I have a (seeming) contradiction and I can't seem to figure out the (obvious) mistake in my reasoning. The following (related to my previous question):
(1) $\omega$, defined to be the least infinite ordinal, by definition has the same cardinality in every model of set theory (model of ZF or ZFC I'm not sure it's true in every theory), namely it is countably infinite. I think one can prove the countability by showing that its cardinality is less equals the cardinality of every infinite set (give an injection and apply Cantor-Schroeder-Bernstein). Then use the existence of a countably infinite set which follows from the axiom of infinity.
(2) We can force $\omega$ to be finite! To this end use the Levy collapse to collapse $\aleph_0$ to $42$. Let's call this new model $M[G]$.
Now, presumably, $M[G]$ is no longer a model of ZF(C).
1.Is this correct? Is $M[G]$ no longer a model of ZFC?
2.And if yes: why not? Because it doesn't have an element representing the natural numbers?
2.a)If yes: how do I know that it doesn't?
3.And more generally: how do I know, after applying forcing to a model $M$ of ZFC, whether the resulting model $M[G]$ is still a model of ZFC or not?
Thanks for your help!