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  "body": "\n## Summary\n\nOrdinal analysis is the part of [[proof theory|N4EDJNMTEF]] that assigns to a formal theory its [[proof-theoretic ordinal|C72RBH0CDR]].\nIt measures the strength of a theory by the transfinite inductions that the theory can prove.\nIt started with Gentzen's consistency proof for [[Peano arithmetic|0QFT7Z76N5]], which gives the ordinal $\\varepsilon_0$.\nAn ordinal analysis usually also yields a consistency proof and a description of the provably total functions of the theory.\n\n## Explanations\n\n### L1 human\n\nTheories of arithmetic and of sets differ in strength, and ordinal analysis gives a ruler to measure them.\nThe marks on the ruler are countable ordinals, written in a finite notation system.\nTo place a theory, one shows that it proves induction up to every mark below some ordinal, and that it cannot reach that ordinal.\nFor Peano arithmetic the answer is $\\varepsilon_0$.\nStronger theories, such as parts of second-order arithmetic, get much larger ordinals.\n",
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