User blog comment:PsiCubed2/Help Me Understand Ordinals Beyond the BHO/@comment-27516045-20170220175334

@Psi: I will use a notation to help you. Rather than psi(e(W+1)) we write psi(psi_1(0)).

The psi_1 function is like the normal psi function, but it uses fixed points of a > W^a. So the limit of psi(psi_1(0)^psi_1(0)^...) is psi(psi_1(1)) = psi(e(W+2)). Then psi(psi_1(a)) for an ordinal a, and the limit of that is psi(psi_1(W)) = psi(e(W*2)). This allows us to go up to psi(psi_1(psi_1(...))) where we use psi(psi_1(W_2)) (or just psi(W_2)). Then there is psi(psi_1(W_2+1)), psi(psi_1(W_2+W)), psi(psi_1(W_2*w)) psi(psi_1(W_2^2)), psi(psi_1(W_2^W_2)), and the limit of that is psi(psi_1(psi_2(0))) or just psi(psi_2(0)) (generally you can just write the innermost layer, and even that isn't necessary if there is no ambiguity, so psi(psi_1(psi_2(W_3))) is commonly written psi(W_3)). The limit of psi(psi_1(psi_2(psi_3(...)))) or just psi(W_n) is psi(W_w).

this is just one variant; using the theta-function gets different results, but similar concept.

Do you understand this? If you do, I will explain further.