Cantor normal form theorem

The Cantor normal form theorem is a theorem about ordinal arithmetic proven by Georg Cantor in 1897. It states that for every ordinal [math]\displaystyle{ \alpha\gt 0 }[/math], there exist unique ordinals [math]\displaystyle{ \alpha_0,\alpha_1,\ldots,\alpha_n }[/math] for some [math]\displaystyle{ n\in\mathbb N }[/math], such that [math]\displaystyle{ \alpha_0\ge\alpha_1\ge\ldots\alpha_n }[/math] and [math]\displaystyle{ \omega^{\alpha_0}+\omega^{\alpha_1}+\cdots+\omega^{\alpha_n}=\alpha }[/math].