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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 β , γ < α {\displaystyle \beta ,\gamma <\alpha } , we have β + γ < α . {\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 ω β {\displaystyle \omega ^{\beta }} for some ordinal β {\displaystyle \beta }.
From the continuity of addition in its right argument, we get that if β < α {\displaystyle \beta <\alpha } and α is additively indecomposable, then β + α = α . {\displaystyle \beta +\alpha =\alpha.}
Obviously 1 is additively indecomposable, since 0 + 0 < 1. {\displaystyle 0+0<1.} No finite ordinal other than 1 {\displaystyle 1} is additively indecomposable. Also, ω {\displaystyle \omega } is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal is additively indecomposable.
The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by ω α {\displaystyle \omega ^{\alpha }}.