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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 are also called gamma numbers or additive principal numbers. 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 ${\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\omega }}}}}$ is therefore the first fixed point of the sequence ${\displaystyle \omega ,\omega ^{\omega }\!,\omega ^{\omega ^{\omega }}\!\!,\ldots }$

## Multiplicatively indecomposable

A similar notion can be defined for multiplication. If α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable. 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (also called delta numbers) are those of the form ${\displaystyle \omega ^{\omega ^{\alpha }}\,}$ for any ordinal α. Every epsilon number is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable. The delta numbers (other than 2) are the same as the prime ordinals that are limits.

## Higher indecomposables

Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of ${\displaystyle \varepsilon _{\alpha }}$), and so on. Therefore, the Feferman-Schutte ordinal ${\displaystyle \Gamma _{0}}$ (fixed point of ${\displaystyle \varphi _{\alpha }(0)}$) is the first ordinal which is ${\displaystyle \uparrow ^{n}}$-indecomposable for all ${\displaystyle n}$, where ${\displaystyle \uparrow }$ denotes Knuth's up-arrow notation.