Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any , we have Additively indecomposable ordinals were named the gamma numbers by Cantor,^p.20 and are also called additive principal numbers. The class of additively indecomposable ordinals may be denoted, from the German "Hauptzahl". The additively indecomposable ordinals are precisely those ordinals of the form for some ordinal.
From the continuity of addition in its right argument, we get that if and α is additively indecomposable, then
Obviously 1 is additively indecomposable, since No finite ordinal other than is additively indecomposable. Also, 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.
The derivative of is written Ordinals of this form are called epsilon numbers. The number is therefore the first fixed point of the sequence

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. The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals are those of the form for any ordinal α''. Every epsilon number is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal is additively indecomposable. The delta numbers 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, and so on. Therefore, is the first ordinal which is -indecomposable for all, where denotes Knuth's [up-arrow notation].