User blog comment:Moooosey/Simple psi-- a fairly simple ocf/@comment-39541634-20191026213457


 * 1) 7 is simply the original Madore Psi augmented by the variation you've introduced in #6.

Madore Psi, of-course, is far stronger then your original system. It reaches the BHO which is the limit of ψ(Ω^Ω^Ω^...) - both with and without your variation (assuming my analysis of #6 was correct).

So the limit of #7 is the BHO.


 * 1) 7a is ill-defined, because the set of ordinals with the property x=αx (for a given α) is unbounded.

I suppose you meant something else. Something along the lines of alllowing "εα" as an operation in addition to "+" and exponentiation. This will, indeed, work as an extention of the original Madore Psi, because now you could write things like ψ(εΩ+1). The limit of this specific extension (using εα as a permissible function) would be ψ(ζΩ+1). In extended Madore we could write this as ψ(ψ1(Ω2)).

You could make this even stronger by allowing φ(α,β) instead of εα (with the limit being ψ(ΓΩ+1). Even stronger functions (like multi-variable φ's) would extend the limit even higher. Eventually, though, in order to satiate our hunger for ever-growing functions, we'll have no choice but to add another OCF-based function to are list of permitted functions. This is precisely how multi-ψ OCFs (like Buchholz or extended Madore) work.

At any rate, the variation you borrowed from #6 is still just as useless here as it was at #7.


 * 1) 7b is interesting. It gives rise to pretty unusual behavior:

ψ(ε₀) = ε₀xωω (because we've artificially added ε₀ to are list of permitted base ordinals)

But

ψ(Ω) is still only ε₀, because there's still no way to construct ε₀ from just 0,1,Ω and ψ(a) for countable a. So we get the peculiar result that ψ(Ω)<ψ(ε₀) even though Ω>ε₀.

The limit, of-course, would still be εω (or the BHO if you add exponentiation).

P.S.

Thanks for the opportunity you're giving me to practice these concepts. I've just realized that analyzing nonstandard OCF extensions is one of the best ways to make sure I understand everything myself.