User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130619141922

Your rules A1 to A3 don't specify how to handle arithmetic operations for all possible ordinals. For example, the don't specify that \(\omega + \omega^2 = \omega^2\).

Your rule B2 gives a general rule for \(\alpha\) and \(\beta\) when they are limit ordinals, so it should work when \(\alpha = \omega\) and \(\beta = \omega^2\), but it doesn't.

Regarding Cantor's normal form, every ordinal needs to be expressed some way. Cantor's normal form gives a canonical way to express every ordinal, so it's very useful.