User blog comment:Alemagno12/An extremely fast-growing OCF/@comment-5029411-20170726203426/@comment-30754445-20170727053958

I think I understand how the last part is supposed to works.

ΨLa(x) works like a fundamental sequence of the ordinal a. Writing it in this way is bit unusual, but I don't see anything wrong with it.

It's just like we can write:

ε₀[n] = ω↑↑n

without needing to define ε₀ first (indeed, the above line can serve as a definition of the ordinal ε₀)

What's less clear, is how he gets from L(x)=Ωx for x<ω to L(ω)=ψ(ψ𝐈(0)). It seems that the ΨL thing is supposed to act as some kind of extension to the L, so we'll have:

ΨLL(ω)(a) = Ωa for any ordinal.

And:

L(ω) = sup [ ΨLL(ω)(1), ΨLL(ω)(ΨLL(ω)(1)), ΨLL(ω)(ΨLL(ω)(ΨLL(ω)(1))), ... ] = ψ(ψ𝐈(0))

I certainly don't see how all the above stem from Alemagno's definitions, but it is fairly obvious that this was his intention.