User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-25601061-20180729214112/@comment-35470197-20180802014528

1. I heard it here.

https://twitter.com/rpakr_googology/status/1009709564269219840

2. If you mean that you allow no axioms, your extention of the weak compact OCF is never well-defined, because the weakly compact OCF is defined by heavily using axioms. It is something like regarding \(\infty\) as a natural number just saying that "it is an extension of Graham number".

If you mean that you allow any choice of non-specific axioms, then you can assume \(0 = 1\). Then you will get the largest number around the world.

Anyway, do you know the definition of the notions of an ordinal and a cardinal? I guess that what you call an "OCF" is just a countable set with expansion rules, which might not be an ordinal notation. In order to present ordinals or enjoy FGH, you need well-founded relations on your notation so that the transfinite induction works.