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 ${p}$.

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 ${A}$ be the set of axioms of an algebraically closed field. For each prime ${p}$, let ${R_p}$ be the sentence ${1 + 1 + \dots + 1 = 0}$ ( ${p}$ times).

Suppose ${S}$ 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 ${A \cup \{ \neg R_p \} }$ implies ${S}$; a finite subset of these axioms thus must imply ${S}$ (this is a consequence of the compactness theorem). In particular, ${S}$ is implied by ${A}$ together with some finite number of the conditions ${\{ \neg R_p\}}$, which means it is true in algebraically closed fields of characteristics outside a finite number of primes.

Conversely, suppose ${S}$ is true in algebraically closed fields of arbitrarily high characteristic. Then any finite subset of ${A \cup \{ \neg R_p \} \cup S}$ is satisfiable, so by compactness, the whole collection is satisfiable. Therefore, ${S}$ holds in some algebraically closed field of characteristic zero.

Here is another application:

Corollary 2 Let ${P(X_1, \dots, X_n ) \in \mathbb{Z}[X_1, \dots, X_n]}$ be absolutely irreducible, i.e. irreducible in ${\mathbb{C}[X_1, \dots, X_n]}$. Then ${\bar{P}}$ is absolutely irreducible in ${ \mathbb{F}_p[X_1, \dots, X_n]}$ for almost all primes ${p}$.

Indeed, the statement about irreducibility can be made into a first-order property. Note that absolute irreducibility is necessary! Consider ${X^2-2}$; this is irreducible in ${\mathbb{Q}[X]}$, but it is reducible in ${\mathbb{F}_p[X]}$ if ${2}$ is a quadratic residue modulo ${p}$. 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 ${V}$ be an algebraic set defined over an algebraically closed field ${F}$, let ${P: V \rightarrow V}$ be a morphism. Suppose it is injective. Then it is surjective.

Consider the statement ${S_{m,d,n}}$: “Whenever ${Q_1, \dots, Q_m, P_1, \dots, P_n}$ are polynomials of degree at most ${d}$ in ${n}$-variables satisfying the following condition: for all ${x_1, \dots, x_n, y_1, \dots, y_n \in F}$ with ${Q_1(x)=Q_2(x)=\dots=Q_m(x)=Q_1(y)=\dots=Q_m(y)=0}$ and some ${x_i \neq y_i}$, then some ${P_j(x) \neq P_j(y)}$, we have the following conclusion: for any ${z_1, \dots, z_n}$ satisfying each ${Q_j}$, there exist ${w_1, \dots, w_n}$ satisfying each ${Q_j}$ such that ${P}$ maps ${w}$ to ${z}$.” 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 ${S_{m,d,n}}$, one obtains the theorem.

Source: David Marker, Model Theory: An Introduction