---
tag: 7TFCZSVX5R
kind: concept
name: Additively indecomposable ordinal
aliases: [additive-principal, additive-principal-ordinal, additively-indecomposable, additively-principal, additively-principal-ordinal, delta-number, indecomposable-ordinal]
sources:
  - wikipedia: en:17202420@r1340030283
    locator: Additively indecomposable ordinal
    note: adopted
    mention: wikipedia://en/17202420
    sections: [Summary]
same_as:
  wikidata: Q4681349
  wikipedia: Additively indecomposable ordinal
inventory: [WE0CFEX6SD]
---

## Summary

In set theory, a branch of mathematics, an **additively indecomposable ordinal** *α* is any [[ordinal number|0RDJK7EQYY]] 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,<sup>p.20</sup> 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 $\beta$.