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  "body": "\n## Statement\n\nLet $S \\in L_\\alpha$, and let $T$ be any constructible subset of $S$. Then there is some $\\beta$ with $T \\in L_{\\beta+1}$, so $T = \\{x \\in L_\\beta : x \\in S \\wedge \\Phi(x, z_i)\\} = \\{x \\in S : \\Phi(x, z_i)\\}$, for some formula $\\Phi$ and some $z_i$ drawn from $L_\\beta$. By the downward Löwenheim–Skolem theorem and Mostowski collapse, there must be some transitive set $K$ containing $L_\\alpha$ and some $w_i$, and having the same first-order theory as $L_\\beta$ with the $w_i$ substituted for the $z_i$; and this $K$ will have the same cardinal as $L_\\alpha$. Since $V = L$ is true in $L_\\beta$, it is also true in $K$, so $K = L_\\gamma$ for some $\\gamma$ having the same cardinal as $\\alpha$. And $T = \\{x \\in L_\\beta : x \\in S \\wedge \\Phi(x, z_i)\\} = \\{x \\in L_\\gamma : x \\in S \\wedge \\Phi(x, w_i)\\}$ because $L_\\beta$ and $L_\\gamma$ have the same theory. So $T$ is in fact in $L_{\\gamma+1}$.\n\nSo all the constructible subsets of an infinite set $S$ have ranks with (at most) the same cardinal $\\kappa$ as the rank of $S$; it follows that if $\\delta$ is the initial ordinal for $\\kappa^+$, then $L \\cap \\mathcal{P}(S) \\subseteq L_\\delta$ serves as the \"power set\" of $S$ within $L$ Thus this \"power set\" $L \\cap \\mathcal{P}(S) \\in L_{\\delta+1}$. And this in turn means that the \"power set\" of $S$ has cardinal at most $\\vert\\delta\\vert$. Assuming $S$ itself has cardinal $\\kappa$, the \"power set\" must then have cardinal exactly $\\kappa^+$. But this is precisely the generalized continuum hypothesis relativized to $L$.\n",
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        "locator": "Constructible universe, section The generalized continuum hypothesis holds in L",
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