User blog comment:MachineGunSuper/The Hierarchy Replacement Function/@comment-32783837-20171223145318/@comment-30754445-20171224051538

What do you mean "interpertation"?

Your specific definition get stuck at ε0. There's no way to "interpert" it in another way (and if there were multiple interpertations, that would make the notation ill-defined).

And it doesn't matter that the standard ordinal notation doesn't recognize tetration. You've defined ordinal tetration yourself, and by writing the rule β{α+1}n = β{α}(β{α+1}n-1), you've set ω↑↑(ω+1) to be ε0.

This can be by-passed, though, by redifining that rule do this:

(i) If β{α+1}n-1 < β{α+1}n then:

(β{α+1}n) = β{α}(β{α+1}n-1)

(ii) If β{α+1}n-1 = β{α+1}n then:

(β{α+1}n) = β{α}[(β{α+1}n-1)+1]

Somewhat artificial, but it escapes the fixed-point trap (in this definition ω↑↑(ω+1) will be ω↑(ε0+1) which is exactly what w want). I think that this version really does get to ω{ω}ω = φ(ω,0) and unless this gets stuck at some later point (I'm not sure whether it does or doesn't) then this definition gives:

ω{α}ω = φ(α,0). for all ω < α < Γ₀

and:

{ω,ω,1,2} = lim (ω{ω}ω, ω{ω{ω}ω}ω, ω{ω{ω{ω}ω}ω}ω, ...) = Γ₀