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Topic: Compactness theorem

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	Compactness theorem - Wikipedia, the free encyclopedia
		The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i.e., has a model, if and only if every finite subset of it is satisfiable.
		The compactness theorem for the propositional calculus is in a sense equivalent to the topological compactness of Stone spaces; hence the theorem's name.
		One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it.
	en.wikipedia.org /wiki/Compactness_theorem (327 words)

	Compactness theorem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		From the theorem it follows for instance that if some first-order sentence holds for every (A piece of land cleared of trees and usually enclosed) field of (A distinguishing quality) characteristic zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p.
		Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows.
		Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found, i.e., proofs that refer to truth but not to provability.
	www.absoluteastronomy.com /encyclopedia/c/co/compactness_theorem.htm (331 words)

	Compactness (Site not responding. Last check: 2007-10-07)
		A note on the discrete compactness property and the de Rham complex...
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		Compactness of pseudohermitian structures with integral bounds on curvature...
	www.scienceoxygen.com /math/339.html (247 words)

	Learn more about Model theory in the online encyclopedia. (Site not responding. Last check: 2007-10-07)
		The compactness theorem states that a set of sentences S is satisfiable, i.e., has a model, if every finite subset of S is satisfiable.
		In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof; in the context of model theory however, this proof is somewhat more difficult.
		This is expressed in the Lowenheim-Skolem theorems - which state that any theory with an infinite model A has models of all infinite cardinalities (at least that of the language) which agree with A on all sentences - they are "elementarily equivalent".
	www.onlineencyclopedia.org /m/mo/model_theory.html (797 words)

	Foundations of Mathematics
		Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries - by Marvin J. Greenberg
		On Herbrand's Theorem - by Samuel R. Buss
		First-order languages, elementary maps, compactness theorem, diagram lemma, Lyndon interpolation theorem, omitting types theorem
	sakharov.net /foundation_rt.html (2708 words)

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