User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333

The equations in Rule B2 do not work in general.

Take \(\alpha = \omega + 1\) and \(\beta = \omega\):

\((\omega + 1 + \omega) [n] = (\omega + \omega) [n] = \omega + n \neq \omega + 1 + \omega [n] = \omega + n + 1\)

\(((\omega + 1) \omega) [n] = (\omega * \omega)[n] = \omega * n \neq (\omega + 1) (\omega [n]) = (\omega + 1) n = \omega * n + 1\)

\((\omega + 1) ^ {\omega}) [n] = (\omega ^ {\omega}) [n] = \omega^n \neq (\omega + 1) ^ {\omega [n]} = (\omega + 1)^n \).