I hereby propose the usage of letters beyond P as an ordinal ruler. This is an semi-formal extension of my letter notation (yes, I'm aware none of this is formally defined until I actually define a specific notation for these scales of numbers. I'm not claiming otherwise. It's just a suggestion for a continuation, which - in the mean time - serve as a rough ruler)

For integers n>2, 0<k<10:

Qn = [ε₀]n = [ω↑↑n]10

Q(n+log k) = [a tower of n ω's topped by k] 10

Rn = [Γ₀]n

Define the function ξ(α) = φ(α,0) (with φ being the Veblen function)

R(n+k/10) = [ξ(n-1)(ω↑↑k)]10

Sn = [SVO]n = [φ(1,0,0,...,0)]10 with n 0's

Tn = [LVO]n = [ψ(Ω↑↑3)]n = ...

Vn = [BHO]n = [ψ(Ω↑↑(n+1))]10 =

Wn = [ψ(ψω(0)]n = [ψ(ψ(n)(0)]10

X3 = [ψ(ψΩ(0)]10

X4 = [ψ(ψΩ_Ω(0)]10




X10 = [ψ(ψI(0)]10

And for 1<n<2 we have:

<letter>n = <previous letter>(10n-1)

Where the letters are chosen from the set E,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X.

Now we can approximate numbers with the form Xk (with k>1). For k very close to 1 we can choose it to be anything:

X1.001 ~ W1.0023 ~ V1.0053 ~ T1.012 ~ S1.028 ~ R1.069 ~ Q1.17 ~ P1.48 ~ N3 = {10,10,10,3} in BEAF

For number higher than P10 this ruler is no longer continuous, but it can still be useful:

X1.01 ~ W1.023 ~ V1.055 ~ T1.135 ~ S1.365 ~ R2.32 = R(2+3.2/10) which is between [φ(ω↑↑3,0)]10 and [φ(ω↑↑4,0)]10.

(actually in this specific case it is easy to interpolate further and write R(2+(3+log 2)/10) = [φ(ω↑ω↑ω2,0)]10 so X1.01 would be between [φ(ω↑↑3,0)]10 and [φ(ω↑ω↑ω2,0)]10)