Extensions of fast-growing hierarchy In recent weeks, I have been fascinated by the possible extensions of the fast-growing hierarchy. But is there a way to define it for all recursive ordinals? I saw a statement of this sort on googology, but they don't provide any additional details.
I have the following questions:  


*

*Is there an extension of FGH to all recursive ordinals (and beyond)?  

*I have also read that the Busy Beaver function is on level $\ f_{\omega_1^{CK}}(n)$ of FGH. Do you know some proof of this?

 A: The first question is easy, if the ordinal has a well ordered set, then yes.  So if made by recursion, then it is well ordered (some caveats there), but broadly yes.  Extends to all up to the Church Kleene Ordinal
I do not think the Busy Beaver Function is limited by the Church Kleene Ordinal.  The Busy Beaver Function is on the order of uncountability as it cannot be calculated by any Turing Machine.  
The Church Kleene Ordinal is countable and is the supremum of all recursive ordinals.
https://googology.wikia.org/wiki/Church-Kleene_ordinal
The Busy Beaver Function quickly breaks the bounds of even ZFC theory
"Wythagoras proved Σ(38)>fω2(167),  Σ(64)>fω2(4,098) and Σ(85)>fε0(1,907).
Wythagoras also showed that Σ(61,6)≫H(3) and LittlePeng9 proved Σ(134,7)≫U(3) where H and U are Chris Bird's functions. Wythagoras has also shown that Σ(38,3)≫fε0(374,676,379).
BB(1,919) is the smallest known value of the busy beaver sequence that is independent of ZFC. The first such TM constructed had 7,918 states."
