User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130619222626

The simplest counterexample is \(\alpha = \omega\) and \(\beta = \omega^2\).

Yes, we want to ignore such examples, but the rules must say when and how we ignore things. That is where Cantor normal form comes in to play, an ordinal is expressed in a "proper" from precisely when it is in Cantor normal form.