Diagonal intersection

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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[edit]

References[edit]

  • 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.