Additively indecomposable ordinal

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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 ${\displaystyle \omega ,\omega ^{\omega }\!,\omega ^{\omega ^{\omega }}\!\!,\ldots }$