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 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 is replaced by a finite field, and one then deduces it for (and maps by an inductive limit argument. It then holds for algebraically closed fields of nonzero characteristic, because is a complete theory—any first-order statement true in one algebraically closed field of characteristic 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 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 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 ). Then is true in algebraically closed fields of characteristic zero if and only if is true algebraically closed fields of arbitrarily high (or sufficiently high, ) characteristic . (more…)