User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191006023842/@comment-31580368-20191009153803

This correspondence is just intuitive and based on an intuitive understanding of the structure of large countable ordinals.

Expressions in TON are not admissible ordinals, and all Ω and C is just terms. But for example in 1st order system Ω behaves like a diagonalizer just like Ω in Feferman's θ-OCF. Limit of expressions that we can do in 1st order system is PTO of KPω, where α|Lα⊧KPω is 1st admissible. The expressions of the 2nd order system (in Main Ordinal Notation System) are so strong that for their analysis we need to compare the diagonalizer expressions with α|Lα - model of KP+some extention, that have same PTO.

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