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  "body": "\n## Summary\n\nThe analytical hierarchy classifies sets of natural numbers by the quantifiers over sets of numbers needed to define them in [[second-order arithmetic|DC60FYCSPM]].\nIts levels are the classes $\\Sigma^1_n$, $\\Pi^1_n$ and $\\Delta^1_n$.\nIt extends the [[arithmetical hierarchy|7CVJCQKY41]], whose sets all sit in $\\Delta^1_1$.\nIt is the effective, or lightface, counterpart of the [[projective hierarchy|8DZTNVTGY2]] of descriptive set theory.\n\n## Explanations\n\n### L1 human\n\nArithmetic can talk about numbers; second-order arithmetic can also talk about infinite sets of numbers.\nA definition that says \"there is a set $X$ such that ...\" or \"for every set $X$ ...\" can describe far more complicated sets.\nFor example, \"this computable ordering of numbers has no infinite descending sequence\" says something about all infinite sequences.\nThe analytical hierarchy counts how many alternations of such set quantifiers a definition needs.\n",
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