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.^[1]

Definition

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

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