Compactness theorem
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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 hence, the theorem's name. Likewise, it is analogous to the finite intersection property characterizat...
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 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 iff every finite subcollection has a non-empty intersection.
