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 $\indecomp$ .

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 epsilon numbers. The number $\epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}$ is therefore the first fixed point of the series $\omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots$