This is a continuation of the discussion here. Here I am attempting to write a reasonable set of rules for an ordinal notation up to ψ(ψᵢ(0)) without consulting any reference materials. If it works as intended, the functions here will be very similar to the various versions of ψ that pop up in various places on this wiki.

Of-course, feedback (and especially correction of mistakes) would be most welcome.

Anyway, here goes:

1. To avoid confusion with all the different versions of ψ on this wiki, I'll notate my functions with "A" instead of ψ.

2. Aβ is an ordinal function whose output is always between Ωβ and Ωβ+1

3. For α,β<I we define:

C(β,α,0) = Ωβ ∪ {Ωβ}

C(β,α,n+1) = { γ | [γ=λ+μ ∨ γ=λμ ∨ γ=ωμ ∨ γ=Ωμ ∨ [γ=Aμ(λ) ∧λ<α]] ∧ λ,μ∈C(β,α,n) }

C(β,α,ω) = ⋃ C(β,α,k)

or in plain English:

C(β,α,ω) contains all the ordinals which can be constructed from the ordinals less than Ωβ, by using any finite combination of the following:

(a) ordinal addition

(b) ordinal multiplication

(c) the function x→ωx

(d) the function x→Ωx

(e) the functions Ax(y) with y<α

4. For α,β<I,  Aβ(α) = { the smallest ordinal not in C(β,α,ω) }

5. Define B(1)=Ω and B(n+1)=ΩB(n).

6. Then, if I've done everything right, we should have:

ψ(ψᵢ(0)) = sup [A0(B(n))] for n∈ℕ.

How am I doing so far? The next step will be to try and find formal rules for the fundamental sequences, but I don't want to even attempt that before I'm 100% sure that the above is correct (by "correct" I mean that (a) the above definition makes mathematical sense and (b) the final claim given in statement #6 is true).