Ulam matrix explained

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

l F be a λ -complete filter on λ . An Ulam matrix is a collection of subsets A _{\alpha} of λ indexed by \alpha \in \kappa, \beta \in λ such that

• If

\beta \ne \gamma \in λ then A _{\alpha} and A _{\alpha} are disjoint.

• For each

\beta \in λ , the union over \alpha \in \kappa of the sets A _{\alpha} , c up\left\{A _{\alpha} :\alpha \in \kappa\right\} , is in the filter l F