Barwise Compactness Theorem

The Barwise compactness theorem, proved by Jon Barwise in his 1967 Stanford doctoral dissertation Infinitary Logic and Admissible Sets, generalizes the ordinary compactness theorem of first-order logic to a class of infinitary languages, formal systems that allow infinitely long conjunctions and disjunctions. Working within the framework of admissible sets, Barwise identified the conditions a language and a set of sentences must satisfy for the same compactness conclusion, that a set of sentences is satisfiable if every admissible subset of it is satisfiable, to continue to hold. The result is distinct from the classical compactness theorem for first-order logic, which it extends rather than restates, and is a foundational tool in the model theory of infinitary logic.