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  "body": "\n## Description\n\nFor a set $X$, let $\\operatorname{Def}(X)$ be the set of all subsets of $X$ that are definable in the structure $(X, \\in)$.\nHere a definition is a first-order formula with parameters from $X$.\nDefine sets $L_\\alpha$ by recursion on the [[ordinals|0YSCSX5YZC]] $\\alpha$:\n$$L_0 = \\emptyset, \\qquad L_{\\alpha + 1} = \\operatorname{Def}(L_\\alpha), \\qquad L_\\lambda = \\bigcup_{\\alpha \u003c \\lambda} L_\\alpha \\text{ for a limit ordinal } \\lambda.$$\nThe *constructible universe* is the [[class|Y48FA69563]] $L = \\bigcup_{\\alpha} L_\\alpha$, where $\\alpha$ runs over all ordinals.\nA set is *constructible* when it lies in $L$.\n\nThe class $L$ is an [[inner model|G78W36EGMB]] of [[ZF|0HE5WDQ9FJ]].\nEach $L_\\alpha$ is transitive, so $L$ is transitive.\nThe ordinal $\\alpha$ lies in $L_{\\alpha + 1}$, because it is the set of ordinals of $L_\\alpha$, and this set is definable over $L_\\alpha$.\nSo $L$ contains every ordinal.\nThe axioms of ZF hold in $L$; the hardest case is the separation schema, which uses a reflection argument.\n\n## Remarks\n\nFor finite $n$ one has $L_n = V_n$, so $L_\\omega = V_\\omega$ is the set of hereditarily finite sets.\nAt the next level they differ: $L_{\\omega + 1}$ is countable, because there are only countably many formulas with finitely many parameters from the countable set $L_\\omega$, while $V_{\\omega + 1}$ is uncountable.\nThe class $L$ satisfies the [[axiom of constructibility|4QKRPRCSXH]] $V = L$, and it is contained in every inner model of ZF.\n[[Gödel's theorem|Q9176JRJ5M]] says that the axiom of choice and the generalized continuum hypothesis hold in $L$.\n\n## Explanations\n\n### L1 human\n\nThe constructible universe is the smallest world of sets that one can build in stages using only what can be described.\nAt each stage one keeps exactly the subsets of the sets built so far that a formula can define.\nNothing is added by arbitrary choice, so the world is very thin and very orderly.\nFor example, at the finite stages every subset can be described, so all finite sets of finite sets appear.\nAt stage $\\omega + 1$ only countably many sets of natural numbers appear, although there are uncountably many in all.\n",
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