User blog comment:Hyp cos/Catching Function - Better Definition/@comment-5529393-20140628065455

Very interesting. I think a somewhat simpler definition is:

$$B_0 (\beta) = \beta + 1$$

$$B_{\alpha+1} (\beta) = B_\alpha^\beta (\beta) $$

if $$\alpha$$ is a limit ordinal, $$B_\alpha (\beta) = \sup \{B(\gamma, \beta):\gamma < \alpha\}$$

$$S = \{ \alpha : B(\alpha,\omega) = \alpha \} $$

C is the enumerating function of S.

I had previously considered this ordinal version of the fast-growing hierarchy (and I think it has appeared here somewhere), but don't think I considered fixed points of the function.

Is there some way to turn these large ordinals into fast growing functions? (I'm not counting plugging them into the FGH since that requires fundamental sequences, which would require a full ordinal notation of all ordinals up to the ordinal in question.)