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:
- Iff $F_n(\alpha)$ exists for every finite $n$ then $F_\omega(\alpha)$ exists.
- 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 Erdos cordinal (I call a cardinal which is an ordinal a cordinal, in case AC is false), Mahlo cardinal, Ramsey, Rowbottom, Jonsonn, Inaccessible, Strongly Inaccessible, Limit, Weakly compact, ethereal, subtle (assuming each of these exists)
- Every other minimum cordinal 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)
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.