User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851/@comment-39605890-20191208003022

> By the way, do you have any evidence that the FGH (or its variant) can be fast-growing

I am not quite sure. Keep in mind that it is consistent with ZFC that if you define a FGH over *all* the countable ordinals, there is still a function that outgrows all the functions in that FGH. I am guessing this extends to HOD, meaning it is consistent there is a HOD function that outgrows the HOD FGH. It is probably "on par" with other FGH's though, I'm guessing. The properties of FGH's usually are not very dependent on the

Probably if I were to fix a specific ordinal, it would be the first ordinal not definable by a $$\Sigma_100$$ formula in the language of first order set theory, relativized to HOD.