User blog comment:Emlightened/Early Birthday Present For Deedlit/@comment-5529393-20170730212008/@comment-27513631-20170730231148

I believe that to extract the notation we would want to do something very similar to the normal \(\psi_\pi\) function, but collapsing below \(\pi\) only when \(\pi = \Omega\gamma + \delta\) with \(\delta<\Omega\) and \(\text{cof }\Omega\gamma \leq \Omega\).

Collapsing below inaccessibility is required; collapsing below Mahloness may not be, I believe. The collapses below inaccessibility are used to create normal cardinals at various limits or \(\alpha\)-Mahlo cardinals and various limits thereof and are useful for maximising the amount of ordinals we need to collapse. However, typically collapsing below a Mahlo cardinal is just used to create stronger layers of inaccessibility, and this is exactly something which can be done with ordinals less than \(\Omega^+\). However, to unlock that, we furtherly require collapsing down to the cardinality \(\Omega\), so this is only avoidable with collapsing functions with that codomain. (Although it may be possible that the benefit is eventually rendered moot, in the same way that the SGH catches up with the FGH.)

Free variables fixed. The basic idea is that if we remove the cardinality restrictions on that o notation used for inaccessibility by forcing, then the resulting o function would gain values higher than \(\Omega^+\) for input \(\Omega\). It's a bit less direct to use, and I may have slipped in the definition, but we don't need to worry about adding more cardinals above \(\Omega^{++}\) in that one.

And no, I haven't. And honestly, I didn't know that much. Ordinal analyses that are made do get subsequently remade or simplified, so it is likely possible that an ordinal notation system can be approximately translated into this one, and then use the translation as a basis for stronger systems.