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 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 f_H) 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.

Multiplicatively indecomposable

See also