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  "body": "\n## Statement\n\nLet $T$ be a consistent first-order [[theory|92SWXST1ZM]] in the language of arithmetic whose set of axioms is [[computably enumerable|TN9G24VABN]].\n\n1. (First incompleteness theorem.) If $T$ contains [[Robinson arithmetic|3K1AKGXF8A]] $\\mathsf{Q}$, then there is a [[sentence|DKFGQTRYZE]] $G$ such that $T \\nvdash G$ and $T \\nvdash \\neg G$. So $T$ is not [[complete|655K8Y7VS8]].\n2. (Second incompleteness theorem.) If $T$ contains the [[Peano axioms|0QFT7Z76N5]] $\\mathsf{PA}$, then $T \\nvdash \\mathrm{Con}(T)$. Here $\\mathrm{Prov}_T$ is the standard provability predicate of $T$ for a [[Gödel numbering|MFJXPAF0R0]], and $\\mathrm{Con}(T)$ is $\\neg \\mathrm{Prov}_T(\\ulcorner 0 = S0 \\urcorner)$.\n\n## Remarks\n\nProof idea for 1: by the [[diagonal lemma|3PRW76471J]] there is a sentence $G$ with $T \\vdash G \\leftrightarrow \\neg \\mathrm{Prov}_T(\\ulcorner G \\urcorner)$.\nIf $T \\vdash G$, then $T \\vdash \\mathrm{Prov}_T(\\ulcorner G \\urcorner)$, and $T$ is inconsistent.\nGödel's argument that $T \\nvdash \\neg G$ used $\\omega$-consistency; Rosser's variant of the sentence needs only consistency, which gives the form stated here.\nProof idea for 2: the argument for \"if $T$ is consistent then $T \\nvdash G$\" can be carried out inside $T$, so $T \\vdash \\mathrm{Con}(T) \\to G$; thus $T \\vdash \\mathrm{Con}(T)$ would give $T \\vdash G$.\n\nThe hypotheses matter.\nThe theory of the true sentences of $\\N$ is complete and consistent, but its axioms are not computably enumerable.\n[[Presburger arithmetic|B66JYY3K58]], with addition but no multiplication, is consistent, computably axiomatized and complete.\n\n## Explanations\n\n### L1 human\n\nThe first theorem says that any reasonable list of axioms for arithmetic leaves some statements undecided: neither they nor their negations can be proved.\nThe second says that such a system cannot prove its own consistency.\nSo no single computer-checkable set of axioms captures all truths about the natural numbers.\n\n### L2 human\n\nThe proofs turn statements about proofs into statements about numbers by a [[Gödel numbering|MFJXPAF0R0]], and then use the [[diagonal lemma|3PRW76471J]] to build a self-referential sentence.\nThe first theorem implies that every consistent computably axiomatized extension of $\\mathsf{Q}$ is incomplete, and with more work that such theories are undecidable.\nThe second theorem ended the hope of Hilbert's program to prove the consistency of mathematics by finitary means within mathematics itself.\nConcrete true statements unprovable in $\\mathsf{PA}$ include [[Goodstein's theorem|XQMNJF241J]] and the [[Paris–Harrington theorem|509ZWA0R26]].\n",
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