Editing Compactness theorem (section)

{{mvar|}}{{Short description|Theorem in mathematical logic}}
In [[mathematical logic]], the '''compactness theorem''' states that a [[Set (mathematics)|set]] of [[First-order predicate calculus|first-order]] [[Sentence (mathematical logic)|sentences]] has a [[Model (model theory)|model]] if and only if every [[Finite set|finite]] [[subset]] of it has a model. This theorem is an important tool in [[model theory]], as it provides a useful (but generally not [[effective method|effective]]) method for constructing models of any set of sentences that is finitely [[Consistency|consistent]].

The compactness theorem for the [[propositional calculus]] is a consequence of [[Tychonoff's theorem]] (which says that the [[Product topology|product]] of [[compact space]]s is compact) applied to compact [[Stone space]]s,{{sfn|Truss|1997}} hence the theorem's name. Likewise, it is analogous to the [[finite intersection property]] characterization of compactness in [[topological space]]s: a collection of [[closed set]]s in a compact space has a [[Empty set|non-empty]] [[Intersection (set theory)|intersection]] if every finite subcollection has a non-empty intersection.

The compactness theorem is one of the two key properties, along with the downward [[Löwenheim–Skolem theorem]], that is used in [[Lindström's theorem]] to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.<ref>J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [https://projecteuclid.org/euclid.pl/1235417263#toc], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.{{doi|10.2307/2274031}} {{JSTOR|2274031}}</ref>