How to prove that CC($\mathbb R$) is true in permutation models

Question

I've familiarised myself with models of set theory and am beginning to understand the basics, but am still very far away from being a proper model theorist. I currently live under the impression that I understand the idea of the construction of the first and second Fraenkel models.

Hence, unfortunately, I must ask how one proves that the axiom of countable choice for subsets of $\mathbb R$ holds in permutation models.

As far as I'm aware, it is not necessary to assume the axiom of choice when constructing permutation models.

There is a particular issue that caused me some trouble. When we start constructing some permutation model, we assume that we already have some set theory as a metatheory. In this set theory, the atoms are mere sets, and thus, if we start building up the von Neumann hierarchy with atoms, all atoms will appear at some point in the kernel, which is a problem, because atoms should not be in the kernel.

What am I missing here?

EDIT: Or are the atoms greater than some strongly inaccessible cardinal?

EDIT 2: Or do we start from atoms and the emtpy set and do a construction analogous to the constructible universe?

Answer 1

Trivially.

Since permutation models preserve choice in the pure sets (or, kernel), and since $\Bbb R$ is in bijection with $\cal P(\omega)$, it must be well-orderable.

Answer 2

I think that I may have discovered an important point here. For if we just assume that AC holds in the meta-theory, then all the statements we prove using permutation models are of the form "If the axiom of choice is true, then statement XYZ is not implied by the axioms." But this is unsatisfactory, because we don't know what happens if AC is not true.

Thus, in my upcoming paper I'll first construct the "ZFA version" of the constructible universe, and then take my permutation models to be subclasses of this universe.

EDIT: Or alternatively, if that will not give me enough sets, I will construct a ZFA model inside the constructible universe.