User blog comment:Syst3ms/A formal definition for UNOCF/@comment-35470197-20180728080043/@comment-35470197-20180728085137

I mean that the usual cofinality is not necessarily computable in a recursive way.

If you use the explicitly cofinality in the definition of \(\psi_{\kappa}\), then you might fail to define a system of fundamental sequences against the associated ordinal notation.

Recall that when you compute the fundamental sequence of \(\psi_{\alpha}(\psi_{\beta}(\gamma))\) for Buchholz's psi function, you use the recursive computation of the cofinality.