* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Compactness Theorem for First-Order Logic
Science of Logic wikipedia , lookup
Quantum logic wikipedia , lookup
Law of thought wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Gödel's incompleteness theorems wikipedia , lookup
Peano axioms wikipedia , lookup
Infinitesimal wikipedia , lookup
Propositional calculus wikipedia , lookup
Intuitionistic logic wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Interpretation (logic) wikipedia , lookup
Laws of Form wikipedia , lookup
Curry–Howard correspondence wikipedia , lookup
Infinite monkey theorem wikipedia , lookup
Mathematical proof wikipedia , lookup
List of first-order theories wikipedia , lookup
First-order logic wikipedia , lookup
Mathematical logic wikipedia , lookup
Compactness Theorem for First-Order Logic CAS 701 Presentation Exercise 30 part a Presented by : Shahram Siavash Date : November 11, 2002 Department of Computing and Software McMaster University Compactness Theorem Let G be any set of formulas of first-order logic. Then G is satisfiable if every finite subset of G is satisfiable. † † † Proof (Using Completeness Theorem) • If G is finite, then the proof is obvious since G is a finite subset of itself so it’s satisfiable. † † • Proof by contradiction. Suppose G is unsatisfiable. • By Godel’s completeness theorem, G is inconsistent in F (sound and † complete formal system for first-order logic) † G |-F j, ÿj • There is a proof in F from G for both j and ÿj † 2 † † • A proof consists of a finite sequence of formulas. • Let P = set of formulas from G which are used in the proof • P is finite and is a subset of G • P |-F j, ÿj • P |= j, ÿj † † since F is sound † This means that P is unsatisfiable. Contradicting the assumption, so G is † satisfiable. † Question Overview - Use the compactness theorem to prove that every first-order theory that has arbitrarily large finite models, has an infinite model. • Let T = (L, G) 3 † • Define f _k = $ x1 $ x 2 ...$ x k .(x1 ≠ x 2 ) Ÿ (x1 ≠ x 3 ) Ÿ ...(x k-1 ≠ x k ) S = G »{f _ k | k ≥ 2} † † includes infinite number of formulas • Any model satisfying S will be infinite. • Let A be a finite subset of S and K be the maximum index of f_k in A. • Every model with cardinality greater than or equal to K that satisfies G will satisfy A. † • Every finite subset of S is satisfiable • By using compactness theorem, we conclude that S is satisfiable and it’s model should be infinite. • Every model that satisfies S, will also satisfy G so T has an infinite model. 4 † References: 1. http://cheng.ececs.uc.edu/cs543/4-29.html 2. http://www.trentu.ca/academic/math/sb/pcml/pcml-i-15.pdf 5