In Thoralf Skolem's Remarks on Axiomatized Set Theory (van Heijenoort translation), Skolem says:
There is no contradiction at all if a set $M$ of the domain $B$ is nondenumerable in the sense of the axiomatization; for this means merely that within $B$ there occurs no one-to-one mapping $\Phi$ of $M$ onto $Z_0$ (Zermelo's number sequence). Nevertheless there exists the possibility of numbering all objects in $B$, and therefore also the elements of $M$, by means of the positive integers...Thus, axiomatizing set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatization...In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. But this would in all cases lead to a circular introduction of the higher infinities; that is, on an axiomatic basis higher infinities exist only in a relative sense.
And the abstract of this paper on Skolem's work says:
Skolem's critique of set theory is seen as part of a larger argument to the effect that no conclusive evidence has been given for the existence of uncountable sets. Some replies to Skolem are discussed and are shown not to affect Skolem's position, since they all presuppose the existence of uncountable sets.
All of this sounds like an assertion that no definitively "legitimate" model of set theory is ever "really" uncountable. Or to (try to) interpret it more precisely, that: For any fully "legitimate" (i.e. "guaranteed" to exist) model $M$ of $\mathsf{ZF(C)}$, when $M$ is viewed "from the outside" there is always some way of establishing a one-to-one correspondence between the sets of $M$ and the (standard) natural numbers.
- Note that I'm being intentionally vague about what exactly it means to "view $M$ from the outside" and what exactly is "some way" of establishing a one-to-one correspondence with $\mathbb{N}$. Maybe this is done within some larger model of set theory, maybe by some mode of reasoning not strictly founded in set theory.
Is my description/interpretation here an accurate characterization of Skolem's views? Also (maybe more importantly), does it describe a view that's at all common among set theorists currently?