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 ${\displaystyle \alpha >0}$, there exist unique ordinals ${\displaystyle \alpha _{0},\alpha _{1},\ldots ,\alpha _{n}}$ for some ${\displaystyle n\in \mathbb {N} }$, such that ${\displaystyle \alpha _{0}\geq \alpha _{1}\geq \ldots \alpha _{n}}$ and ${\displaystyle \omega ^{\alpha _{0}}+\omega ^{\alpha _{1}}+\cdots +\omega ^{\alpha _{n}}=\alpha }$.^[1]

References

1. M. Rathjen, [The Art of Ordinal Analysis https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf] (p.4).