In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.[1]

Definition

Suppose that κ and λ are cardinal numbers, and let be a -complete filter on . An Ulam matrix is a collection of subsets of indexed by such that

  • If then and are disjoint.
  • For each , the union over of the sets , is in the filter .

References

  1. Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York: Springer-Verlag, p. 131, ISBN 978-3-540-44085-7, Zbl 1007.03002
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