User blog:Ikosarakt1/Fast-growing hierarchy

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

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: \(n>1\)

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

\(\alpha^{n+1} = \alpha^n*\alpha\)

Rule A3. Condition: \(n=1\)

\(\alpha*1 = \alpha\)

\(\alpha^1 = \alpha\)

Also, some basic fundamental sequence rule that will be useful.

Rule B1. Condition: \(\alpha = \omega\)

\(\omega[n] = n\)

Rule B2. Condition: \(\alpha\) and \(\beta\) = T.

\((\alpha+\beta)[n] = \alpha+\beta[n]\)

\((\alpha*\beta)[n] = \alpha*\beta[n]\)

\((\alpha^{\beta})[n] = \alpha^{\beta[n]}\)

When we set \(\alpha = \beta = \omega\), we can easily define fundamental sequences for all ordinals up to \(\epsilon_0\). Below are some examples:

\(\omega*2[n] = (\omega+\omega)[n] = \omega+\omega[n] = \omega+n

(More soon.)