SAT that occur in practice, such as in articial intelligence, circuit design and automatic theoremproving, can actually be solved rather eciently using heuristical SAT-solvers. Such algorithms are not believed to beecient on all SAT instances, but experimentally these algorithms tend to work well for many practical applications.

4.1 Basic denitions and terminologyA propositional logic formula, also calledBoolean expression, is built from variables, operators AND (conjunction,also denoted by ), OR (disjunction, ), NOT (negation, ), and parentheses. A formula is said to be satisable ifit can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The Booleansatisability problem (SAT) is, given a formula, to check whether it is satisable. This decision problem is ofcentral importance in various areas of computer science, including theoretical computer science, complexity theory,algorithmics, cryptography and articial intelligence.There are several special cases of the Boolean satisability problem in which the formulas are required to have aparticular structure. A literal is either a variable, then called positive l