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  "body": "\n## Definition\n\nLet $\\kappa$ be a [[cardinal|0EBACQFAZW]].\nFor an ordinal $\\gamma$, let $V_\\gamma$ be the set of sets of rank less than $\\gamma$ in the [[von Neumann universe|3SZADQXVZJ]].\nThe cardinal $\\kappa$ is a *Woodin cardinal* if for every function $f \\colon \\kappa \\to \\kappa$ there is an ordinal $\\alpha \u003c \\kappa$ such that:\n\n1. $f(\\beta) \u003c \\alpha$ for every $\\beta \u003c \\alpha$.\n2. There is an [[elementary embedding|CGH8C8EA9N]] $j \\colon V \\to M$, with $M$ a transitive class, whose critical point is $\\alpha$.\n3. $V_{j(f)(\\alpha)} \\subseteq M$.\n\n## Remarks\n\nIf $\\kappa$ is a Woodin cardinal, then $\\kappa$ is Mahlo, and there are $\\kappa$ measurable cardinals below $\\kappa$.\nEvery supercompact cardinal is a Woodin cardinal.\nA non-example: a measurable cardinal $\\kappa$ that is the least measurable cardinal is not Woodin, since a Woodin cardinal has measurable cardinals below it.\nMartin and Steel proved that if there are $n$ Woodin cardinals with a measurable cardinal above them all, then every $\\Pi^1_{n+1}$ set of reals is determined.\n",
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