Explanation of how models can differ on $\omega$? Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets.
So, there is a $\omega^V\in V$.  A set $A\in V$  is countable iff a bijection $f\in V$ exists between $A$ and $\omega^V$.
By the downward Löwenheim–Skolem Theorem, there is a countable model $M$ of ZFC such that the domain M of $M$ is in $V$ and a bijection $B\in V$ exists between M and $\omega^V$.
Since $M$ is a model of ZFC, there must be some $\omega^M\in M$ and some $\omega_1^M\in M$.
There is no guarantee that $\omega^M$ is $\omega^V$, and no guarantee that $\omega_1^M$ is $\omega_1^V$.
My questions:  


*

*Can someone explain when/how/why/under what conditions this divergence (between models taking different sets to be $\omega$) happens?

*When/how/why/under what conditions is there a guarantee that sets will agree on $\omega$?


To the first question, consider $M$ to be a model of ZFC which is a set in the universe $V$. Suppose that $M$ and $V$ agree on $\omega$, we can take an ultrapower of $M$ by a free ultrafilter on the real $\omega$. This happens in $V$, and externally to $M$.
The result is a model, $N$, which externally speaking has a different version of $\omega$. Namely, $\omega^N$ is the ultraproduct of $\omega^M$ by a free ultrafilter over a countable set, so it is not well-founded and therefore cannot be the real $\omega$.
On the other hand, we can require that the models we work with are $\omega$-models, namely their copy of $\omega$ is isomorphic (from the external, and real point of view) to the real $\omega$. In which case even if the model does not fully agree on how $\omega$ looks like, its agrees on its behavior (again, from an external point of view). If we have a transitive $\omega$-model then as sets $\omega^M$ and the real $\omega$ of $V$ are the same set.
