User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130619135918

Unfortunately, any infinite ordinal can be "unproper". The ordinal \(\omega*2\) can be expressed as \(\omega + \omega\) or as \(\omega + 1 + \omega\), and neither is preferred over the other, unless we specify it to be so. So you have to specify it in your rules.

The usual way to deal with this is to define the set P of additively principal ordinals. These are the ordinals of the form \(\omega^{\alpha}\). By Cantor's Normal Form Theorem, every ordinal can be uniquely expressed in the form

\(\omega^{a_1} + \omega^{a_2} + \ldots + \omega^{a_m}\), where \(a_1 \geq a_2 \geq \ldots \geq a_m\).

Then we can specify that

\((\omega^{a_1} + \omega^{a_2} + \ldots + \omega^{a_m+1})[n] = \omega^{a_1} + \omega^{a_2} + \ldots + \omega^{a_m} * n \)

and, when \(a_m\) is a limit ordinal,

\((\omega^{a_1} + \omega^{a_2} + \ldots + \omega^{a_m})[n] = \omega^{a_1} + \omega^{a_2} + \ldots + \omega^{a_m[n]}\).

We can then discard the multiplication and exponentiation rules, as they don't necessarily work even for principal ordinals.