Additively indecomposable ordinal information

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (August 2021) (Learn how and when to remove this message)

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any ${\displaystyle \beta ,\gamma <\alpha }$, we have ${\displaystyle \beta +\gamma <\alpha .}$ Additively indecomposable ordinals were named the gamma numbers by Cantor,^[1]p.20 and are also called additive principal numbers. The class of additively indecomposable ordinals may be denoted ${\displaystyle \mathbb {H} }$, from the German "Hauptzahl".^[2] The additively indecomposable ordinals are precisely those ordinals of the form ${\displaystyle \omega ^{\beta }}$ for some ordinal ${\displaystyle \beta }$.

From the continuity of addition in its right argument, we get that if ${\displaystyle \beta <\alpha }$ and α is additively indecomposable, then ${\displaystyle \beta +\alpha =\alpha .}$

Obviously 1 is additively indecomposable, since ${\displaystyle 0+0<1.}$ No finite ordinal other than ${\displaystyle 1}$ is additively indecomposable. Also, ${\displaystyle \omega }$ is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.

The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by ${\displaystyle \omega ^{\alpha }}$.

The derivative of ${\displaystyle \omega ^{\alpha }}$ (which enumerates its fixed points) is written ${\displaystyle \varepsilon _{\alpha }}$ Ordinals of this form (that is, fixed points of ${\displaystyle \omega ^{\alpha }}$) are called epsilon numbers. The number ${\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdots }}}}$ is therefore the first fixed point of the sequence ${\displaystyle \omega ,\omega ^{\omega }\!,\omega ^{\omega ^{\omega }}\!\!,\ldots }$