So, I don't know much about countable models of set theory, other than that they exist. To me, their existence is a very weird thing (and a reason to move away from first-order formulations). Here is one thing that really weirds me out:
We have definitions of infinity, some set is infinite if it can be put into 1-1 correspondence with the natural numbers or a set is infinite if it can be put into 1-1 correspondence with a proper subset of itself (Dedekind infinite, if I recall).
How do these definitions work when we're in a countable model of ZFC? In particular, given that we know that the powerset of a countably infinite set will be uncountably infinite, how are we to understand this given that our model is countable? What does "uncountable" mean in such a context?