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  "body": "\n## Statement\n\nLet $\\mathcal{N} = (\\N, 0, S, +, \\cdot)$ be the standard model of arithmetic on the [[natural numbers|0MWKJ44V5A]], and fix a [[Gödel numbering|MFJXPAF0R0]] of the [[formulas|HYJ10AF6RT]] of the language of arithmetic.\nThere is no formula $\\mathrm{True}(x)$ with one free variable such that, for every [[sentence|DKFGQTRYZE]] $\\varphi$,\n$$\\mathcal{N} \\models \\mathrm{True}(\\ulcorner \\varphi \\urcorner) \\iff \\mathcal{N} \\models \\varphi.$$\nIn other words, the set of codes of sentences true in $\\mathcal{N}$ is not definable in $\\mathcal{N}$.\n\n## Remarks\n\nProof idea: suppose $\\mathrm{True}(x)$ exists.\nThe [[diagonal lemma|3PRW76471J]], applied in the theory of all sentences true in $\\mathcal{N}$, gives a sentence $\\lambda$ with $\\mathcal{N} \\models \\lambda \\leftrightarrow \\neg \\mathrm{True}(\\ulcorner \\lambda \\urcorner)$.\nThen $\\lambda$ is true exactly when it is not true, a contradiction.\nThis is a precise form of the liar paradox.\n\nBy contrast, for each $n$ the set of true $\\Sigma_n$ sentences is definable in $\\mathcal{N}$; only full truth is not.\nA syntactic form also holds: if $T$ is a consistent theory that contains Robinson arithmetic, there is no formula $\\mathrm{True}(x)$ with $T \\vdash \\mathrm{True}(\\ulcorner \\varphi \\urcorner) \\leftrightarrow \\varphi$ for every sentence $\\varphi$.\n\n## Explanations\n\n### L1 human\n\nArithmetic can talk about its own formulas through codes, but it cannot define which of them are true.\nAny attempt runs into the sentence \"this sentence is false\".\nSo the notion of arithmetical truth lies outside arithmetic itself.\n\n### L2 human\n\nCompare with [[Gödel's incompleteness theorems|X9WYNYE29C]]: provability in a computably axiomatized theory is definable in arithmetic, but truth is not.\nSo truth and provability differ, which gives another proof of the first incompleteness theorem.\nIn terms of the [[arithmetical hierarchy|7CVJCQKY41]], the set of true sentences is not $\\Sigma^0_n$ for any $n$.\n",
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    "name": "Tarski's undefinability theorem",
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  "markdown": "https://math2.tamnd.com/node/YRV6SYK2BY.md",
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  "name": "Tarski's undefinability theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/YRV/YRV6SYK2BY.md",
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