How large is the supremum of all countable ordinals that have absolute definitions? This answer contains the following statement:

it's easy to check that $\omega^2$, $\omega^\omega$, $\epsilon_0$, and even "big" countable ordinals like $\omega_1^{CK}$ (= the first "non-computable" ordinal) have absolute definitions.

My question is: how large is the supremum $\alpha$ of all countable ordinals that have absolute definitions? And how to prove that $\alpha$ does not have an absolute definition? Do we need at least two systems, $S_1$ (that allows to formalize the notion of absolute definitions) and $S_2$ (that allows to define $\alpha$) in order to prove this? Does the value ("height") of $\alpha$ depend on the background set theory that we use?
 A: There is a lot of surprising subtlety here!
(As the author of the linked answer, I can tell you what I had in mind in that specific context: say that a "really absolute ordinal" is an ordinal $\alpha$ such that there is a formula $\varphi$ which defines $\alpha$ in all transitive models of $\mathsf{ZFC}$ and such that $\mathsf{ZFC}$ proves that $\varphi$ defines a countable ordinal. This is a sharpening of Notion $1$ below to include a countability assumption, which was important for the linked question but doesn't seem to be particularly important for your OP so I'm dropping it here. If you're curious, though, the supremum of the ordinals of this type is just $\omega_1^{L_{\min\{\alpha: L_\alpha\models\mathsf{ZFC}\}}}$.)
Throughout, I'll reason in a somewhat-strong background theory $T$ - say, $\mathsf{ZFC}$ + "Every set is contained in a transitive model of $\mathsf{ZFC}$."

Let's start with a simple notion:

*

*A $\mathsf{ZFC}$-absolute ordinal is an ordinal $\theta$ such that for some formula $\varphi(x)$, we have $\varphi^M=\theta$ for every transitive model $M\models\mathsf{ZFC}$.

This is however a fairly boring notion: the $\mathsf{ZFC}$-absolute ordinals are just the ordinals $<\alpha$ where $\alpha$ is the least ordinal with $L_\alpha\models\mathsf{ZFC}$. But in a sense, this $\alpha$ itself is defined in an absolute way! It's only in "pathologically small" transitive models that we fail to see $\alpha$.
Can we do better?


*An almost-$\mathsf{ZFC}$-absolute ordinal is an ordinal $\theta$ such that for some formula $\varphi(x)$ and some set $c$, every transitive model $M\models\mathsf{ZFC}$ with $c\in M$ has $\varphi^M=\theta$.

Here we get much more variety! For example, if $\theta$ is almost-$\mathsf{ZFC}$-absolute, then so is $\alpha_\theta$ = the $\theta$th index of a level of $L$ satisfying $\mathsf{ZFC}$. But at this point we have a separate problem:

Is $\omega_1$ almost-$\mathsf{ZFC}$-absolute?

It certainly shouldn't be! But if $\omega_1^L=\omega_1$, then - taking $c=\omega_1$ - we do have that $\omega_1$ is almost-$\mathsf{ZFC}$-absolute via the formula "$x$ is the smallest uncountable ordinal." At this point the role of our background theory $T$ comes into focus: in order to get a good theory of almost-absoluteness, we'll need $T$ to be rather strong ($\omega_1$ really shouldn't be almost-absolute, and thinking along these lines motivates including some large cardinal axioms in $T$). But at this point we're running into real issues of background theory strength vs. phenomenon being studied. And all of this hasn't even touched the question of what happens if we change the "subject" theory from $\mathsf{ZFC}$ to something else.
