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 $\alpha >0$, there exist unique ordinals ${\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n$ for some $n\in \mathbb{\mathbb{N}}$, such that ${\alpha }_0\ge {\alpha }_1\ge \dots {\alpha }_n$ and ${\omega }^{{\alpha }_0}+{\omega }^{{\alpha }_1}+\cdots +{\omega }^{{\alpha }_n}=\alpha$.