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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nA subset $A \\subseteq P$ is cofinal in $P$ if for each $p \\in P$ there is $a \\in A$ with $p \\le a$.\nLet $\\alpha$ be an [[ordinal|0RDJK7EQYY]], ordered by $\\in$.\nThe cofinality $\\operatorname{cf}(\\alpha)$ of $\\alpha$ is the least ordinal $\\beta$ such that $\\alpha$ has a cofinal subset of order type $\\beta$.\n\n## Remarks\n\nEquivalently, $\\operatorname{cf}(\\alpha)$ is the least $\\beta$ for which there is a [[function|0XHDTE5YMZ]] $f \\colon \\beta \\to \\alpha$ whose range is cofinal in $\\alpha$.\nExample: $\\operatorname{cf}(\\aleph_\\omega) = \\omega$.\nCheck: the set $\\set{\\aleph_n : n \u003c \\omega}$ has order type $\\omega$ and is cofinal in $\\aleph_\\omega$, and no finite subset of a limit ordinal is cofinal.\nNon-example of a cofinal subset: a finite subset $A$ of $\\omega$ is not cofinal, since $\\max A + 1$ has no element of $A$ above it.\nFor a successor ordinal $\\alpha + 1$, the set $\\set{\\alpha}$ is cofinal, so $\\operatorname{cf}(\\alpha + 1) = 1$, and $\\operatorname{cf}(0) = 0$.\nAlways $\\operatorname{cf}(\\operatorname{cf}(\\alpha)) = \\operatorname{cf}(\\alpha)$, and the cofinality of a limit ordinal is a regular cardinal.\nIn ZFC, König's theorem gives $\\operatorname{cf}(2^\\kappa) \u003e \\kappa$ for each infinite cardinal $\\kappa$.\n",
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