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
β + γ < α .
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
From the continuity of addition in its right argument, we get that if
and α is additively indecomposable, then
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
(which enumerates its fixed points) is written
Ordinals of this form (that is, fixed points of
) are called epsilon numbers.
is therefore the first fixed point of the sequence
A similar notion can be defined for multiplication.
The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2.
Besides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantor[1]p.20) are those of the form
The delta numbers (other than 2) are the same as the prime ordinals that are limits.
Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of
denotes Knuth's up-arrow notation.
[citation needed] This article incorporates material from Additively indecomposable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.