# Measurable cardinal

## Definition

A cardinal $\kappa$ is *measurable* if it is uncountable and there is a $\kappa$-complete non-principal ultrafilter on $\kappa$.

Measurable cardinals were first defined by Stanislaw Ulam.

## Elementary embedding characterization

A cardinal $\kappa$ is measurable if and only if there is a transitive class $M$ and an elementary embedding $j : V \to M$ with critical point $\kappa$, meaning that $\kappa$ is the least ordinal with $j(\kappa) \ne \kappa$.

This characterization was used by Dana Scott to prove that the existence of measurable cardinals is inconsistent with $V = L$: if $\kappa$ is the least measurable cardinal of $V$ then $j(\kappa)$ is the least measurable cardinal of $M$, but if $V = L$ then $M = L$ also, so $j(\kappa) = \kappa$, a contradiction.

## Related cardinal notions

Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of strong compactness and supercompactness respectively.)

Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal (also defined by Ulam.)

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