User blog comment:P進大好きbot/ZFC-variant of Rayo's number/@comment-39605890-20191204031223/@comment-35470197-20191204060827

Partially yes. Although eval itself is not fast growing as you guessed, CoRayo is fast growing.

Let f be a defining formula of a natural number in ZF set theory. I mean, ZFC proves that there exists a unique natural natural number satisfying f. Take the least (with respect to Goedel numbering) finite segment S of ZFC which can prove f. Then any V_α satisfying S satisies that there uniquely exists a natural number satisfying f. If there is a provable inequality betwee definable natural numbers in ZFC, we can choose S so that the inequality holds.

Of course, the value defined by f might differ up to the universe. The point is I can choose an arbitrary big finite segment of ZFC. Say, if n is 10↑^{10}10, then S can be chosen to be a finite segment of ZFC consisting of 10↑^{10}10 formulae. So roughly speaking, I approximate ZFC by finitely axiomisable set theory (bounded by n), and use the satisfaction at the least model as a truth predicate in order to name a large number.

It is not an evidence, but I would like to say that there are few "evidences" to ensure the largeness of uncomputable large numbers. For example, nobody knows whether second order busybeaver function is stronger than ω_1^{CK} in FGH or not.