User blog comment:P進大好きbot/New Googological Ruler/@comment-39541634-20190722124029/@comment-35470197-20190723054042

Thank you for the explanation.

Although I do not really understand Deedlit's OCF, I guess that it is true. Rathjen's standard OCF is much stronger than many googologists expect. (I personally experienced many times that when they talk about Rathjen's OCF, then it was supposed to behave something like UNOCF...) Employing strong base functions like φ gives strong diagonalisations.

Maybe it is stronger than Rathjen's simplified OCF because ψ_{Ω_1}(ψ_{χ_{M+1}(0)}(0)) in the standard OCF coincides with PTO(KPM), which is expressed as ψ^{Ω_1}(ε_{M+1}) in the simplified OCF. According to Deedlit's explanation, Deedlit's OCF based on M coincides with the notation obtained by replacing superscripts in the simplified OCF by subscripts. If they are true, then Rathjen's standard OCF is stronger than Deedlit's OCF with M. (Maybe several explanations in the OCF originally given by Deedlit are wrong, because he seemed to be comfounding the simplified OCF with the standard OCF.)

But it might be surprising that even if we drop φ from the definition of the standard OCF, the limit of the notation will not change. (Of course, ψ_{Ω_1}(ψ_I(0)) will change, though.) It is roughly because Rathjen employed another strong base function χ in the definition of ψ. Unlike simple diagonalisations of higher inaccessibility, Rathjen's χ itself is defined as a sort of an OCF. Namely, it is a function collapsing higher inaccessibility. Therefore Rathjen's standard notation is a system including two sorts of OCFs.