Ulam matrix

In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Ulam (1930) in his 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 F be a λ-complete filter on λ. An Ulam matrix is a collection of subsets A_αβ of λ indexed by α in κ, β in λ such that

• If β is not γ then A_αβ and A_αγ are disjoint.
• For each β the union of the sets A_αβ is in the filter F.

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

• Ulam, Stanisław (1930), "Zur Masstheorie in der allgemeinen Mengenlehre", Fundamenta Mathematicae, 16 (1): 140–150