User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190914073943

Hmm...? Maybe we are referring to distict theories by the word "meta theory". I meant the theory in which you want to define the syntax of the KP set theories. I guess that you meant the KP set theories.

In order to clarify, I call the former theory "the base theory", and the latter theories "the formalised theories". So now, you are considering the formalised theory \(T_x\) consisting of (the Goedel numbers of) formulae \(\phi\) satisfying \(L_x \models \phi\) for each ordinal \(x\) in the base theory. Am I understanding your intension correctly?

Then you have the same issue again. The inequality \(x < y\) of ordinals in the base thoery does not imply the inequality \(T_x < T_y\) of the strength of the formalised theories. I guess that you want to create a new hierarchy of formalised theories indexed by ordinals in the base theory, and hence the current construction is still not suitable for this purpose.