User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851/@comment-39605890-20191204031834

Yes, any ordinal definable fundamental sequence is HOD since pairs of ordinals are HOD.

> I have another question. Is there a specific way to choose a large ordinal below ω_1^{HOD}? Unlike ω_1^{CK}, it can be uncountable, and hence it looks a non-trivial question.

Yes, that is indeed the "catch". This construction is not useful to Googology without that.

I am thinking that you would go into HOD (which is an inner model), and ask it for large countable ordinals. If HOD believes an ordinal is countable, than in reality is will be less than $$ω_1^{HOD}$$. For example, it is consistent with ZFC that HOD thinks that what it thinks is $$ω_1^{HOD}$$ (i.e. $${ω_1^{HOD}}^{HOD}$$) is countable. If that is the case, you could use $${ω_1^{HOD}}^{HOD}$$. Maybe some ordinals listed in (https://en.wikipedia.org/wiki/Large_countable_ordinal#Beyond_recursive_ordinals), internalized to HOD, would work as well. (Some of those ordinals may be OD anyways, without needing to ask HOD. That would be even better.)