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  "body": "\n## Summary\n\nA supercompact cardinal is a cardinal $\\kappa$ that is the critical point of many [[elementary embeddings|CGH8C8EA9N]] $j \\colon V \\to M$.\nFor each length, one of them has a target $M$ closed under sequences of that length.\nIt strengthens the embedding form of [[measurability|AKBWT323RQ]], which asks only for closure under sequences of length $\\kappa$.\nSupercompact cardinals are [[Woodin cardinals|CMC2C1D53W]] and lie high in the hierarchy of [[large cardinals|X1FG7PSCTY]].\nThey are used to prove the consistency of strong axioms such as the [[proper forcing axiom|35XTXNEWQ9]].\n\n## Explanations\n\n### L1 human\n\nA measurable cardinal gives an embedding of the universe into a copy of itself that keeps sequences of length $\\kappa$.\nA supercompact cardinal gives such embeddings that keep sequences of every length, as long as one fixes the length in advance.\nSo the copy looks more and more like the whole universe.\nThis makes supercompact cardinals very strong, and many consistency results use them.\n",
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