User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-24509095-20140507175906/@comment-5529393-20140508084719

Well, it's up to your taste what ordinal notation you use, but what's so bad about $$\varphi$$ being included? The values of $$\psi_\alpha(\beta)$$ are the values that are "inaccessible" using the closure functions, so they sort of plug in to the "holes" in the ordinals. If $$\varphi$$ is included you only need to fill in the strongly critical ordinals.

What's wrong with just using $$\varphi(1,I+1)$$ to denote that ordinal?