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  "body": "\n## Definition\n\nIdentify each [[cardinal number|0QKPNZKNEA]] of a well-orderable set with the least [[ordinal|0RDJK7EQYY]] of that cardinality.\nFor an infinite cardinal $\\kappa$, let $\\kappa^+$ be the least cardinal greater than $\\kappa$.\nThe aleph numbers $\\aleph_\\alpha$, for ordinals $\\alpha$, are defined by [[transfinite recursion|0ZHQMQ2C1N]]:\n\n1. $\\aleph_0 = \\omega$, the set of natural numbers.\n2. $\\aleph_{\\alpha + 1} = (\\aleph_\\alpha)^+$.\n3. $\\aleph_\\lambda = \\sup_{\\beta \u003c \\lambda} \\aleph_\\beta$ for each nonzero limit ordinal $\\lambda$, the [[supremum|0BTMQ04556]] of the earlier alephs.\n\n## Remarks\n\nExample: $\\aleph_1$ is the set of countable ordinals, which is the least uncountable ordinal $\\omega_1$.\nCheck: each countable ordinal is smaller than $\\omega_1$, and $\\omega_1$ itself is uncountable, so it is the least cardinal above $\\aleph_0$.\nNon-example: $\\omega + 1$ is an infinite ordinal but not an aleph, since it has the same cardinality as the smaller ordinal $\\omega$.\nThe cardinal $\\kappa^+$ exists in ZF by Hartogs' theorem, without the axiom of choice.\nThe map $\\alpha \\mapsto \\aleph_\\alpha$ is a normal function, and every infinite well-orderable cardinal is $\\aleph_\\alpha$ for exactly one $\\alpha$.\nThe ordinal $\\aleph_\\alpha$ is also written $\\omega_\\alpha$ when it is viewed as an ordinal.\n",
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