# Difference between revisions of "Limit ordinal"

A limit ordinal is an ordinal that is neither $0$ nor a successor ordinal. Some authors classify zero as a limit ordinal.

## Properties

All limit ordinals are equal to their union.

All limit ordinals contain an ordinal $\alpha$ if and only if they contain $\alpha + 1$.

$\omega$ is the smallest nonzero limit ordinal, and the smallest ordinal of infinite cardinality.

$(\omega + \omega)$, also written $( \omega \cdot 2 )$, is the next limit ordinal. $( \omega \cdot \alpha )$ is a limit ordinal for any ordinal $\alpha$.