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  "body": "\n## Definition\n\nLet $\\kappa$ be a [[cardinal|0EBACQFAZW]].\nThe cardinal $\\kappa$ is *(strongly) inaccessible* if:\n\n1. $\\kappa$ is [[uncountable|0W7FBYS1F1]].\n2. $\\kappa$ is [[regular|13ZV2F8JAR]]: $\\kappa$ is not the union of fewer than $\\kappa$ sets each of size less than $\\kappa$.\n3. $\\kappa$ is a strong limit: $2^\\lambda \u003c \\kappa$ for every cardinal $\\lambda \u003c \\kappa$, where $2^\\lambda$ is the cardinality of the [[power set|00E0YZCERH]] of a set of size $\\lambda$.\n\nThe cardinal $\\kappa$ is *weakly inaccessible* if it satisfies clauses 1 and 2 and is a limit cardinal: $\\lambda^+ \u003c \\kappa$ for every cardinal $\\lambda \u003c \\kappa$.\n\n## Remarks\n\nThe cardinal $\\aleph_0$ satisfies clauses 2 and 3, since a union of finitely many finite sets is finite and $2^n$ is finite; it fails only clause 1.\nThe cardinal $\\aleph_1$ is regular but not inaccessible: it fails clause 3, since $2^{\\aleph_0} \\ge \\aleph_1$.\nEvery inaccessible cardinal is weakly inaccessible, and under the generalized continuum hypothesis the two notions agree.\nIf $\\kappa$ is inaccessible, then $V_\\kappa$, the set of sets of rank less than $\\kappa$, is a model of ZFC.\nBy Gödel's second incompleteness theorem, ZFC therefore cannot prove that an inaccessible cardinal exists, if ZFC is consistent.\nIn Mathlib the notion is `Cardinal.IsInaccessible`.\n",
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