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

Obviously $1inmathbb\{H\}$, since No finite ordinal other than $1$ is in $mathbb\{H\}.$ Also, $omegainmathbb\{H\}$, 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 sequence $omega,omega^omega!,omega^\{omega^omega\}!!,ldots$

See also

* Ordinal arithmetic