Overflow Theorem

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Theorem

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

Then $F$ has an infinite model.

Corollary

The class of finite models is not $\Delta$-elementary.

That is:

there is no set of formulas $F$ such that $F$ is satisfied by a model $\MM$

if and only if:

$\MM$ is finite.

Proof

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

$\exists x_1 \exists x_2 \ldots \exists x_n: \paren {x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n - 1} \ne x_n}$

Then $\mathbf A_i$ is true in a structure $\AA$ if and only if $\AA$ has at least $n$ elements.

Take:

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

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 $\MM$.

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

So $A$ has an infinite model.

$\blacksquare$

Also see

• Upward Löwenheim-Skolem Theorem

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.7$ First-Order Logic Semantics: Theorem $\mathrm{II.7.17}$