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
Canonical name	AdditivelyIndecomposable
Date of creation	2013-03-22 13:29:04
Last modified on	2013-03-22 13:29:04
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Definition
Classification	msc 03F15
Classification	msc 03E10
Related topic	OrdinalArithmetic
Defines	epsilon number
Defines	epsilon zero