User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851/@comment-35470197-20191208013608

> 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.

Of course, I know. FGH can be broken if one chooses very weak fundamental sequences as I verified here, and also can be arbitrarily strong if one chooses very strong fundamental sequences as I essentially verified here. That is why I am asking this question. For example, we do not even known whether \(f_{\omega_1^{\textrm{CK}}\) with respect to Kleene's \(\mathcal{O}\) and a specific algorithm to enumerate Turing machines is stronger than \(f_{\varepsilon_0}\) or not. (Professor Kihara has shown a way to enumerate Turing machines to make \(f_{\omega_1^{\textrm{CK}}\) weaker than \(f_{\omega^4}\).) Also, I verified several properties on FGH here including how to rearrange it.

> definable by a $ \Sigma_100 $ formula in the language of first order set theory, relativized to HOD.

But since HOD is proper, it is really hard to explicitly write down the truth predicate, isn't it?