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.

Let ${\displaystyle A}$ be a countable admissible set. Let ${\displaystyle L}$ be an ${\displaystyle A}$-finite relational language. Suppose ${\displaystyle \Gamma }$ is a set of ${\displaystyle L_{A}}$-sentences, where ${\displaystyle \Gamma }$ is a ${\displaystyle \Sigma _{1}}$ set with parameters from ${\displaystyle A}$, and every ${\displaystyle A}$-finite subset of ${\displaystyle \Gamma }$ is satisfiable. Then ${\displaystyle \Gamma }$ is satisfiable.