User blog comment:P進大好きbot/New Googological Ruler/@comment-39541634-20190722124029/@comment-35470197-20190722144554

> difference

I received the same question from Scorcher007 below, and hence it might be confusion. They are not the same ordinals. I do not know why many googologists believe ψ_0(Ω_Ω_Ω_…) in extended Buchholz's OCF coincides with ψ_{Ω_1}(ψ_I(0)) in Rathjen's standard OCF. I guessed that it is because of this article, which incorrectly stated that the least omega fixed point is presented as \(\psi_I(0)\) with respect to Rathjen's standard OCF.

In Rathjen's standard OCF, the least fixed point is simply described as Φ_1(0) even without using ψ. The capital Veblen function Φ is defined by the formula obtained by replacing AP, which coincides with the class of ordinals presented as powers of ω, in the defining formula of Veblen function \(\varphi\) by the class of uncountable cardinals. Namely, we have Φ_0(β) = ℵ_{1+β}\), and Φ_1(β) = (1+β)-th omega fixed point.

Therefore ψ_0(Ω_Ω_Ω_…) in extended Buchholz's OCF is strictly smaller than ψ_{Ω_1}(ψ_I(0)) = ψ_{Ω_1}(Φ_{Φ_…(0)}(0)). I think that this presentation answers the second question, because ψ_{Ω_1} commutes with sup.

Although I have not seriously computed the precise correspondence, I guess that ψ_0(Ω_Ω_Ω_…) in extended Buchholz's OCF coincides with ψ_Ω_1(Ω_Ω_Ω_…) = ψ_{Ω_1}(Φ_1(0)) in Rathjen's standard OCF, because cardinals are fixed point of φ_α(0). I think that this presentation answers the first question.

> P.S.

It is my pleasure!