I hope this is not a duplicate, if so please let me know.
So in MacLane Moerdijk‘s book Sheaves in Geometry and Logic it is shown that there is a boolean, two-valued topos with nno and choice, in which the continuum hypothesis ($\vert \Bbb N \vert < \vert \cal{P}(\Bbb N)\vert$) fails, ie. that it satisfies NCH. But they only state that Gödel has proven the consistency of CH with ZFC, so that in summary CH is independent from ZFC. I find this kind of unsatisfying, since Gödel worked in a setting quite different to topos theory and would like to know, whether CH is consistent with boolean two-valued topoi with nno and choice (which I will call $\mathsf{Set}$–like). In other words
Is there a topos theoretic proof of the consistency of CH with $\mathsf{Set}$-like topoi (written down somewhere)?
As far as I can tell (I have absolutely no background in set theory) Gödel‘s original proof modifies and plays with the membership relation of ZF. This doesn‘t sound like it carries over to topos theory.
A very naive idea would be, that maybe any $\mathsf{Set}$-like topos $\cal{E}$ has a $\mathsf{Set}$-like subtopos, which satisfies CH. But this would kind of imply that the free $\mathsf{Set}$–like topos (if such a thing even exists) has to satisfy CH. I don‘t know why, but this feels like too strong of a statement.
Another idea would be that, given a $\mathsf{Set}$–like topos $\mathcal{E}$ satisfying NCH, we can turn it into a $\mathsf{Set}$-like topos satisfying CH, by forcing (no pun intended) all strict cardinal inequalities $\Bbb{N} < K$ (for those $K$ strictly smaller than $\cal{P}(\Bbb N)$) to become isomorphisms. Maybe we can modify the filter quotient construction (which I don‘t really know yet) or make a localization / sheafification kind of argument. But since I am fairly new to topos theory, I don’t have made up my mind yet and would like to ask for references or other inputs.
Thank you for your time.