ZFC,unprovability of existence of a countable model,Skolem construction and paradox

Question

The well-known Skolem construction yields a countable model of ZFC,elemetarily equivalent to the universe of sets $V$.Why this construction is not a proof of existence of models of ZFC,as such proofs cannot exist by Godel's theorems?

Answer 1

We can have a model $V$ of ZFC, which is uncountable, but which does not think there exist any countable models of ZFC: Externally, Lowenheim-Skolem gives us a countable elementary submodel $M\subseteq V$, but there's no reason to believe $M\in V$, so $V$ might not "see" that such an $M$ exists at all.

Note also that $V$ may "think" that $V\not\models ZFC$! This is because if $V$ has nonstandard natural numbers, this will yield nonstandard axioms of what $V$ thinks is $ZFC$, and it's possible that one of these might be false in $V$ (actually, this phrase doesn't make sense - the sentence in question will be nonstandard, so infinite in length, so it's not even clear what "false in $V$" means - but this can be made precise and true with some effort). So even ignoring the fact that $V$ can't construct its own theory, $V$ might think "$ZFC$ is false," even though $V$ satisfies every actual axiom of $ZFC$. (Lest you think this is a dodge, by the way, note that nonstandard natural numbers are crucial to Godel's theorem, so this is really very much in the right spirit.)