Editing Measurable cardinal (section)

{{Short description|Set theory concept}}
In [[mathematics]], a '''measurable cardinal''' is a certain kind of [[large cardinal]] number. In order to define the concept, one introduces a two-valued [[measure (mathematics)|measure]] on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', it can be described as a subdivision of all of its [[subset]]s into large and small sets such that ''κ'' itself is large, ∅ and all [[singleton (mathematics)|singleton]]s {''α''} (with ''α''&nbsp;∈&nbsp;''κ'') are small, [[set complement|complement]]s of small sets are large and vice versa. The [[intersection]] of fewer than ''κ'' large sets is again large.<ref>{{harvnb|Maddy|1988}}</ref>

It turns out that [[uncountable]] cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from [[ZFC]].<ref>{{harvnb|Jech|2002}}</ref>

The concept of a measurable cardinal was introduced by [[Stanisław Ulam]] in 1930.<ref>{{harvnb|Ulam|1930}}</ref>