Measurable Cardinal

A measurable cardinal is a large cardinal, an infinite cardinal number whose existence cannot be proved from the standard Zermelo-Fraenkel axioms of set theory alone, defined by the existence of a special kind of measure, a way of assigning either zero or one to every subset of the cardinal so that the assignment behaves consistently under combining even infinitely many subsets at once, a much stronger requirement than the more familiar measures used in ordinary probability and analysis. The notion was introduced in 1930 by the Polish mathematician Stanislaw Ulam, who showed as part of the same work that a measurable cardinal, if one exists at all, must already be enormously large, far exceeding the more modest large cardinals, such as inaccessible cardinals, that had already been studied by that point. Measurable cardinals are closely tied to the technique of elementary embeddings, since the existence of a measurable cardinal turns out to be equivalent to the existence of a certain structure-preserving embedding of the entire mathematical universe into a smaller inner model, a reformulation developed by the American mathematician Dana Scott that became the standard modern approach to the concept. Measurable cardinals sit well below still larger notions such as Woodin cardinals in the large cardinal hierarchy, and their assumed existence has significant consequences elsewhere in set theory, including implying that every set of real numbers definable in a certain broad sense is Lebesgue measurable.