# 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 $\beta ,\gamma <\alpha$ , we have $\beta +\gamma <\alpha .$ Additively indecomposable ordinals are also called gamma numbers. The additively indecomposable ordinals are precisely those ordinals of the form $\omega ^{\beta }$ for some ordinal $\beta$ .

From the continuity of addition in its right argument, we get that if $\beta <\alpha$ and α is additively indecomposable, then $\beta +\alpha =\alpha .$ Obviously 1 is additively indecomposable, since $0+0<1.$ No finite ordinal other than $1$ is additively indecomposable. Also, $\omega$ 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 $\omega ^{\alpha }$ .

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