User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-11227630-20130927104812/@comment-5529393-20131002063245

Figuring out general rules for fundamental sequences actually gets pretty hard for the higher notations. I guess the notation with a weakly inaccessible cardinal shouldn't be too bad, but the higher ones will be pretty challenging I guess.

However, one can "cheat" and define fundamental sequences using the following trick. I will give an example using the notation with a weakly inaccessible cardinal:

Define \(N(\alpha)\) for ordinal notations \(\alpha\) as follows.

\(N(0) = N(I) = 0\)

\(N(\alpha + \beta) = N(\varphi(\alpha,\beta)) = N(\psi_{\alpha}(\beta) = N(\alpha) + N(\beta) + 1\)

\(N(\Omega_\alpha) = N(\alpha) + 1\)

Then, for an ordinal \(\alpha\) whose ordinal notations are \(\alpha_i\), we define \(N(\alpha) = \min_i (N(\alpha_i))\).

Finally, for an ordinal \(\alpha\) we define

\(\alpha[n] = \max \lbrace \beta | N(\beta) \le N(\alpha) + n \rbrace \).

This is a perfectly well-defined definition, so we can use FGH on all these stronger ordinal notations. But, this definition of fundamental sequence isn't totally satisfactory, since we can't easily see what the fundamental sequences are. So there is still some value in creating explicit fundamental sequences for these higher ordinal notations.

You mention Ordinal Notations III as needing fundamental sequences, but I define them at the bottom of that blog post.