Given an ordinal $\alpha$, I define $F_{n}(\alpha)$ as follows:

• $F_0(\alpha)=\alpha$
• $F_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the language of $\{\in\}$ has $(\mathrm{V}\models\phi(S,F_0(\alpha),F_1(\alpha)...F_{n}(\alpha)))\Leftrightarrow S=\beta$
• $F_\omega(\alpha)=\mathrm{sup}\{F_{n}(\alpha):n<\omega\}$

The existance of Reinhardt cardinals implies that $F_1(\alpha)$ exists for every sufficiently small $\alpha$. In fact, every Reinhardt cardinal is $F_1(\alpha)$ for some ordinal $\alpha$. (Yes, even though Reinhardt cardinals are not consistent with AC, they are all ordinals.)

However, this is the only information I could find on this. Could somebody else tell me what this is called if it is already named?

Is it inconsistent to say $F_n(\alpha)$ exists for $n\leq\omega$? Which $n$ is it inconsistent for?

Information gathered post-question:

1. If there is a first-order $\phi$ such that $(\mathrm{V}\models\phi(S))\Leftrightarrow S=\alpha$, then $F_n(\alpha)=F_n(0)$ for every $n$. Here are some $\alpha$ for which this is true:
   • The smallest Erdős initial ordinal, Mahlo cardinal, Ramsey, Rowbottom, Jonsonn, Inaccessible, Strongly Inaccessible, Limit, Weakly compact, ethereal, subtle (assuming each of these exists)
   • Every other minimum initial ordinal for a first-order large cardinal axiom
   • The second of each of those axioms, the third, the $\alpha$-th for $\alpha$ on this list (assuming they exist)
   • Every $\alpha<\omega_1^{CK}$ (hint for proving this: Kleene's $\mathcal{O}$ is first-order definable)
   • Every $\aleph_{\alpha}<\aleph_{\omega_1^{CK}}$ and $\beth_{\alpha}<\beth_{\omega_1^{CK}}$(a corrolary from above)
2. $F_\alpha(\beta)$ is countable or $\omega_1$ (assuming ZF)

Honestly I think this is the first time $\beth_{\omega_1^{CK}}$ has been used in a theorem ever. If I'm wrong, please tell me.