Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (because ordinals are well-ordered), so let $\alpha_1$ be the smallest ordinal in $X_1$.
So, I have 2 questions:
- Is the class $X_1$ actually well-defined?
- If it is well-defined, what can we determine about the actual value of $\alpha_1$?
We could also ask the same question for $\aleph_\beta$ (for any ordinal $\beta$) instead of $\aleph_1$, which I'd also be interested in.
My thoughts:
There are two points I think to verify that $X_1$ is well-defined. The first is that "is a transitive model of ZF(C)" is definable in the language of ZF(C), and the second is that "$M$ thinks $\alpha$ is $\aleph_1$" needs to be definable.
Being transitive is definable, so really, the first part reduces to whether "being a model of ZF(C)" is definable. I'm a little unsure of whether or not it is, because ZFC is not finitely axiomatizable, but NBG is, which makes me think the idea behind this question might still be salvageable even if it's not definable in ZF(C).
As for that "$M$ thinks $\alpha$ is $\aleph_1$" is definable, it seems very true, because countability is definable, and then "every ordinal less than $\alpha$ is countable, but $\alpha$ is not countable", which should definitely be definable.
So, that means, that part 2 makes sense to talk about. We know that (at least for small enough $\beta$), $\alpha_\beta$ must be countable, because countable models of ZF(C) exist (by Lowenheim Skolem), which is already interesting. I suspect they are all fairly big countable ordinals, almost certainly larger than $\omega_1^{CK}$.