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**Can a model be countable from its own "perspective"?**

I'm reading about Skolem. And I'm wondering about the result of the paradox: that countability (at least in first-order formulations) is relative. Now, even when we state Skolem's theory -- if a first-order theory has an infinite model then it has a countable model -- from what "perspective" is this model countable? From another model? Absolutely?

If we say that a model M is countable "from its own perspective" that means there is some bijection (set) B in the domain of M containing every set in the domain of M (including B) and the set of natural numbers. But that can't happen, because then B would be a member of itself.

So, when we casually talk about countability, we take it to be an absolute notion. But, what does this say about Skolem's Paradox?