TITLE: Are all countable, nonstandard models of arithmetic given by ultrapowers?
QUESTION [7 upvotes]: Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest).  The only way I've seen to construct a nonstandard model is through taking an ultraproduct or, equivalently, using the compactness theorem.  My question is wether or not these are all the models there are?  There are continuum many ultrafilters and continuum many nonstandard, countable models, but I don't know if there's a surjective correspondence.

REPLY [4 votes]: It could be mentioned perhaps that Skolem's non-standard model of arithmetic (1933-1934) is countable and is a kind of a "definable" version of the ultraproduct construction.  Namely, Skolem only uses definable sequences in his construction.  The advantage of his model is that it is constructed without using the axiom of choice.