User blog:Ikosarakt1/Fast-growing hierarchy

First, we have three main rules for FGH. I shall use S for successor ordinal and L for limit ordinal and [n] for n-th term of the fundamental sequence. \(\alpha\) can stand for either

Rule M1. Condition: (\(\alpha = 0\))

\(f_0(n) = n+1\)

Rule M2. Condition: (\(\alpha = S\))

\(f_{\alpha+1}(n) = f^n(n)\)

Rule M3. Condition: (\(\alpha = L\))

\(f_{\alpha}(n) = f_{\alpha[n]}(n)\)

It order to get to \(\epsilon_0\) and for further purposes, I define the ordinal arithmetic:

Rule A1. Condition: \(\alpha\) is a transfinite ordinal.

\(1+\alpha = \alpha\)

Rule A2. Condition: \(n>1\)

\(\alpha*(n+1) = \alpha*n+\alpha\)

Rule A3. Condition