Another surprising connection: The Ax-Kochen theorem that for each positive integer $d$ there is a finite set $Y_d$ of prime numbers, such that if $p$ is any prime not in $Y_d$ then every homogeneous polynomial of degree $d$ over the $p$-adic numbers in at least $d^2+1$ variables has a nontrivial zero.

This was proved using model theory.

Another surprising connection is the Ax-Kochen theorem. Let $\mathcal{F}_{p,n,d}$ denote the set of homogeneous polynomials ("forms") in $n$ variables over the $p$-adics $\mathbb{Q}_p$ of degree $d$. The Ax-Kochen theorem is: For every positive integer $d$ there is a finite set $Y_d$ of "bad" prime numbers such that if $p$ is a "good" prime for $d$ (i.e. not in $Y_d$) then every $f \in \mathcal{F}_{p,d^2+1,d}$ has a non-trivial zero.

This was proved using model theory.