additively indecomposableAdditivelyIndecomposable2003-02-23 22:58:322005-03-03 17:01:41Definitionepsilon numberepsilon zero\usepackage{amssymb}
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\def\indecomp{\mathbb{H}}An ordinal $\alpha$ is called \emph{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 $\indecomp$.
Obviously $1\in\indecomp$, since $0+0<1$.
No finite ordinal other than $1$ is in $\indecomp$.
Also, $\omega\in\indecomp$, since the sum of two finite ordinals is still finite.
More generally, every infinite cardinal is in $\indecomp$.
$\indecomp$ is closed and unbounded, so the enumerating function of $\indecomp$ is normal.
In fact, $f_\indecomp(\alpha)=\omega^\alpha$.
The derivative $f_\indecomp^\prime(\alpha)$ is written $\epsilon_\alpha$.
Ordinals of this form (that is, fixed points of $f_\indecomp$) are called \emph{epsilon numbers}.
The number $\epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}$ is therefore the first fixed point of the series
$\omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots$