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  "body": "\n## Statement\n\nLet $T$ be a [[theory|92SWXST1ZM]] in the language of arithmetic that contains the [[Peano axioms|0QFT7Z76N5]] and whose set of axioms is [[computably enumerable|TN9G24VABN]].\nLet $\\mathrm{Prov}_T(x)$ be the standard provability predicate of $T$ for a [[Gödel numbering|MFJXPAF0R0]].\nFor every [[sentence|DKFGQTRYZE]] $\\varphi$, if $T \\vdash \\mathrm{Prov}_T(\\ulcorner \\varphi \\urcorner) \\to \\varphi$, then $T \\vdash \\varphi$.\n\n## Remarks\n\nProof idea: by the [[diagonal lemma|3PRW76471J]] there is $\\psi$ with $T \\vdash \\psi \\leftrightarrow (\\mathrm{Prov}_T(\\ulcorner \\psi \\urcorner) \\to \\varphi)$.\nThe derivability conditions of $\\mathrm{Prov}_T$, which $T$ proves, give $T \\vdash \\mathrm{Prov}_T(\\ulcorner \\psi \\urcorner) \\to \\mathrm{Prov}_T(\\ulcorner \\varphi \\urcorner)$.\nWith the hypothesis this yields $T \\vdash \\mathrm{Prov}_T(\\ulcorner \\psi \\urcorner) \\to \\varphi$, hence $T \\vdash \\psi$, hence $T \\vdash \\mathrm{Prov}_T(\\ulcorner \\psi \\urcorner)$, and finally $T \\vdash \\varphi$.\n\nExample: take $\\varphi$ to be $0 = S0$.\nIf $T$ proved $\\mathrm{Con}(T)$, that is, $\\neg \\mathrm{Prov}_T(\\ulcorner 0 = S0 \\urcorner)$, then $T$ would prove $\\mathrm{Prov}_T(\\ulcorner 0 = S0 \\urcorner) \\to 0 = S0$, and so $T \\vdash 0 = S0$.\nSo a consistent $T$ does not prove its own consistency, which is the second of [[Gödel's incompleteness theorems|X9WYNYE29C]].\nAnother consequence: the sentence that says \"I am provable\" is provable, which answers a question of Henkin.\n\n## Explanations\n\n### L1 human\n\nA theory can never prove \"if I prove $\\varphi$, then $\\varphi$ is true\" unless it already proves $\\varphi$ itself.\nSo a theory cannot vouch for its own soundness, even one sentence at a time.\n\n### L2 human\n\nLöb's theorem generalizes the second incompleteness theorem, as the example shows.\nIts formalized version, $\\square(\\square p \\to p) \\to \\square p$, is the characteristic axiom of the modal [[provability logic|WJK0YA4GP2]] $\\mathsf{GL}$, read with $\\square$ as \"it is provable in $T$ that\".\n",
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    "name": "Löb's theorem",
    "role": "theorem",
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      "wikidata": "Q204884",
      "wikipedia": "Löb's theorem"
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    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Löb's_theorem"
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    "tag": "3HD30YG3DG",
    "uses": [
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  "link": "https://math2.tamnd.com/node/3HD30YG3DG/",
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  "markdown": "https://math2.tamnd.com/node/3HD30YG3DG.md",
  "msc": [
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  "name": "Löb's theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/3HD/3HD30YG3DG.md",
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  "tag": "3HD30YG3DG",
  "uri": "m2://3HD30YG3DG",
  "url": "https://math2.tamnd.com/wiki/lobs-theorem/",
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    "tag": "3HD30YG3DG",
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