In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, the Barwise compactness theorem, named after Jon Barwise

Jon Barwise

Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used....

, is a generalization of the usual compactness theorem

Compactness theorem

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...

 for first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.

Statement of the theorem

Let T be a set of sentences of a first-order language L. If for each finite subset of T there is a first-order structure making this subset of T true, then there is a first-order structure ℳ that makes all the sentences of T true.
(Language, Proof and Logic)

External links

• http://plato.stanford.edu/entries/logic-infinitary/#5 Stanford Encyclopedia of Philosophy, "Infinitary Logic", Section 5, "Sublanguages of L(ω1,ω) and the Barwise Compactness Theorem"