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  "body": "\n## Definition\n\nAssume ZFC, so that each set has a [[cardinal number|0QKPNZKNEA]] and $2^\\kappa$ is the cardinality of the [[power set|0BC7KX51Z8]] of a set of cardinality $\\kappa$.\nThe beth numbers $\\beth_\\alpha$, for [[ordinals|0RDJK7EQYY]] $\\alpha$, are defined by [[transfinite recursion|0ZHQMQ2C1N]]:\n\n1. $\\beth_0 = \\aleph_0$, the cardinality of $\\N$, see [[aleph-null|0V2H5PGJ9B]].\n2. $\\beth_{\\alpha + 1} = 2^{\\beth_\\alpha}$.\n3. $\\beth_\\lambda = \\sup_{\\beta \u003c \\lambda} \\beth_\\beta$ for each nonzero limit ordinal $\\lambda$, the [[supremum|0BTMQ04556]] of the earlier beths.\n\n## Remarks\n\nExample: $\\beth_1 = 2^{\\aleph_0}$ is the cardinality of $\\R$, since $\\R$ is in bijection with the power set of $\\N$.\n$\\beth_2 = 2^{2^{\\aleph_0}}$ is the cardinality of the power set of $\\R$, and also of the set of all functions $\\R \\to \\R$.\nNon-example: $\\aleph_1$ is a beth number only if $\\aleph_1 = \\beth_1$, which is the continuum hypothesis and is not provable in ZFC.\nBy Cantor's theorem $\\beth_{\\alpha+1} \u003e \\beth_\\alpha$, and by induction $\\beth_\\alpha \\ge \\aleph_\\alpha$ for every $\\alpha$.\nAnother notation is $\\beth_\\alpha = \\beth_\\alpha(\\aleph_0)$; more generally $\\beth_0(\\kappa) = \\kappa$ and $\\beth_{\\alpha+1}(\\kappa) = 2^{\\beth_\\alpha(\\kappa)}$.\n",
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