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  "body": "\n## Statement\n\nLet PA be first-order [[Peano arithmetic|0QFT7Z76N5]], and let PRA be primitive recursive arithmetic.\nFix a standard [[primitive recursive|5NVA2JRHPV]] notation system for the ordinals below [[epsilon nought|KT50SMH3EC]], ordered by a primitive recursive relation $\\prec$.\nThen the statement that PA is consistent is provable in PRA extended by the rule of [[transfinite induction|0ZHQMQ2C1N]] along $\\prec$ for quantifier-free formulas.\n\n## Remarks\n\nGentzen published the proof in 1936, and a second proof in 1938.\nIn the second proof, one assigns to each derivation of a sequent in a sequent calculus for PA an ordinal below $\\varepsilon_0$.\nOne then gives a primitive recursive reduction step that turns any derivation of the empty sequent into another derivation of the empty sequent with a smaller ordinal.\nTransfinite induction along $\\prec$ shows that no derivation of the empty sequent exists.\n\nBy [[Gödel's second incompleteness theorem|X9WYNYE29C]], PA does not prove its own consistency.\nSo PA does not prove the transfinite induction used here, although every part of the proof other than this induction is finitary.\nGentzen later showed directly that PA proves transfinite induction along $\\prec$ up to each ordinal below $\\varepsilon_0$, but not up to $\\varepsilon_0$.\nThus $\\varepsilon_0$ is exactly the length of induction that PA cannot reach.\n\n## Explanations\n\n### L1 human\n\nGödel showed that arithmetic cannot prove its own freedom from contradiction.\nGentzen found what extra principle is enough to prove it.\nThe principle is induction along an ordering of length $\\varepsilon_0$, which is easy to describe with finite symbols.\nEvery other step of the proof uses only simple, finite reasoning about formulas.\n\n### L2 human\n\nThe proof assigns ordinals to proofs and shows that reduction steps make the ordinal go down.\nSince the ordinals below $\\varepsilon_0$ are well-ordered, the reduction must stop, and so no proof of a contradiction can exist.\nThis is the first ordinal analysis: it shows that $\\varepsilon_0$ is the proof-theoretic ordinal of PA.\nThe method extends to stronger theories, which get larger ordinals.\n",
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