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An additive principal number (also known as an additively indecomposable ordinal[1]) 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 $$\textrm{AP}$$.[2]

## Sources

1. PlanetMath, Additively Indecomposable
2. M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal. Archive for Mathematical Logic 29(4) 249-263 (1990).

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