Measurable cardinal

Summary

A measurable cardinal is an uncountable cardinal $\kappa$ that carries a nonprincipal ultrafilter closed under intersections of fewer than $\kappa$ sets. Equivalently, $\kappa$ is the critical point of an elementary embedding $j \colon V \to M$ of the universe into a transitive class. Measurable cardinals are weakly compact, and their existence contradicts the axiom of constructibility. The name comes from the measure problem of Banach and Ulam: a two-valued measure on all subsets of $\kappa$.