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  "body": "\n## Definition\n\nWork in [[ZF|0HE5WDQ9FJ]], and let $M$ be a [[class|Y48FA69563]].\nThe class $M$ is an *inner model* if:\n\n1. $M$ is transitive: if $x \\in M$ and $y \\in x$, then $y \\in M$.\n2. $M$ contains every [[ordinal|0YSCSX5YZC]].\n3. $(M, \\in)$ satisfies each axiom of ZF, that is, the relativization $\\varphi^M$ holds for each axiom $\\varphi$ of ZF.\n\n## Remarks\n\nThe universe $V$ and the constructible universe $L$ are inner models.\nFor $L$, clause 1 holds because each level $L_\\alpha$ is transitive, and clause 2 holds because $\\alpha \\in L_{\\alpha + 1}$.\nThe set $L_{\\omega_1}$ is a non-example: it is transitive, but it fails clause 2, since $\\omega_1 \\notin L_{\\omega_1}$.\nClause 3 is a schema, one statement for each axiom, because satisfaction in a proper class is not expressed by a single formula.\nA transitive proper class that satisfies ZF contains all ordinals, so some authors drop clause 2 and ask instead that $M$ be a proper class.\nSome authors ask for the axioms of ZFC; the name \"inner model of ZFC\" makes this choice explicit.\n",
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