User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190914114819

> Well, is there any way to define a theory like KP based on its L_x model

Also, you hope that larger ordinals yield stronger theory, right? One possible way is to consider the arithmetic PA+Ind(α) for each recursive ordinal α. Since Ind(α) is an axiom schema which is (non-strictly) increasing, it gives such a hierarchy. Is it what you want?

> I want to understand if there is a way to use large countable ordinals in countable googology without OCF.

Actually, there are, but I am not fully understood them. For example, TON is defined without an actual OCF, and if TON is well-founded, its ordinal type gives a large ordinal. As far as I know, TON is the onely example of such an ordinal notation appearing googology which is created without an OCF but is (partially) verified the well-foundedness. In mathematics outside googology, there are several known methods, but nobody has brought them to googology because they seem highly difficult.