User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130620134719

Well, it's easiest to define \(\alpha = \omega\). You could use "\(\alpha\) isn't the sum, product, or exponential of some other ordinals", as that will make \(alpha\) either \(\omega\) or an epsilon number. Then you must make sure that \(\beta\) isn't a greater epsilon number, so add the requirement \(\alpha^\beta \neq \beta\). Also, you must require that \(\alpha\) cannot be replaced by a greater epsilon number.

Also, you probably want to change rule PB1, to say "use previous rules" or something like that.

You don't need the multiplcation rule in full generality, however you must have a rule for \(\alpha^\beta\) with \(\beta\) is a successor ordinal. Note that you need the same requirements on \(\alpha\).