User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190718022500

Firstly, I completely agree that without a definition standard form and recursive system of fundamental sequences, fPTO(theory)(n) or hPTO(theory)(n) type expressions don't make sense. The statement "if we could create fundamental sequences" is like saying "Hello Utter Oblivion".

But I'm still interested in the question of whether it is possible to build a scale for measuring computable functions based on theories.

Is it possible to use a system like Transcendental Integer?

For example:

Θ(t) is computable function which assigns to each n∈N the least natural number greater than or equal to the halting times of Turing machines with input 0 whose terminations admit formal proofs of length ≤(n = 10100) under the axiom of t-therory. t-theory take from the list below.