User blog comment:MachineGunSuper/The Hierarchy Replacement Function/@comment-32783837-20171223145318

It's not easy to express something beyond tetration in SGH using normal ordinals, so normally this would be very, very difficult to "use correctly".

This can be made easier, though.

Scrap the ε0, ζ0, hierarchy system and just continue through hyper-operators, so ε0 would just be expressed as ω↑↑ω. Then, continue with ω↑↑(ω+1), ω↑↑(ω2), >ω↑↑ω2, ω↑↑ωω, ω↑↑↑3, ω↑↑↑ω, ω↑↑↑↑ω, etc.

Then, define ω{ω}ω as the limit of the sequence {ωω, ω↑↑ω, ω↑↑↑ω, ω↑↑↑↑ω, ω↑↑↑↑↑ω, ω↑↑↑↑↑↑ω, ...}, call that in FGH, and the result will be perfectly well-defined, probably comparable to φ(ω,0) using normal ordinal notations.

Actually, why stop there? We can continue to define hyper-operator ordinals in FGH using the following rules:

β{α+1}1 = β

β{α+1}n = β{α}(β{α+1}n-1)

β{α}n = β{α[x]}n where x is the number plugged into FGH if α is a limit ordinal

β{n}α = β{n}α[x] where x is the number plugged into FGH if α is a limit ordinal

Then, we can continue to define ω{ω+1}ω, ω{ω2}ω, ω{ωω}ω, ω{ω{ω}ω}ω, etc.

and then continue through BEAF operators (i.e. something like {ω,ω,1,1,ω})...

then we can call something like fω↑↑ω & ω(10) and that would yield a pretty bulky result. Not sure how strong that'd be compared to normal ordinals, probably somewhere between LVO and BHO.