If there is countable transitive model of ZFC, this model cannot capture all ordinals of ZFC. But we use it for stuffs like forcing.
But this seems to violate ZFC's axioms – for example, power set axiom (take, $\omega = \aleph_0$ and apply power set operator.).
So how can we use countable transitive model of ZFC, then?
Best Answer
You are mixing internal and external point of views.
If $M$ is a countable transitive model of ZFC then it doesn't know that it is countable, and it certainly don't know that all its members are countable. That is to say, there are many sets $x\in M$ such that $M\models\aleph_0<|x|$. So while that in the full universe there is a bijection between $x$ and $\omega$ it is not an element of $M$.
When we use c.t.m for forcing we work internally when we define the forcing poset, and we argue externally when we prove that there is a generic set, and that the construction results in another countable transitive model of ZFC.
Somewhat related: