User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851/@comment-35470197-20191204081646

> 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.

Exactly, but it is not used to define a large number in ZFC set theory because of the consistency.

> Maybe some ordinals listed in … internalzed to HOD

Right. I know there are many known non-recursive ordinals, because they are already theoretically used to show certain properties of ordinal notations, which of course apar in computable googology.

I wondered whether you have a specific way to define a large ordinal (internalised in ω_1^{HOD} or just known to be hereditary ordinal definable) using your strategy, but using known large ordinals are actually good solutions, because you can combine your system with other strong systems such as ITTM. Your system itself is quite novel and astonishing for me. (Although HOD is already used in the definition of Little Bigeddn in the axiom, the cool point of your googological use of HOD is that you are working in ZFC set theory.)

By the way, do you have any evidence that the FGH (or its variant) can be fast-growing, if you compare it to higher order busybeaver functions or FGH aplied to Kleene's O? I created a googological ruler here, and is interested in the level of your system if you fix a sufficiently large known countable ordinal. I note that the ruler does not scale the precise size. If you want to know the precise definition of the level, see the section 3.2.