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  "body": "\n## Description\n\nLet $L$ be the [[constructible universe|6TF2JBTPPJ]].\nSuppose that there is a closed unbounded class $I$ of [[ordinals|0YSCSX5YZC]] with the following property for every uncountable cardinal $\\kappa$.\nThe set $I \\cap \\kappa$ has order type $\\kappa$ and is a set of indiscernibles for the structure $(L_\\kappa, \\in)$.\nEvery element of $L_\\kappa$ is definable in $L_\\kappa$ from finitely many elements of $I \\cap \\kappa$.\nSuch a class is unique when it exists, and its elements are the *Silver indiscernibles*.\nIn this case *zero sharp* is the set\n$$0^\\sharp = \\set{\\ulcorner \\varphi \\urcorner : L_{\\aleph_\\omega} \\models \\varphi[\\aleph_1, \\ldots, \\aleph_n]}.$$\nHere $\\varphi(x_1, \\ldots, x_n)$ runs over the formulas of set theory, and $\\ulcorner \\varphi \\urcorner$ is the Gödel number of $\\varphi$.\nOne says that $0^\\sharp$ exists when such a class $I$ exists.\n\n## Remarks\n\nSince $0^\\sharp$ is a set of natural numbers, it is often treated as a real number.\nIf $0^\\sharp$ exists, then every uncountable cardinal of $V$ is a Silver indiscernible, and hence an inaccessible cardinal in $L$.\nIf $0^\\sharp$ exists, then $0^\\sharp \\notin L$, so $V \\neq L$.\nSilver proved that $0^\\sharp$ exists if there is a [[measurable cardinal|AKBWT323RQ]].\nIf $0^\\sharp$ does not exist, then [[Jensen's covering theorem|RJE1W5NRE3]] shows that $L$ is close to $V$.\nIf ZFC is consistent, it does not prove that $0^\\sharp$ exists, since $0^\\sharp$ does not exist in $L$.\n\n## Explanations\n\n### L1 human\n\nZero sharp is a single set of natural numbers that, if it exists, describes all truths about the constructible universe.\nIt exists only if the full universe is much richer than the constructible universe.\nIts existence marks a sharp line: either the constructible universe is close to everything, or zero sharp exists and the constructible universe is very thin.\nIts existence follows from large cardinals such as a measurable cardinal, but it cannot be proved from the usual axioms.\n",
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