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  "body": "\n## Definition\n\nWork in [[ZF|0HE5WDQ9FJ]], and let $L = \\bigcup_\\alpha L_\\alpha$ be the [[constructible universe|6TF2JBTPPJ]], where $\\alpha$ runs over the [[ordinals|0YSCSX5YZC]].\nThe *axiom of constructibility* is the sentence\n$$\\forall x \\, \\exists \\alpha \\, (\\alpha \\text{ is an ordinal} \\wedge x \\in L_\\alpha).$$\nIt is written $V = L$, where $V$ is the class of all sets.\n\n## Remarks\n\nThe sentence holds inside $L$ itself: the construction of $L_\\alpha$ is absolute for transitive models of ZF that contain $\\alpha$, so the class $L$ computed inside $L$ is again $L$.\nThis gives the relative consistency of $V = L$ with ZF.\nFor a non-example, take a forcing extension $L[c]$ of $L$ by a Cohen real $c$; the real $c$ is not in $L$, so $V = L$ fails in $L[c]$.\nUnder $V = L$ there is a well-order of the reals that is $\\Delta^1_2$ in the [[projective hierarchy|8DZTNVTGY2]].\nScott proved that $V = L$ implies that there is no [[measurable cardinal|AKBWT323RQ]].\n",
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