Compactness Theorem

Theorem
Let $\LL$ be the language of predicate logic.

Let $T$ be a set of $\LL$-sentences.

Then $T$ is satisfiable $T$ is finitely satisfiable.

Proof
By definition, $T$ is finitely satisfiable means that every finite subset of $T$ is satisfiable.

Because the direction:
 * $T$ satisfiable implies $T$ finitely satisfiable

is trivial, the proofs below all justify the converse:
 * $T$ finitely satisfiable implies $T$ satisfiable.

Also see

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