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 The class of additively indecomposable ordinals (aka gamma numbers) is sometimes denoted
From the continuity of addition in its right argument, we get that if and α is additively indecomposable, then
is closed and unbounded, so the enumerating function of is normal. In fact,
The derivative (which enumerates fixed points of fH) is written Ordinals of this form (that is, fixed points of ) are called epsilon numbers. The number is therefore the first fixed point of the sequence
A similar notion can be defined for multiplication. The multiplicatively indecomposable ordinals (aka delta numbers) 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.
- Sierpiński, Wacław (1958), Cardinal and ordinal numbers., Polska Akademia Nauk Monografie Matematyczne 34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787