Editing Measurable cardinal (section)

== Definition ==
Formally, a measurable cardinal is an uncountable [[cardinal number]] ''κ'' such that there exists a ''κ''-additive, non-trivial, 0-1-valued [[measure (mathematics)|measure]] ''μ'' on the [[power set]] of&nbsp;''κ.''

Here, κ-additive means: For every ''λ''&nbsp;<&nbsp;''κ'' and every ''λ''-sized set {''A''<sub>''β''</sub>}<sub>''β''<''λ''</sub> of pairwise disjoint subsets ''A''<sub>''β''</sub>&nbsp;⊆&nbsp;''κ,'' we have
:''μ''(⋃<sub>''β''<''λ''</sub>&nbsp;''A''<sub>''β''</sub>)&nbsp;= Σ<sub>''β''<''λ''</sub>&nbsp;''μ''(''A''<sub>''β''</sub>).

Equivalently, ''κ'' is a measurable cardinal if and only if it is an uncountable cardinal with a ''κ''-complete, non-principal [[ultrafilter]].  This means that the intersection of any ''strictly less than'' ''κ''-many sets in the ultrafilter, is also in the ultrafilter.

Equivalently, ''κ'' is measurable means that it is the [[critical point (set theory)|critical point]] of a non-trivial [[elementary embedding]] of the [[universe (set theory)|universe]] ''V'' into a [[transitive class]] ''M''. This equivalence is due to [[Jerome Keisler]] and [[Dana Scott]], and uses the [[ultraproduct|ultrapower]] construction from [[model theory]]. Since ''V'' is a [[proper class]], a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called [[Scott's trick]].