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  "body": "\n## Summary\n\nThe cofinality of a limit [[ordinal|0YSCSX5YZC]] is the least length of a sequence of smaller ordinals that climbs all the way up to it.\nThe [[precise definition|NTE5C260S1]] uses cofinal subsets, and it applies also to partially ordered sets.\nCofinality splits infinite cardinals into [[regular|13ZV2F8JAR]] and singular ones, and it constrains the possible values of cardinal exponentiation.\n\n## Explanations\n\n### L1 human\n\nTo reach the top of $\\omega$, the natural numbers, you need infinitely many steps $0, 1, 2, \\ldots$, so its cofinality is $\\omega$.\nThe much bigger cardinal $\\aleph_\\omega$ can also be reached in $\\omega$ steps, through $\\aleph_0, \\aleph_1, \\aleph_2, \\ldots$, so its cofinality is also $\\omega$.\nIn contrast, $\\aleph_1$ cannot be reached by any countable sequence of countable ordinals, since a countable union of countable sets is countable.\nSo the cofinality of $\\aleph_1$ is $\\aleph_1$ itself.\nCofinality measures how hard it is to climb to the top.\n",
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