User blog comment:Edwin Shade/The Grand List Of Transfinite Ordinals/@comment-30754445-20171130012046/@comment-32213734-20171201061459

Chronolegends, I'm trying to understand how Ψ1, Ψ2,... work. For now, I'm not trying to change any closure functions. φ1, φ2,... do not change anything, but may help to calculate Ψ1, Ψ2,... (of course, if the idea is right). I guess, this idea may work out, but only for finite number of cardinalities (i. e. not beyond Ωω), since it can not be an infinite number of nesting levels of Ψα functions. There we should use another variant of OCF, where Ωα can be directly in Ψ. For example, Ψ(Ω2) instead of Ψ(Ψ1(Ω2)), and Ψ(Ω3) instead of Ψ(Ψ1(Ψ2(Ω3))). How to deal with it, I don't know.

You meant ωΩ + 1 = Ω? I think, ωα + 1 is not less than α + 1, and α + 1 > α. So, ωα + 1 > α. If α = Ω then ωΩ + 1 = Ωω > Ω. I meant Ω = ωΩ = εΩ = ζΩ = ηΩ = φ(Ω,0) = ΓΩ etc. Generally, Ω = φ((any ordinals < Ω),Ω) = φ((any ordinals < Ω),Ω,(zeros)). So,

Ω = ωΩ

ΩΩ = (ωΩ)Ω = ωΩ 2 = ω(ω Ω)2 = ωω Ω2

ΩΩ Ω = (ωΩ)Ω Ω = ωΩ·Ω Ω = ωΩ 1 + Ω = ωΩ Ω = ωω Ω 2 = ωω ω Ω2

ΩΩ Ω Ω = ωω ω Ω 2   = ωω ω ω Ω2

and so on.