User blog comment:Googology Noob/Ordinal FGH, with an actual definition!/@comment-1605058-20151219081414/@comment-27173506-20151219140949

Definition: 000 is a string of zeroes.

alpha[w] = alpha

w(m+1)[alpha] = wm+alpha. If wm+alpha = alpha, then w(m+1)[alpha] = alpha+wm

w^m+1[alpha] = (w^m)alpha

w^w[alpha] = w^alpha. If w^alpha = alpha, then w^w[alpha] = w^alpha+1

epsilon_0[alpha]. For finite alpha, epsilon_0[alpha] = w^^alpha. Otherwise, epsilon_0[w*alpha] = epsilon_alpha. If epsilon_alpha = alpha, then epsilon_0[w*alpha] = epsilon_alpha+1. epsilon_0[(w*alpha)+beta+1] = w^(epsilon_0[(w*alpha)+beta])+1

phi(m+1,000,0)[alpha] = For finite alpha, phi(m+1,000,0)[alpha+1] = phi(m,phi(m+1,000,0)[alpha])) and phi(m+1,000,0)[1] = phi(m,000,0). Otherwise, phi(m+1,000,0)[w*alpha] = phi(m+1,000,alpha). If phi(m+1,000,alpha) = alpha, then phi(m+1,000,0)[w*alpha] = phi(m+1,000,alpha+1).

phi(w,000,0)[alpha] = phi(alpha,000,0). If phi(alpha,000,0) = alpha, then phi(w,000,0)[alpha] = phi(alpha,000,1).

That is, if I'm not mistaken, all the ordinals I used. Did I miss something?

I agree that its definition is not perfect, but I think it's pretty good. If you can think of something to improve, please tell me!