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m (Undo revision 320132 by PowerFlameX (talk) The set of cofinal subsets is not totally ordered, and hence I doubt the standardness of this terminology. Please add a source if you want to share your knowledge.) Tag: Undo |
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| − | The cofinality of an ordinal is the length of the shortest possible sequence leading up to it from below. For example, the cofinality of \(\omega2\) is \(\omega\) because of the sequence \(\omega,\omega+1,\omega+2,\omega+3,\cdots\). |
+ | The '''cofinality''' of an ordinal is the length of the shortest possible sequence leading up to it from below{{citation needed}}. For example, the cofinality of \(\omega2\) is \(\omega\) because of the sequence \(\omega,\omega+1,\omega+2,\omega+3,\cdots\). |
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| + | The cofinality of any countable limit ordinal is [[ω]]<ref>P. Nadathur, [http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Nadathur.pdf To Infinity And Beyond] (p.27) (accessed 2021-03-15)</ref>. The cofinality of any successor ordinal is 1<ref>P. Nadathur, [http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Nadathur.pdf To Infinity And Beyond] (p.27) (accessed 2021-03-15)</ref>. The cofinality of 0 is 0. |
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== Regular == |
== Regular == |
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Revision as of 22:35, 27 March 2021
The cofinality of an ordinal is the length of the shortest possible sequence leading up to it from below[citation needed]. For example, the cofinality of \(\omega2\) is \(\omega\) because of the sequence \(\omega,\omega+1,\omega+2,\omega+3,\cdots\).
The cofinality of any countable limit ordinal is ω[1]. The cofinality of any successor ordinal is 1[2]. The cofinality of 0 is 0.
Regular
An ordinal is called regular if its cofinality equals itself.
- ↑ P. Nadathur, To Infinity And Beyond (p.27) (accessed 2021-03-15)
- ↑ P. Nadathur, To Infinity And Beyond (p.27) (accessed 2021-03-15)