Cantor normal form

Ordinal Normal Form (Cantor).

For ordinal numbers $\alpha \mathrm{\ge }\mathrm{2}$ and $\gamma \mathrm{\ge }\mathrm{1}$ there is a unique $n$ such that there exist unique ${\beta }_{\mathrm{0}}\mathrm{>}\mathrm{\cdots }\mathrm{>}{\beta }_n$ and such that $\gamma \mathrm{=}{\alpha }^{{\beta }_{\mathrm{0}}}\mathrm{\cdot }{\delta }_{\mathrm{0}}\mathrm{+}\mathrm{\cdots }\mathrm{+}{\alpha }^{{\beta }_n}\mathrm{\cdot }{\delta }_n$.

This theorem is often referred to as the Cantor Normal Form of $\gamma$ in the base of $\alpha$.