# Barwise compactness theorem

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.