User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130619143731

It's not really about proper versus unproper ordinals, but rather proper versus unproper _expressions_. Any infinite ordinal can be expressed in a proper or an unproper expression. So you want to express your ordinals only in proper form - but you have to somehow specify that in your rules! If you just say \(\alpha + \beta\), there's nothing preventing \(\alpha\) from being \(\omega\) and \(\beta\) from being \(\omega^2\). If you set rules that prevents that from happening, then you're good.

That's where Cantor normal form is useful - basically, Cantor normal form is the "proper" way to express an ordinal.