---
tag: AKBWT323RQ
kind: concept
name: Measurable cardinal
msc: ["03E55"]
specializes: [1RHTVMH34G]
sources:
  - url: https://en.wikipedia.org/wiki/Measurable_cardinal
same_as:
  wikidata: Q925445
  wikipedia: Measurable cardinal
---

## Summary

A measurable cardinal is an uncountable cardinal $\kappa$ that carries a nonprincipal [[ultrafilter|J64Y3JC0RE]] closed under intersections of fewer than $\kappa$ sets.
Equivalently, $\kappa$ is the critical point of an [[elementary embedding|CGH8C8EA9N]] $j \colon V \to M$ of the universe into a transitive class.
Measurable cardinals are [[weakly compact|1RHTVMH34G]], and their existence contradicts the [[axiom of constructibility|4QKRPRCSXH]].
The name comes from the measure problem of Banach and Ulam: a two-valued measure on all subsets of $\kappa$.

## Explanations

### L1 human

An ultrafilter on a set decides, for each subset, whether it is "large" or "small".
On a measurable cardinal there is such a decision that is very robust: the intersection of fewer than $\kappa$ large sets is still large, and single points are small.
On the natural numbers one only gets this for finitely many sets at a time.
This robust notion of "large" makes $\kappa$ so big that its existence changes the whole universe; for example, not every set can be constructible.