In set theory, a branch of mathematics, an additively indecomposable ordinal is any ordinal number Additively indecomposable ordinals were named the gamma numbers by Cantor,p.20 and are also called additive principal numbers. The class, from the German "Hauptzahl"..

From the continuity and

No finite 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.

Ordinals of this form (that is, fixed points) are called epsilon numbers is therefore the first fixed point of the sequence

Multiplicatively indecomposable

A similar notion can be defined for multiplication. If is greater than the multiplicative identity, 1, and < and < imply · < , then is multiplicatively indecomposable. The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantorp.20 for any ordinal . 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.

Higher indecomposables

Exponentially indecomposable ordinals are equal to the epsilon numbers. Tetration, as usually defined, gets stuck at limit ordinals, so the notion of indecomposable beyond exponentiation is not useful.

See also

• Ordinal arithmetic

References