User blog comment:Syst3ms/A formal definition for UNOCF/@comment-35470197-20180728080043/@comment-35470197-20180728095828

> I don't see why it matters to know the cofinality of ψ κ (α) ψκ(α). Why not just compute it and then see what its cofinality is ?

If you want to define an ordinal notation associated to the OCF with a system of fundamental sequences in a recursive way, you need to give a way to compute the cofinality of any ordinal term also in a recursive way. Since the ordinal corresponding to an ordinal term is not described in arithmetic, the relation \(\in\) and the function \(\textrm{cof}\) should be translated into recursive ones.

Of course, if you do not want to construct an ordinal notation with a system of fundamental sequences from the OCF, it does not matter.

> Then, what I mean by ↦ is a fixed point. For example, α↦ψ(α) is β such that ψ(β)=β

You mean the first fixed point, right? It is usually written \(\textrm{enum}(\psi(\beta) = \beta)(0)\), \(\psi^{\infty}(0)\) (by the Kleene fixed point theorem because \(\psi\) is scott continuous below \(\kappa\)), or something like that.

Anyway, is \(\ddot \kappa\) always greater than or equal to \(\kappa\), right? (If not, then it conflicts the limit rule.)