@Justin : I am not sure what you mean. My reasoning in my question was wrong, because I forgot that notions like "finite" are not absolute since the saturated model $N$ won't be transitive. The saturated model $N$ of course still must satisfy Foundation. To tell you the truth I can't really visualize clearly how $N$ can satisfy Foundation and at the same time have elements like $v'$, but I guess the whole construction of finite ordinals in $N$ must be strange enough to permit such thing.

@Andreas Blass, Andres Caicedo : Thank you very much for your fast response. Indeed I forgot to notice that since $N$ won't be transitive (even if $M$ is) then most of the concepts like "finite" etc won't be absolute. Just to be sure I understand though... If we had extended a countable transitive model of ZFC $M$ to some $M[G]$ through forcing, since $M[G]$ is also c.t.m. of ZFC, we wouldn't be able to add any natural numbers in $M[G]$.