in mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. this theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent. the compactness theorem for the propositional calculus is a consequence of tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact stone spaces;see truss (1997). hence, the theorem's name. likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. the compactness theorem is one of the two key properties, along with the downward lÃ¶wenheimâskolem theorem, that is used in lindstrÃ¶m'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. ==history== kurt gÃ¶del proved the countable compactness theorem in 1930. anatoly maltsev proved the uncountable case in 1936.vaught, robert l.: alfred tarski's work in model theory. j. symbolic logic 51 (1986), no. 4, 869â882 robinson, a.: non-standard analysis. north-holland publishing co., amsterdam 1966. page 48. == applications == the compactness theorem has many applications in model theory; a few typical results are sketched here. the compactness theorem implies robinson's principle: if a first-order sentence holds in every field of characteristic zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. this can be seen as follows: suppose Ï is a sentence that holds in every field of characteristic zero. then its negation Â¬Ï, together with the field axioms and the infinite sequence of sentences 1+1&nbsp;â &nbsp;0, 1+1+1&nbsp;â &nbsp;0, â¦, is not satisfiable (because there is no field of characteristic 0 in which Â¬Ï holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). therefore, there is a finite subset a of these sentences that is not satisfiable. we can assume that a contains Â¬Ï, the field axioms, and, for some k, the first k sentences of the form 1+1+...+1&nbsp;â &nbsp;0 (because adding more sentences doesn't change unsatisfiability). let b contain all the sentences of a except Â¬Ï. then any model of b is a field of characteristic greater than k, and Â¬Ï together with b is not satisfiable. this means that Ï must hold in
Metatheorems Model_theory Theorems_in_the_foundations_of_mathematics