Measurable cardinal

In mathematics, specifically in set theory, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal ${\displaystyle \kappa }$, or more generally on any set. For a cardinal ${\displaystyle \kappa }$, it can be described as a subdivision of all of its subsets into large and small sets such that ${\displaystyle \kappa }$ itself is large, the empty set and all singletons ${\displaystyle \{\alpha \}}$ with ${\displaystyle \alpha \in \kappa }$ are small, complements of small sets are large and vice versa. The intersection of fewer than ${\displaystyle \kappa }$ large sets is again large.

It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC.

The concept of a measurable cardinal was introduced by Stanisław Ulam in 1930.