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 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.

Formally, a measurable cardinal is an uncountable cardinal number ${\displaystyle \kappa }$ such that there exists a ${\displaystyle \kappa }$-additive, non-trivial, 0-1-valued measure ${\displaystyle \mu }$ on the power set of ${\displaystyle \kappa }$.

Here, ${\displaystyle \kappa }$-additive means that for every ${\displaystyle \lambda <\kappa }$ and every ${\displaystyle \lambda }$-sized collection ${\displaystyle \{A_{\beta }\}_{\beta <\lambda }}$ of pairwise disjoint subsets ${\displaystyle A_{\beta }\subseteq \kappa }$, we have

Equivalently, ${\displaystyle \kappa }$ is a measurable cardinal if and only if it is an uncountable cardinal with a ${\displaystyle \kappa }$-complete, non-principal ultrafilter. This means that the intersection of any strictly less than ${\displaystyle \kappa }$-many sets in the ultrafilter is also in the ultrafilter.

Equivalently, ${\displaystyle \kappa }$ is measurable if it is the critical point of a non-trivial elementary embedding of the universe ${\displaystyle V}$ into a transitive class ${\displaystyle M}$. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since ${\displaystyle V}$ 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.

It is trivial to note that if ${\displaystyle \kappa }$ admits a non-trivial ${\displaystyle \kappa }$-additive measure, then ${\displaystyle \kappa }$ must be regular: by non-triviality and ${\displaystyle \kappa }$-additivity, any subset of cardinality less than ${\displaystyle \kappa }$ must have measure 0, and then by ${\displaystyle \kappa }$-additivity again, this means that the entire set must not be a union of fewer than ${\displaystyle \kappa }$ sets of cardinality less than ${\displaystyle \kappa }$. Finally, if ${\displaystyle \lambda <\kappa }$ then it can't be the case that ${\displaystyle \kappa \leq 2^{\lambda }}$. If this were the case, we could identify ${\displaystyle \kappa }$ with some collection of 0-1 sequences of length ${\displaystyle \kappa }$. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these ${\displaystyle \lambda }$-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the axiom of choice, we can infer that ${\displaystyle \kappa }$ is a strong limit cardinal, which completes the proof of its inaccessibility.