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  "body": "\n## Definition\n\nLet $\\kappa$ be an infinite [[cardinal|0QKPNZKNEA]], identified with the least ordinal of its cardinality.\nThe cardinal $\\kappa$ is regular if its [[cofinality|NTE5C260S1]] is $\\kappa$, that is, $\\operatorname{cf}(\\kappa) = \\kappa$.\nIt is singular if $\\operatorname{cf}(\\kappa) \u003c \\kappa$.\n\n## Remarks\n\nExample: in ZFC, every infinite successor cardinal $\\kappa^+$ is regular.\nCheck: if $\\kappa^+$ were the union of fewer than $\\kappa^+$ sets each of size less than $\\kappa^+$, it would have size at most $\\kappa \\cdot \\kappa = \\kappa$, using the axiom of choice.\nNon-example: $\\aleph_\\omega$ is singular, since $\\set{\\aleph_n : n \u003c \\omega}$ is a cofinal subset of order type $\\omega$.\nEquivalently, in ZFC, $\\kappa$ is regular if and only if it is not the union of fewer than $\\kappa$ sets each of cardinality less than $\\kappa$.\n$\\aleph_0$ is regular, and a limit cardinal $\\aleph_\\lambda$ with $\\lambda$ a limit ordinal has cofinality $\\operatorname{cf}(\\lambda)$.\nA regular uncountable limit cardinal is called weakly inaccessible; its existence cannot be proved in ZFC, if ZFC is consistent.\n",
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