We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?)
I suspect that the theory of Galois groups is not a first-order theory, because otherwise we could apply Loewenheim-Skolem's theorem, since there exists an infinite Galois group (e.g. $\hat{\mathbb Z}$), and thus obtain a countable model, i.e. a countable Galois group. So I can't apply anything of the model theory I know (i. e. only first-order).
Maybe an ultraproduct construction could help? Thank you very much.