User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-35470197-20180805082043

@PsiCubed2

I guess that \(C\) is supposed to play a similar role to \(\Phi\), \(\chi\), and \(I\) in the definition of Rathjen's OCF.

Roughly, I guess the following:
 * The hierarchy \(I\) is used to give big (but accessible) cardinals in Rathjen's OCF. (Unlike \(\Phi\), \(I\) and \(\chi\) are actually necessary to define Rathjen's OCF.)
 * Imitating this, they need a certain stronger hierachy \(M\) to give big (but smaller than the first weakly Mahlo cardinal) weakly inacessible cardinal.
 * Imitating this, they need a certain stronger hierarchy \(K\) to give big (but smaller than the first weakly compact cardinal) weakly Mahlo cardinal.
 * Imitating this, they need a certain multi-variable hierarchy \(C\) to iterate this process, as the multi-variable Veblen hierarchy did against Veblen hierarcy.

My guess is presently bad, because the first three (finite) steps are given in explicitly without reasons and there is no way to actually extend this process transfinitely. Just replacing the hierarchy of weakly Mahlo by more reasonable (transfinitely extendable) choice such as \(\Lambda_0\) would not work, because it lacks good reflection properties.

What they are doing with \(C\) seems to give sufficient presentability of ordinals, in order to define an OCF \(\psi_{\kappa}\) as the system of ordinals which are not presented by lower terms.