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 $\omega_1^V$.
- 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 $\omega$?