Ulam matrix

Term in mathematical set theory

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 F {\displaystyle {\mathcal {F}}} be a λ {\displaystyle \lambda } -complete filter on λ {\displaystyle \lambda } . An Ulam matrix is a collection of subsets A α β {\displaystyle A_{\alpha \beta }} of λ {\displaystyle \lambda } indexed by α ∈ κ , β ∈ λ {\displaystyle \alpha \in \kappa ,\beta \in \lambda } such that

  • If β ≠ γ ∈ λ {\displaystyle \beta \neq \gamma \in \lambda } then A α β {\displaystyle A_{\alpha \beta }} and A α γ {\displaystyle A_{\alpha \gamma }} are disjoint.
  • For each β ∈ λ {\displaystyle \beta \in \lambda } , the union over α ∈ κ {\displaystyle \alpha \in \kappa } of the sets A α β , ⋃ { A α β : α ∈ κ } {\displaystyle A_{\alpha \beta },\,\bigcup \left\{A_{\alpha \beta }:\alpha \in \kappa \right\}} , is in the filter F {\displaystyle {\mathcal {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


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