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  "body": "\n## Statement\n\nAssume ZFC, and assume that [[zero sharp|HCTNBZZQ0W]] does not exist.\nThen for every uncountable set $X$ of [[ordinals|0YSCSX5YZC]] there is a set $Y$ in the [[constructible universe|6TF2JBTPPJ]] $L$ such that $X \\subseteq Y$ and $\\abs{Y} = \\abs{X}$.\n\n## Remarks\n\nThe proof by Jensen uses the fine structure theory of $L$; one takes an elementary substructure that contains $X$ and analyses its transitive collapse.\nA consequence: if $0^\\sharp$ does not exist, then every singular cardinal $\\kappa$ of $V$ is singular in $L$, and $(\\kappa^+)^L = \\kappa^+$.\nIt also follows that the singular cardinals hypothesis holds when $0^\\sharp$ does not exist.\nThe hypothesis is necessary.\nIf $0^\\sharp$ exists, then $\\aleph_\\omega$ is a regular cardinal in $L$.\nThen the set $X = \\omega_1 \\cup \\set{\\aleph_n : n \u003c \\omega}$ of size $\\aleph_1$ cannot be covered.\nIndeed, a set $Y \\in L$ with $\\abs{Y} = \\aleph_1$ has order type less than $\\omega_2$.\nSo $Y \\cap \\aleph_\\omega$ has $L$-cardinality less than $\\aleph_\\omega$, and it is bounded in $\\aleph_\\omega$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that, unless zero sharp exists, the constructible universe is a good approximation of the full universe.\nEvery uncountable set of ordinals sits inside a constructible set of the same size.\nSo the constructible universe cannot miss much: it has the same large singular cardinals and the same successors of them.\nThis gives a dichotomy: either the constructible universe is close to everything, or it is very thin.\n\n### L2 human\n\nThe theorem is the model for many covering results for larger inner models, such as the Dodd–Jensen core model and the [[core model|A6ZJ8MSCAY]] in general.\nIt turns the nonexistence of a large cardinal object into strong combinatorial consequences, such as the singular cardinals hypothesis.\nRead the other way, it gives lower bounds for consistency strength.\nIf the singular cardinals hypothesis fails, then $0^\\sharp$ exists, and in fact much stronger large cardinal hypotheses hold in inner models.\n",
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