User blog comment:Ynought/What grows faster?/@comment-35470197-20190309234110

It seems almost impossible for us to go beyond Friedman's functions, because they are so strong that they are unprovably total under ZFC set theory but pointwise well-defined. They are greater than functions corresponding to ANY recursive ordinal below PTO(ZFC) with respect to FGH.

Since the most common way to create a computable large number here is to apply a recursion through a notation with a correspondence to ordinals whose limit is MUCH smaller than PTO(ZFC), the resulting computable large function is MUCH smaller than them.

On the other hand, my strongest computable large function directly goes beyond PTO(T) with respect to FGH, where T is the sete theory we are working in, and seems to go beyond Greedy clique sequence when T = HUGE+. I note that Greedy clique sequence is a system whose pointwise well-definedness is proved under HUGE+. Since the omega consistency of HUGE+ is not provable under suitable set theories, it does not ensure that the pointwise well-definedness is provable under ZFC. Therefore we actually need to work in HUGE+ in order to argue this topic. Questioning "which one is large?" without setting axioms is non-sense.