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.



Do these ordinals exist?
For the definition of $F\_{n+1}$ to make sense, we need, in addition to the usual axiomatic apparatus of ZFC, a notion of "satisfaction of formulas in $V$." If we have this additional notion and if we allow it to occur in replacement axioms, then we can prove that $F\_n(\alpha)$ exists and is countable for every $n$ and $\alpha$, and therefore $F\_\omega(\alpha)$ also exists and is countable. The argument is essentially as in Zetapology's answer, with the additional apparatus replacing the "Clearly" claim (which isn't justified without some definability and some use of replacement axioms). 


