Overflow Theorem

Theorem
Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size.

Then $F$ has an infinite model.

Proof
For each $n$, let $\mathbf A_n$ be the formula:


 * $\exists x_1 \exists x_2 \ldots \exists x_n: \left({x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n-1} \ne x_n }\right)$

Then $\mathbf A_i$ is true in a structure $\mathcal A$ iff $\mathcal A$ has at least $n$ elements.

Take:
 * $\displaystyle \Gamma := A \cup \bigcup_{i \mathop = 1}^\infty A_i$

Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.

From the Compactness Theorem, $\Gamma$ is satisfiable in some model $\mathcal M$.

But since $\mathcal M \models A_i$ for each $i$, $\mathcal M$ must be infinite.

So $A$ has an infinite model.

Also see

 * Upward Löwenheim-Skolem Theorem