Skolem "showed" that the notion of countability is relative.

In set theory, if there is a set consisting of a bijection between a set A and the set of naturals N in the domain some model M, then A is countable in M. Now, suppose that we removed the bijections between A and N from M and called this new model M'. According to M', A is uncountable. And so, Skolem "shows" that there is no absolute notion of a set being countable.

However, we ordinarily talk as if there is an absolute notion; and indeed, we wouldn't be struck by Skolem's paradox if we didn't ordinarily treat countability as an absolute notion. One of the things that Skolem's Paradox reveals is the gap between the ordinary English semantics of countability, sets, quantification, membership, etc. on the one hand and the model-theoretic semantics of these notions on the other.

So, in the actual world, which do you think is right -- that countability is relative or absolute?