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  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nFor an ordinal $\\gamma$, let $V_\\gamma$ be the set of sets of rank less than $\\gamma$, as in the [[von Neumann universe|3SZADQXVZJ]].\n\n1. For every ordinal $\\delta$, there is no [[elementary embedding|CGH8C8EA9N]] $j \\colon V_{\\delta + 2} \\to V_{\\delta + 2}$ other than the identity.\n2. There is no elementary embedding $j \\colon V \\to V$ other than the identity, where $j$ is a class in the sense of von Neumann–Bernays–Gödel set theory.\n\n## Remarks\n\nLet $j$ be a nontrivial embedding with critical point $\\kappa_0$, let $\\kappa_{n+1} = j(\\kappa_n)$, and let $\\lambda = \\sup_n \\kappa_n$, so that $j(\\lambda) = \\lambda$.\nThe proof uses the axiom of choice to find a function on the countable subsets of $\\lambda$ that $j$ cannot respect.\nOne such tool is an $\\omega$-Jónsson function, from a theorem of Erdős and Hajnal.\nPart 2 follows from part 1 applied to $\\delta = \\lambda$, after restricting $j$ to $V_{\\lambda + 2}$.\nThe theorem sets an upper limit to large cardinal axioms built from embeddings: the axioms just below it ask for elementary embeddings of $V_\\lambda$ or $V_{\\lambda + 1}$ into itself.\nIt is an open problem whether part 2 holds without the axiom of choice.\n\n## Explanations\n\n### L1 human\n\nMany large cardinals are defined by maps from the universe of sets into a smaller copy of itself that keep all truths.\nOne might hope for the strongest possible version, a map from the universe into itself that is not the identity.\nKunen proved that, with the axiom of choice, such a map cannot exist.\nSo there is a ceiling for this way of building large cardinals.\n\n### L2 human\n\nThe embeddings $j \\colon V \\to M$ of measurable and supercompact cardinals become stronger as $M$ gets closer to $V$.\nKunen's theorem says that $M$ cannot be all of $V$, and more precisely that $M$ cannot contain the set $j[\\lambda]$ for the $\\lambda$ above.\nThe strongest large cardinal axioms in use, such as elementary embeddings $j \\colon V_{\\lambda + 1} \\to V_{\\lambda + 1}$, sit just below the limit given by part 1.\n",
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