Greatest power of omega below an ordinal
= Greatest power of omega below an ordinal
Every nonzero ordinal $\alpha$ has a greatest exponent $\beta$ such that $\omega^\beta\leq\alpha$. It is the leading exponent in the <Cantor normal form> of $\alpha$, and
$$
\alpha=\omega^\beta n+\gamma
$$
for a positive finite integer $n$ and $\gamma<\omega^\beta$.