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

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