Barwise compactness theorem

In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.

Statement

Let be a countable admissible set. Let be an -finite relational language. Suppose is a set of -sentences, where is a set with parameters from , and every -finite subset of is satisfiable. Then is satisfiable.