Model theory often provides a framework from one which one can obtain “finitary” versions of infinitary results, and vice versa.

One spectacular example is the Ax-Grothendieck theorem, which states that an injective polynomial map ${P: \mathbb{C}^n \rightarrow \mathbb{C}^n}$ is surjective. The key idea here is that the theorem for polynomial maps of a fixed degree is a statement of first-order logic, to which the compactness theorem applies. Next, the theorem is trivial when ${\mathbb{C}}$ is replaced by a finite field, and one then deduces it for ${\overline{\mathbb{F}_p}}$ (and maps ${P: \overline{\mathbb{F}_p} \rightarrow \overline{\mathbb{F}_p}}$ by an inductive limit argument. It then holds for algebraically closed fields of nonzero characteristic, because ${ACF_p}$ is a complete theory—any first-order statement true in one algebraically closed field of characteristic ${p}$ is true in any such field. Finally, one appeals to a famous result of Abraham Robinson that any first-order statement true in algebraically closed fields of characteristic ${p>p_0}$ is true in algebraically closed fields of characteristic zero.

There is a discussion of this result and other proofs by Terence Tao here.

For fun, I will formally state and prove Robinson’s theorem.

Theorem 1 (A. Robinson) Let ${S}$ be a statement in first-order logic in the language of fields (i.e., referring to the operations of addition and multiplication, and the constants ${0,1}$). Then ${S}$ is true in algebraically closed fields of characteristic zero if and only if ${S}$ is true algebraically closed fields of arbitrarily high (or sufficiently high, ${p>p_0}$) characteristic ${p}$. (more…)