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  "body": "\n## Definition\n\nLet $T$ be a recursively axiomatized theory that contains elementary arithmetic.\nAn *ordinal analysis* of $T$ is the determination of its [[proof-theoretic ordinal|C19Z3B6V4P]] $\\abs{T}$.\nIt consists of three parts.\n\n1. A computable notation system: a computable well-order $\\prec$ on a set of codes in $\\N$, whose order type is an ordinal $\\alpha$.\n2. A lower bound: for each $\\beta \u003c \\alpha$, a proof in $T$ of [[transfinite induction|0ZHQMQ2C1N]] along the initial segment of $\\prec$ of order type $\\beta$.\n3. An upper bound: a proof that $T$ does not prove transfinite induction along $\\prec$ itself. More precisely, each computable well-order that $T$ proves well-founded has order type below $\\alpha$.\n\nThen $\\abs{T} = \\alpha$.\n\n## Remarks\n\nExample: Gentzen's analysis of [[Peano arithmetic|0QFT7Z76N5]] uses the Cantor normal forms below [[epsilon nought|KT50SMH3EC]] as notation system, and gives $\\abs{\\mathrm{PA}} = \\varepsilon_0$.\nThe upper bound comes from a cut-elimination argument for PA, together with transfinite induction up to $\\varepsilon_0$.\nNon-example: a proof that $T$ is consistent, by itself, is not an ordinal analysis, because it gives no ordinal.\nThe upper bound is usually proved in a weak base theory together with transfinite induction along $\\prec$.\nThis also gives a proof of the consistency of $T$ in that base theory.\n",
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