# additively indecomposable

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 series $\omega,\omega^{\omega}\!,\omega^{\omega^{\omega}}\!\!,\ldots$

Title additively indecomposable AdditivelyIndecomposable 2013-03-22 13:29:04 2013-03-22 13:29:04 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 03F15 msc 03E10 OrdinalArithmetic epsilon number epsilon zero