Additively indecomposable ordinal

Summary

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any $\beta,\gamma<\alpha$, we have $\beta+\gamma<\alpha.$ Additively indecomposable ordinals were named the gamma numbers by Cantor,^p.20 and are also called additive principal numbers. The class of additively indecomposable ordinals may be denoted $\mathbb H$, from the German "Hauptzahl". The additively indecomposable ordinals are precisely those ordinals of the form $\omega^\beta$ for some ordinal $\beta$.