User blog comment:Hyp cos/Attempt of OCF up to Stability/@comment-35470197-20181026043805/@comment-35470197-20181026053801

> Reflection instances, in this definition, are not sets or ordinals, but just "character strings".

Character strings in which ordinals appear should be sets. If they were not sets, how could you define the notion of such character strings...? Ordinals, reflection properties, OCFs, and so on makes sense only in set theory, and set theory can deal only with sets.

> And what do you mean by "an ordinal notation system associated to \(\varphi\)'s"?

There is no canonical way to obtain an ordinal notation system from an OCF, and hence "an ordinal notation system associated to an OCF" roughly means an ordinal notation system which is constructed by using certain properties of the OCF.

For example, Buchholz's OCF \(\psi\) yields an ordinal notation system whose limit is \(\psi_0(\varepsilon_{\Omega_{\omega}+1)\). For more details, see this.

An OCF itself does not generate a computable large number, while an ordinal notation system does through FGH. So if you do not have a way to define a well-defined ordinal notation system associated to your OCF, you do not obtain a well-defined computable large number.

> In the transfinite inductive definition, we first define \(C(0,\beta)\), then \(\psi_\pi(\mathbb X,0)\) where all inductive varibles of \(\mathbb X\) are 0;

I see. It is a transfinite induction on the well-founded partial order on \((\mathbb{X},\alpha\) rexicographically associated to \(IV \mathbb{X} \cup \alpha\) and the tree structure of \(\mathbb{X}\).