Why can't a model "say" of itself that it is countable? Why can't a (standard?) model of ZFC "say of itself" that it is countable?
That is, why is there no bijection $f$ ∈  between  and $\omega^$?
(I've read that it fails regularity, or even without regularity we get Cantor's paradox.  But a direct answer to the question would be most helpful.)
Thanks.
 A: If $\frak M$ is a [standard] model of ZFC then we know several things:


*

*$\frak M$ thinks that $\{x\mid x\notin x\}=\frak M$ is not a set.

*If $f\in\frak M$ and $\frak M$ thinks that $f$ is a function, then the range of $f$ is a set in $\frak M$. (This is an instance of the axiom schema of replacement)

*$\omega^\frak M$ is a set in $\frak M$.


These combined tell us that if $\frak M$ knew about a function from its own $\omega$ onto its entire universe it would violate the second thing in the list above, and will not be a model of ZFC. 
In the case of a standard model, we can also have the contradiction from the fact that if such $f$ was in $\frak M$ then we would have $\frak M\in\frak M$ and that, as you said, would contradict the axiom of regularity (both in the universe and in $\frak M$) but this is in addition to the above argument.
A: If you can prove the existence of uncountable sets in ZFC, and if you can also prove a proposition saying the model is countable, then you have a contradiction in ZFC.  With countable models of ZFC, the statement that a set is uncountable is true in the model if the set is not "internally" countable, i.e. no enumeration of the set is a member of the model.
