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

$\displaystyle\Delta_{\alpha<\delta} X_\alpha,$

is defined to be

$\displaystyle\{\beta<\delta\mid\beta\in \bigcap_{\alpha<\beta} X_\alpha\}.$

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

$\displaystyle\bigcap_{\alpha < \delta} ( [0, \alpha] \cup X_\alpha ),$

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

See also

• 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.

This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.