# Difference between revisions of "Limit ordinal"

From Cantor's Attic

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Trixie Wolf (Talk | contribs) (Fixed a couple of things.) |
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{{DISPLAYTITLE: Limit ordinal}} | {{DISPLAYTITLE: Limit ordinal}} | ||

− | A limit ordinal is an [[ordinal]] that is neither [[ | + | A limit ordinal is an [[ordinal]] that is neither [[zero|$0$]] nor a [[successor ordinal]]. Some authors classify zero as a limit ordinal. |

==Properties== | ==Properties== | ||

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All limit ordinals contain an ordinal $\alpha$ if and only if they contain $\alpha + 1$. | 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 [[cardinal number|cardinality]]. | |

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

## Revision as of 18:27, 10 January 2017

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 $\alpha$.