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  "body": "\n## Definition\n\nLet $\\alpha$ and $\\beta$ be [[ordinal numbers|0RDJK7EQYY]].\nIn the clauses below, $\\lambda$ is a nonzero limit ordinal, and $\\sup$ is the [[supremum|0BTMQ04556]] of a set of ordinals, which is its union.\nThe sum, product and power of ordinals are defined by [[transfinite recursion|0ZHQMQ2C1N]] on $\\beta$:\n\n1. $\\alpha + 0 = \\alpha$, $\\alpha + (\\beta + 1) = (\\alpha + \\beta) + 1$, and $\\alpha + \\lambda = \\sup_{\\beta \u003c \\lambda} (\\alpha + \\beta)$.\n2. $\\alpha \\cdot 0 = 0$, $\\alpha \\cdot (\\beta + 1) = \\alpha \\cdot \\beta + \\alpha$, and $\\alpha \\cdot \\lambda = \\sup_{\\beta \u003c \\lambda} \\alpha \\cdot \\beta$.\n3. $\\alpha^0 = 1$, $\\alpha^{\\beta + 1} = \\alpha^\\beta \\cdot \\alpha$, and $\\alpha^\\lambda = \\sup_{\\beta \u003c \\lambda} \\alpha^\\beta$ when $\\alpha \u003e 0$, with $0^\\lambda = 0$.\n\nHere $\\beta + 1 = \\beta \\cup \\set{\\beta}$ is the successor of $\\beta$.\n\n## Remarks\n\nExample: $\\omega + 1$ is the successor of $\\omega$, and $\\omega \\cdot 2 = \\omega + \\omega$ by clauses 2 and 1.\nNon-example of commutativity: $1 + \\omega = \\sup_{n \u003c \\omega} (1 + n) = \\omega$, while $\\omega + 1 \\ne \\omega$.\nLikewise $2 \\cdot \\omega = \\sup_n 2n = \\omega$, while $\\omega \\cdot 2 = \\omega + \\omega \u003e \\omega$.\nEquivalently, $\\alpha + \\beta$ is the order type of $\\alpha$ followed by $\\beta$.\nAnd $\\alpha \\cdot \\beta$ is the order type of the pairs $(b, a)$ with $b \u003c \\beta$ and $a \u003c \\alpha$, compared first by $b$ and then by $a$.\nAddition and multiplication are associative, and $\\alpha \\cdot (\\beta + \\gamma) = \\alpha \\cdot \\beta + \\alpha \\cdot \\gamma$, but the other distributive law fails.\nOn finite ordinals the operations agree with those of the natural numbers.\nOrdinal exponentiation differs from cardinal exponentiation: $2^\\omega = \\omega$ as ordinals, while $2^{\\aleph_0} \u003e \\aleph_0$ as cardinals.\n",
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