Hello, I’m back (who could have expected?) with my new OCF (*Not* an ordinal notation.).
We follow several conventions from the world of OCFs. First, let
$$\Omega_\nu=\begin{cases} 1 & \mbox{if }n=0 \\
\aleph_\nu & \mbox{otherwise}\end{cases}$$
And let \(\Omega=\Omega_1\).

Then we define the function \(\psi_\nu(\alpha)\) as the least ordinal that cannot be constructed by:

- All ordinals less that \(\Omega_\nu\)
- Summing two previously constructed ordinals
- \(\psi_\beta(\gamma)\) for two previously constructed ordinals \(\beta,\gamma\), subject to \(\gamma<\alpha\)
- \(\eta_\beta(\gamma)\) for two previously contructed ordinals \(\beta,\gamma\) with \(\gamma<\alpha\)

Here, the \(\eta\) function is a function used to define large cardinals similarly. Let \(\Lambda_0=1\), and otherwise \(\Lambda_\alpha\) be sequence of sufficiently large cardinals (not to be confused with large cardinals, though most large cardinals should work.). Let \(C_\nu(\alpha)\) be the set of ordinals constructible by the folllowing:

- All ordinals less than \(\Lambda_\nu\)
- Summing two previously contructed ordinals
- \(\psi_\beta(\gamma)\) for two previously constructed ordinals \(\beta,\gamma\), subject to \(\gamma\le\alpha\)
- \(\eta_\beta(\gamma)\) for two previously contructed ordinals \(\beta,\gamma\) with \(\gamma<\alpha\)