Compactness of First-Order Logic

Theorem
Let $\Gamma$ be any countable set of first-order formulas, and suppose every finite subset of $\Gamma$ is satisfiable.

Then $\Gamma$ is satisfiable.

Proof
Suppose $\Gamma$ is unsatisfiable.

Since consistency implies satisfiability, $\Gamma$ is inconsistent, i.e., it proves a contradiction.

But first-order proofs are by definition finite, so there is some subset $\Gamma_0$ of $\Gamma$ such that $\Gamma_0$ is inconsistent.

Now since satisfiability implies consistency, $\Gamma_0$ is unsatisfiable.

Hence if all finite subsets of $\Gamma$ are satisfiable, so is $\Gamma$.

Consequences

 * Overflow Theorem
 * The Class of Finite Models is not $\Delta$-Elementary
 * Existence of Non-Standard Models of Arithmetic