User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190718022500/@comment-35470197-20190718024533

> Is it possible to use a system like Transcendental Integer?

Partially, yes. There are several problems:
 * 1) The length of a proof heavily depends on how you code theories.
 * 2) There are many choices of formal languages for \(\textrm{ZFC}\) set theory.
 * 3) There are many choices of logical symbols for \(\textrm{ZFC}\) set theory.
 * 4) There are many choices of inferences of first order logic.
 * 5) The length of a proof heavily depends on how you code Turing machines in \(t\).
 * 6) Ther are many choices of Goedel correspondences.
 * 7) There are many choices of enumeration of Turing machines.
 * 8) If \(t\) is sufficiently strong, then the length of a proof just interpretes the length of the code of a Turing machine.
 * 9) If \(t\) can prove anything, i.e. contradiction, then the length of a proof is estimated using the length of the code.
 * 10) If \(t\) is stronger than the base theory, then a Turing machine whose termination can be provable in \(t) does not necessarily actually terminates in the base theory. Namely, there can be a model of the base theory in which such a Turing machine has an infinite loop, even if \(t\) itself is consistent.
 * 11) This is related to the \(\Sigma_1\)-soundedness problem that \(f_{\textrm{PTO}(\textrm{ZFC} + \textrm{I}0)}\) does not work well in \(\textrm{ZFC}\) set theory.

Such problems are usually ignorable if you consider only finitely many theories weaker than the base theory, because the ambiguity of the choices of codings does not contribute the growth rate well. On the other hand, if you consider infinitely many theories, then you need to formalise the coding, and hence the ambiguity causes problems.