User blog comment:Billicusp/So I don't know if anybody noticed.../@comment-27173506-20160318160910/@comment-27513631-20160319141205

Huh, this is the first time \(\)'s worked for me. Go figure :p.

I disagree with \(I(1,0)\) not being reachable recursively; its just the least \(\alpha\) for which \(\alpha=I(\alpha)\), where \(I\) enumerates the fixed points of the regular cardinal enumerating function.

Anyway, deedlit pointed out on his blog post that \(\chi(M^\alpha)=I(1@1+\alpha)\), so there's that.

And I think I also disagree with the \(\chi\) function being unlike the \(\psi\) function as well. If you don't bother to go past the proof-theoretic ordinal of KPM, then you can define an alternate version of the \(\chi\) function as \(\psi_M\) (and a couple of alterations e.g. restricting the output to regular cardinals) which is just as strong; think of it like using the \(\vartheta\) function in place of the \(\theta\) function.