User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191006023842/@comment-35470197-20191009221210

> Expressions in TON are not admissible ordinals, and all Ω and C is just terms.

I just said "ordinal type" but not "ordinal". Please do not comfound them. Moreover, C is not a term. It is a 2-ary function symbol.

> of expressions that we can do in 1st order system is PTO of KPω, where α|Lα⊧KPω is 1st admissible.

But PTO does not characterise the least ordinal such that the corresponding segment of L forms a model.

> we need to compare the diagonalizer expressions with α|Lα - model of KP+some extention, that have same PTO.

I and hyp cos are talking about the canonicity and the uniqueness. Since such extensions are not unique, when you want to extend the correspondence in a canonical way, you need to formally characterise the assignment.

> All these "KP+some extention" can be considered metatheories defined in some stronger basic theory.

As I pointed out, your terminology of "meta theory" seems wrong. You are just referring to formalised theory, which is coded in the base theory. They are not the meta theory of the base theory.