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  "body": "\n## Summary\n\nThe arithmetical hierarchy classifies sets of natural numbers by the number of alternating blocks of unbounded quantifiers needed to define them in first-order arithmetic.\nIts levels are the classes $\\Sigma^0_n$, $\\Pi^0_n$ and $\\Delta^0_n$ for $n \\in \\N$.\nThe first levels are familiar: $\\Delta^0_1$ sets are the [[computable sets|PPGD5KW8JC]] and $\\Sigma^0_1$ sets are the [[computably enumerable sets|TN9G24VABN]].\n[[Post's theorem|NYJX5QXX1T]] links each level to computability relative to iterated [[Turing jumps|60P2XWY5QX]].\n\n## Explanations\n\n### L1 human\n\nSome properties of a number $e$ can be checked by a search that ends when a witness is found: \"there is a $c$ such that ...\".\nThese are the $\\Sigma^0_1$ properties, for example \"program $e$ halts on input $e$\".\nOther properties need two alternating quantifiers.\nAn example is \"for every $x$ there is a $c$ such that program $e$ halts on $x$ within $c$ steps\", which says that program $e$ is total.\nEach extra alternation of \"there is\" and \"for all\" can define strictly more sets.\nThe hierarchy measures how unsolvable a problem is.\n",
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