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  "body": "\n## Description\n\nThe von Neumann universe $V$ is the [[class|Y48FA69563]] of all sets, arranged in stages indexed by the [[ordinal numbers|0YSCSX5YZC]].\nThe stages $V_\\alpha$ are defined by [[transfinite recursion|0ZHQMQ2C1N]] on the ordinal $\\alpha$:\n\n1. $V_0 = \\varnothing$.\n2. $V_{\\alpha + 1} = \\mathcal{P}(V_\\alpha)$, the [[power set|0BC7KX51Z8]] of $V_\\alpha$.\n3. $V_\\lambda = \\bigcup_{\\beta \u003c \\lambda} V_\\beta$ for each limit ordinal $\\lambda$.\n\nThen $V = \\bigcup_{\\alpha} V_\\alpha$, the union over all ordinals.\nThe rank of a set $x$ in $V$ is the least ordinal $\\alpha$ such that $x \\subseteq V_\\alpha$.\n\n## Remarks\n\nEach $V_\\alpha$ is a transitive set, and $V_\\beta \\subseteq V_\\alpha$ when $\\beta \\le \\alpha$.\nThe first stages are $V_1 = \\{\\varnothing\\}$ and $V_2 = \\{\\varnothing, \\{\\varnothing\\}\\}$, and in general $\\abs{V_{n+1}} = 2^{\\abs{V_n}}$ for finite $n$.\nThe stage $V_\\omega$ is the set of hereditarily finite sets, and it is countable.\nOver the other axioms of ZF, the [[axiom of regularity|NFQ9FVCP31]] is equivalent to the statement that every set lies in some $V_\\alpha$.\nIf $\\kappa$ is an [[inaccessible cardinal|NW4DREQ5DQ]], then $V_\\kappa$ is a model of ZFC.\n\n## Explanations\n\n### L1 human\n\nStart with nothing, the empty set.\nAt each step, form all sets whose elements were already built; at a limit step, collect everything built so far.\nThe von Neumann universe is everything that ever appears in this process, and in ZF this is every set.\nFor example, $\\varnothing$ is an element of $V_1$, $\\{\\varnothing\\}$ is an element of $V_2$, and $\\{\\{\\varnothing\\}\\}$ is an element of $V_3$.\nThe picture is often drawn as a V-shaped cone that widens as the stages go up.\n",
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