Additively indecomposable ordinal

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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[edit]

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[edit]

References[edit]

  • Sierpiński, Wacław (1958), Cardinal and ordinal numbers., Polska Akademia Nauk Monografie Matematyczne 34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787 

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