Fodor's lemma: Difference between revisions

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'''Diagonal intersection''' is a term used in [[mathematics]], especially in [[set theory]].
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If <math>\displaystyle\delta</math> is an [[ordinal number]] and <math>\displaystyle\langle X_\alpha \mid \alpha<\delta\rangle</math>
is a [[sequence]] of subsets of <math>\displaystyle\delta</math>, then the ''diagonal intersection'', denoted by
 
:<math>\displaystyle\Delta_{\alpha<\delta} X_\alpha,</math>
 
is defined to be
 
:<math>\displaystyle\{\beta<\delta\mid\beta\in \bigcap_{\alpha<\beta} X_\alpha\}.</math>
 
That is, an [[ordinal number|ordinal]] <math>\displaystyle\beta</math> is in the diagonal intersection <math>\displaystyle\Delta_{\alpha<\delta} X_\alpha</math> if and only if it is contained in the first <math>\displaystyle\beta</math> members of the sequence. This is the same as
 
:<math>\displaystyle\bigcap_{\alpha < \delta} ( [0, \alpha] \cup X_\alpha ),</math>
 
where the closed interval from 0 to <math>\displaystyle\alpha</math> is used to
avoid restricting the range of the intersection.
 
==See also==
*[[Fodor's lemma]]
*[[Club set]]
*[[Club filter]]
 
==References==
* Thomas Jech, ''Set Theory'', The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003, page 92.
* Akihiro Kanamori, ''The Higher Infinite'', Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.
 
{{PlanetMath attribution|id=3233|title=diagonal intersection}}
 
[[Category:Set theory]]
[[Category:Ordinal numbers]]
 
{{mathlogic-stub}}

Latest revision as of 14:46, 9 December 2014

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