User blog comment:Pellucidar12/Large number idea (inspired by Oblivion)/@comment-1605058-20170409074700

Believe it or not, there are many more problems with this than with Oblivion, so let me just run through them.

You use FGH up to ordinals for which no fundamental sequences have been given.

The notion of a "system of mathematics" is far from being well-defined.

Assuming that with a "system of mathematics" you mean something like FOST, then the proof-theoretic ordinal of such a system is not well-defined (you would need an axiomatic theory like ZFC for that).

If you really mean what I think you mean with "using the fewest different symbols", then you are going to have a hard time, because "systems" with finitely many symbols are mostly useless.

Even then, there may be many systems which use the same number of symbols and you specify nothing which says which of these to choose.

You use the notion of ordinal having "growth rate comparable" to a function. If you are to make sense of that using FGH, you again need to specify all the fundamental sequences, and this time you need to do it up to CK0 ordinal (which you are defining right now!).

Once you get to ordinals beyond Church-Kleene ordinal, no "system" (or, actually, axiomatic theory) has such a proof-theoretic ordinal, since these are always equal to at most the CK ordinal.

Good luck fixing all these problems :)