Compactness Theorem. A set of wffs is satisfiable iff every finite subset is satisfiable.

Corollary. If $\Sigma\vDash\tau$ , then there is a finite subset $\Sigma_{0}\subseteq\Sigma$ s.t. $\Sigma_{0}\vDash\tau$ .

We prove the theorem assuming that the corollary is true. Let $\Sigma$ be a set of wffs. If $\Sigma$ is satisfiable, so is every its (finite) subset. The other direction. Suppose that every finite subset of $\Sigma$ is satisfiable. A set of wffs $\Sigma'$ is unsatisfiable iff $\Sigma'\vDash\bot$ (or any other tautologically false wff). If $\Sigma$ were unsatisfiable, then $\Sigma\vDash\bot$ , and, according to the corollary, there would be a finite subset $\Sigma_{0}\subseteq\Sigma$ such that $\Sigma_{0}\vDash\bot$ , meaning that $\Sigma_{0}$ is not satisfiable, and contradicting the assumption. So, $\Sigma$ is satisfiable.