Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any \beta,\gamma, we have \beta+\gamma Additively indecomposable ordinals were named the gamma numbers by Cantor,p.20 and are also called additive principal numbers. The class of additively indecomposable ordinals may be denoted \mathbb H, from the German "Hauptzahl". The additively indecomposable ordinals are precisely those ordinals of the form \omega^\beta for some ordinal \beta.

From the continuity of addition in its right argument, we get that if \beta and α is additively indecomposable, then \beta + \alpha = \alpha.

Obviously 1 is additively indecomposable, since 0+0 No finite ordinal other than 1 is additively indecomposable. Also, \omega is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.

The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by \omega^\alpha.

The derivative of \omega^\alpha (which enumerates its fixed points) is written \varepsilon_\alpha Ordinals of this form (that is, fixed points of \omega^\alpha) are called epsilon numbers. The number \varepsilon_0 = \omega^{\omega^{\omega^{\cdots}}} is therefore the first fixed point of the sequence \omega,\omega^\omega!,\omega^{\omega^\omega}!!,\ldots