Additive principal number

An additive principal number (also known as an additively indecomposable ordinal ) is a non-zero ordinal \(\alpha\) satisfying \(\forall(\beta,\gamma<\alpha)(\beta+\gamma<\alpha)\). In other words, it is a non-zero ordinal closed under addition. 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. However, not all limit ordinals are additive principal numbers, for example \(\omega2\).

The class of all additive principal numbers is often denoted by \(\textrm{AP}\).