additively indecomposable

An ordinal $\alpha$ is called additively indecomposable if it is not $0$ and for any , we have .The set of additively indecomposable ordinals is denoted $ℍ$.

Obviously $1\in ℍ$, since .No finite ordinal other than $1$ is in $ℍ$.Also, $\omega \in ℍ$, since the sum of two finite ordinals is still finite.More generally, every infinite cardinal is in $ℍ$.

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

The derivative $f_{ℍ}^{\prime }(\alpha )$ is written ${ϵ}_{\alpha }$.Ordinals of this form (that is, fixed points of $f_{ℍ}$) are called epsilon numbers.The number ${ϵ}_0={\omega }^{{\omega }^{{\omega }^{{\cdot }^{{\cdot }^{\cdot }}}}}$ is therefore the first fixed point of the series$\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},\mathrm{\dots }$