User blog comment:P進大好きbot/New Issue on Traditional Analyses/@comment-31663462-20190825021820

An evil mathematician joins the fray and says:

Define \(\alpha[n]\) as the largest thing less than \(\alpha\) writable in \(n\) symbols using the given ordinal notation.

>:) even better, this gives you an fgh that is, as a matter of fact, increasing, though not strictly in the ordinal argument, but strictly in the natural argument.

As for the matter of whether or not these are computable functions, indeed they are not. But what they (usually) form are computable notations, meaning given any two ordinals writable in \(n\) symbols, for a reasonable definition of what that means, you can write an algorithm to compare their values, which is all that is required to use the above fundamental sequence.