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  "body": "\n## Summary\n\nA measurable cardinal is an uncountable cardinal $\\kappa$ that carries a nonprincipal [[ultrafilter|J64Y3JC0RE]] closed under intersections of fewer than $\\kappa$ sets.\nEquivalently, $\\kappa$ is the critical point of an [[elementary embedding|CGH8C8EA9N]] $j \\colon V \\to M$ of the universe into a transitive class.\nMeasurable cardinals are [[weakly compact|1RHTVMH34G]], and their existence contradicts the [[axiom of constructibility|4QKRPRCSXH]].\nThe name comes from the measure problem of Banach and Ulam: a two-valued measure on all subsets of $\\kappa$.\n\n## Explanations\n\n### L1 human\n\nAn ultrafilter on a set decides, for each subset, whether it is \"large\" or \"small\".\nOn 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.\nOn the natural numbers one only gets this for finitely many sets at a time.\nThis robust notion of \"large\" makes $\\kappa$ so big that its existence changes the whole universe; for example, not every set can be constructible.\n",
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