Ulam matrix

Updated on Nov 16, 2023

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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.

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

Ulam matrix Wikipedia

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