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 .
One can give a direct result of one half of this theorem using ultraproducts and Los’s theorem, but we can also appeal to the compactness theorem. Let be the set of axioms of an algebraically closed field. For each prime , let be the sentence ( times).
Suppose is true in an algebraically closed field of characteristic zero and hence in all; by Vaught’s test, the theory of algebraically closed fields of a fixed characteristic is complete. Then implies ; a finite subset of these axioms thus must imply (this is a consequence of the compactness theorem). In particular, is implied by together with some finite number of the conditions , which means it is true in algebraically closed fields of characteristics outside a finite number of primes.
Conversely, suppose is true in algebraically closed fields of arbitrarily high characteristic. Then any finite subset of is satisfiable, so by compactness, the whole collection is satisfiable. Therefore, holds in some algebraically closed field of characteristic zero.
Here is another application:
Corollary 2 Let be absolutely irreducible, i.e. irreducible in . Then is absolutely irreducible in for almost all primes .
Indeed, the statement about irreducibility can be made into a first-order property. Note that absolute irreducibility is necessary! Consider ; this is irreducible in , but it is reducible in if is a quadratic residue modulo . In fact, the theory of fields of a specified characteristic is far from complete.
Finally, I will formally prove a stronger variant of the Ax-Grothendieck theorem:
Theorem 3 Let be an algebraic set defined over an algebraically closed field , let be a morphism. Suppose it is injective. Then it is surjective.
Consider the statement : “Whenever are polynomials of degree at most in -variables satisfying the following condition: for all with and some , then some , we have the following conclusion: for any satisfying each , there exist satisfying each such that maps to .” This is clearly a first-order condition, and it holds in each finite field because an injective map on a finite set is surjective! It thus holds in the algebraic closure of each finite field by a direct limit argument. It thus holds in algebraically closed fields of characteristic zero by Robinson’s theorem. Taking the conjunction of all , one obtains the theorem.
Source: David Marker, Model Theory: An Introduction