My understanding of the resolution of Skolem's Paradox is that although in a countable model of ZFC there does not exist a bijection between a countable set and its powerset, we can still construct a bijection working outside the model which is a bijection.
But given this, why is there a reason to believe that this cannot always be done for any model? In other words, how do we know that "uncountable" ever means something other than undefinable and that Cantor's Theorem points to combinatorial properties of levels of infinity rather than just an inherent limitation of definable functions, making everything essentially $\aleph_0$?
EDIT
I realize I probably wasn't clear in what I was asking originally so I'll try again: From what I've learned so far about set theory it seems that bijections within a model can fail for possibly one of two reasons:
- Case I: There are "more" elements in one set than the other. This certainly happens with finite sets of different cardinalities and I see this also in the presentation of the diagonal argument on a proposed enumeration of the real numbers in the interval $(0,1)$
- Case II: The sets are both subsets (at least in an informal sense) of countable sets but a bijection is ruled out for logical reasons: Like if we make a Godel numbering on the von Neumann integers and set $A$ equal to the subset of Godel numbers of all sentences true in that given model. Then $A$ is a subset of a countable set but there can't be a bijection between it and all the von Neumann integers because that would violate Tarski's Theorem. (or so I think). This would also be the case in a countable model, where a countable set and its powerset are both countable viewed externally but no bijection exists internally because of Cantor's Theorem.
So my question is how do we know that Case I is a real possibility (besides for finite sets or a finite and infinite set)? In other words, could it be that cardinal numbers greater than $\aleph_0$ just exist because of extensions of Case II above and aren't really new mathematical objects in their own right?
Hopefully this is clearer, but I'll stop if not.