additively indecomposable

An ordinalMathworldPlanetmathPlanetmath α is called additively indecomposable if it is not 0 and for any β,γ<α, we have β+γ<α. The set of additively indecomposable ordinals is denoted .

Obviously 1, since 0+0<1. No finite ordinal other than 1 is in . Also, ω, since the sum of two finite ordinals is still finite. More generally, every infiniteMathworldPlanetmath cardinal is in .

is closed and unboundedPlanetmathPlanetmath, so the enumerating function of is normal. In fact, f(α)=ωα.

The derivative f(α) is written ϵα. Ordinals of this form (that is, fixed points of f) are called epsilon numbers. The number ϵ0=ωωω is therefore the first fixed point of the series ω,ωω,ωωω,

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