Ulam matrix

Last updated September 20, 2024

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 ${\mathcal {F}}$ be a $\lambda$ -complete filter on $\lambda$ . An Ulam matrix is a collection of subsets $A_{\alpha \beta }$ of $\lambda$ indexed by $\alpha \in \kappa ,\beta \in \lambda$ such that

If $\beta \neq \gamma \in \lambda$ then $A_{\alpha \beta }$ and $A_{\alpha \gamma }$ are disjoint.
For each $\beta \in \lambda$ , the union over $\alpha \in \kappa$ of the sets $A_{\alpha \beta },\,\bigcup \left\{A_{\alpha \beta }:\alpha \in \kappa \right\}$ , is in the filter ${\mathcal {F}}$ .

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References

↑ 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, doi:10.4064/fm-16-1-140-150

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[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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