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

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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