User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130620130233

When you say "\(\alpha_p\) isn't the sum, product, or exponential of some other ordinals", the only ordinal that satisfies this is \(\omega\). (Unless you mean to include ordinals greater than or equal to \(\varepsilon_0\); in that case \(\alpha_p\) must be either \(\omega\) or \(\varepsilon_\gamma\) for some \(\gamma\).)

In that case, the addition rule now works, as it is now equivalent to Cantor normal form. But the multiplication  rule still doesn't work;  take \(\alpha_1 = \alpha_2 = \omega, \beta_1 = \omega + 1, \beta_2 = \omega\) and the rule fails again.

The exponentiation rule still doesn't work; we can take \(\alpha = \omega^2, \beta = \omega\). If you specify that \(\alpha = \omega\), then the rule works.