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 \beta,\gamma<\alpha, we have \beta+\gamma<\alpha. The class of additively indecomposable ordinals (aka gamma numbers) is denoted \mathbb{H}.

From the continuity of addition in its right argument, we get that if \beta < \alpha and α is additively indecomposable, then \beta + \alpha = \alpha.

Obviously 1\in\mathbb{H}, since 0+0<1. No finite ordinal other than 1 is in \mathbb{H}. Also, \omega\in\mathbb{H}, since the sum of two finite ordinals is still finite. More generally, every infinite cardinal is in \mathbb{H}.

\mathbb{H} is closed and unbounded, so the enumerating function of \mathbb{H} is normal. In fact, f_\mathbb{H}(\alpha)=\omega^\alpha.

The derivative f_\mathbb{H}^\prime(\alpha) (which enumerates fixed points of fH) is written \epsilon_\alpha. Ordinals of this form (that is, fixed points of f_\mathbb{H}) are called epsilon numbers. The number \epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}} is therefore the first fixed point of the sequence \omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots

Multiplicatively indecomposable

A similar notion can be defined for multiplication. The multiplicatively indecomposable ordinals (aka delta numbers) are those of the form \omega^{\omega^\alpha} \, 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.

See also

References

This article incorporates material from Additively indecomposable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.