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 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.

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 ${\mathcal {F}}$.

References

• Ulam, Stanisław (1930), "Zur Masstheorie in der allgemeinen Mengenlehre", Fundamenta Mathematicae, 16 (1): 140–150, doi:10.4064/fm-16-1-140-150