Diagonal Intersection

Diagonal intersection is a term used in mathematics, especially in set theory.

If $\displaystyle\delta$ is an ordinal number and $\displaystyle\langle X_\alpha \mid \alpha<\delta\rangle$
is a sequence of subsets of $\displaystyle\delta$, then the ''diagonal intersection'', denoted by

:$\displaystyle\Delta_ X_\alpha,$

is defined to be

:$\displaystyle\.$

That is, an ordinal $\displaystyle\beta$ is in the diagonal intersection $\displaystyle\Delta_ X_\alpha$ if and only if it is contained in the first $\displaystyle\beta$ members of the sequence. This is the same as

:\displaystyle\bigcap_ ( , \alpha\cup X_\alpha ),

where the closed interval from 0 to $\displaystyle\alpha$ is used to
avoid restricting the range of the intersection.

See also

* Club filter
* Club set
* Fodor's lemma

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.

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Ordinal numbers
Set theory