Additive principal number

An additive principal number is a non-zero ordinals \(\alpha\) satisfying \(\forall(\beta,\gamma<\alpha)(\beta+\gamma<\alpha)\). An ordinal is an additive principal number if and only if it is of the form \(\omega^{\beta}\) for some ordinal \(\beta\). In particular, an additive principal number is either \(1\) or a limit ordinal. The class of all aditive principal number is denoted by \(\textrm{AP}\).