Difference between revisions of "Limit ordinal"

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{{DISPLAYTITLE: Limit ordinal}}
 
{{DISPLAYTITLE: Limit ordinal}}
  
A limit ordinal is an [[ordinal]] that is neither [[Zero|$0$]] nor a [[successor ordinal]].
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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 $\alpha + 1$.
  
All limit ordinals contain [[omega|$\omega$]].  $\omega$ is the smallest limit ordinal.
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$\omega$ 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 $\alpha$.
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$(\omega + \omega)$, 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 $\alpha$.