Additively indecomposable ordinal

An ordinal $\alpha$ is called additively indecomposable if it is not 0 and for any $\beta ,\gamma <\alpha$ , we have $\beta +\gamma <\alpha .$ The set of additively indecomposable ordinals is denoted $\mathbb {H} .$

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 $f_{\mathbb {H} }^{\prime }(\alpha )$ 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 $\omega ,\omega ^{\omega }\!,\omega ^{\omega ^{\omega }}\!\!,\ldots$

See also