Yes, I am still working on HaN. I have defined Part V already, which defines <A,★> for arbitrarily large alpha-strings in the second entry and then goes on to define linear alpha-string arrays for any set number of entries. But, I cannot release it until I know how to approximate it in FGH. Unfortunately, the limit of Part IV was ψ(OFP), the limit of Extended Buchholz's function, and I do not know how to express beyond that.
I am capable of defining a simple ordinal function ψI(a) such that ψI(a) = (1+a)th omega fixed point, which is capable of functioning as inputs in extended Buchholz just fine, however I run into two problems:
- This is not an OCF.
- This breaks at the fixed point of a = ψI(a), which I would define as ψI(I) if I knew how, but unfortunately I don't.
This, and I'm generally dissatisfied with only going from ψ(OFP) to ψ(I). This might seem like a strong recursive jump, but consider a modified version of the Veblen hierarchy Φ(a,b) with the base rule Φ(0,b) replaced with Ωb:
OFP = Φ(1,0), and ψI(I) is a mere Φ(2,0). I am NOT comfortable with this. At the very least, I should expect to reach Φ(ω,0) (which should be ψ(Iω)), if not Φ(1,0,0) (which should be ψ(II)), or if we're idealizing, the limit of Φ(1,0,0,0,...) for finitely many "0"s entirely (which should be ψ(IIω)). But I just don't know how.
Until I figure out a possible stronger definition of alpha-string arrays, and either define or learn a proper inaccessible OCF that works up to ψ(inaccessible fixed point), Part V is on hold.
This notation will be complete one day, I promise.