User blog comment:Emlightened/Early Birthday Present For Deedlit/@comment-5529393-20170730212008

Thanks for this.

I'm rather curious how one is to extract a notation from these ordinal collapsing functions. In previous OCFs, the closure functions showed you how to express a particular ordinal. But if you just add all ordinals of various cofinalities, I don't see how one goes about mapping ordinals to expressions. (Of course it can only be a countable subset of ordinals that get mapped to expressions!)

For the first OCF, it looks like Psi(a) will enumerate the regular ordinals until it gets stuck at I, so Psi(Ω) will be I, and Psi(Ωa) will enumerate the weakly inaccessibles. Psi(Ω2a) will enumerate the 1-weakly inaccessibles, and we get an inaccessible hierarchy based with Ω as a diagonalizer. Then, if Psi(Ω+) is the first weakly Mahlo cardinal, Psi(Ω+) will take over as the inaccessible hierarchy diagonalizer once we pass Ω+. So Psi(Ω+) will have the same function as Ξ(2) in the traditional OCF, Psi(Ω+ 2) will have the same function as Ξ(3), and Psi(Ω+ a) will have the same function as Ξ(1+a). So Psi((Ω+)2) will have the same function as Ξ(K). So it does look like Ω+ will provide the same function as K (diagonalizing over various levels of Mahloness), and this OCF has roughly the same strength as the traditional OCF using the weakly compact cardinal K.

However, a big part of the strength of the K notation was that, not only did we have regular ordinal collapsing, we had collapsing to regular cardinals, and collapsing to all the stages of Mahloness. Do we still need that here? I would think so.

I understand little of the second notation. I'm not familiar with Levy collapse or forcing in general. Does $$(o(\beta))^{V[G_{\kappa \leftarrow \lambda}]}$$ stand for what $$o(\beta)$$ gets taken to after the Levy collapse? Also, I'm not sure how you can define the Ψ function with κ and λ as free variables.

Hmm, if you believe that these notations can greatly simplify ordinal analyses, that could be significant. Have you contacted people who do ordinal analysis, like Michael Rathjen? From what I understand, what you want from the ordinal notation is not necessarily the greatest simplicity, but the one that is most compatible with the cut elimination process for the theory under investigation. But, I'm a newbie to ordinal analysis.