Additively closed ordinal
= Additively closed ordinal
A nonzero ordinal $\delta$ is additively closed when $\beta+\gamma<\delta$ for all $\beta,\gamma<\delta$. These are exactly the ordinals $\omega^\alpha$: sufficiency follows from <Cantor normal form>, while necessity follows by splitting any leading coefficient or nonzero tail below $\delta$ into two smaller summands.