Additive principal ordinals

An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a CNF representation) that the additively principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some \(\gamma\). As such, the second infinite additively principal ordinal is \(\omega^2\), the first limit of additively principal ordinals is \(\omega^\omega\), and the first limit of those is \(\omega^{\omega^2}\).

Additively principal ordinals can be generalized to multiplicatively principal ordinals and exponentially principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicatively principal ordinals are to additively principal ordinals as additively principal ordinals are to limit ordinals. However, exponentially principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just the same as the epsilon numbers.