I have a question about undecidable statements in $\mathsf{ZFC}$.
I know there are true statements like: $$X\text{ is independent of }\mathsf{ZFC}.$$
But is it also possible that such a statement exists?
$$\text{"}X\text{ is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}$$
Or even worse:
$$\text{""}X\text{ is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}$$
I think you see what I mean by "nested independence of $\mathsf{ZFC}$". Is it possible that such a true statement exists? If it is possible: What does it mean? And is there an example for such a true statement? Unless such a true statement can exist: Why not?
Another funny case would be is the depth of this nested statement goes to infinity:
$$\begin {align} \mathrm{statement}_0 &= X \\ \mathrm{statement}_{n+1} &= \mathrm{statement}_n\text{ is independent of }\mathsf{ZFC} \\ \mathrm{specialstatement} &= \lim_{n\to\infty}\mathrm{statement}_n \end{align}$$
Does such a $\mathrm{specialstatement}$ exist? And what is the meaning of it?