User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-35470197-20190629153128

> I know this, but it allows me to somehow note the missing intermediate scales.

Actually, it does not allow you to do so. HH heavily depends on the choice of a recursive system of fundamental sequences, and larger ordinals do not necessarily correspond to larger functions. Namely, your level 40 can be smaller than my level 23 if you formalised the definition.

The formulation with eventual domination does not work, either. Maybe you miss the point that you need to fix a base theory. Even under the assumption of the consistency of ZFC, the existence of such a function does not necessarily hold at a model of ZFC.

> Ok, but you table skips scales from 24 to 40 of my list. Can you give not-ill-defined designations of these scales?

As I wrote above, I am not skipping. Just they do not necessarily exist.

Also, one of the aims of this ruler is to set levels so that every level naturally appears in googology. For example, if few can define notable large numbers between levels 24 and 40 in your scale which will be formalised well, I do not have to fill the blank. Do you know any specific candidates of large numbers which will be obviously between them? I guess no.