# 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 $\mathbb{H}$.

Obviously $1\in \mathbb{H}$, since $$.
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 ${\u03f5}_{\alpha}$.
Ordinals of this form (that is, fixed points of ${f}_{\mathbb{H}}$) are called *epsilon numbers*.
The number ${\u03f5}_{0}={\omega}^{{\omega}^{{\omega}^{{\cdot}^{{\cdot}^{\cdot}}}}}$ is therefore the first fixed point of the series
$\omega ,{\omega}^{\omega},{\omega}^{{\omega}^{\omega}},\mathrm{\dots}$

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 |