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  "body": "\n## Definition\n\nLet $\\Sigma$ be a countable alphabet whose symbols are listed effectively, and let $\\Sigma^*$ be the set of finite strings over $\\Sigma$.\nA *Gödel numbering* of $\\Sigma^*$ is a map $\\ulcorner \\cdot \\urcorner \\colon \\Sigma^* \\to \\N$ such that:\n\n1. $\\ulcorner \\cdot \\urcorner$ is injective;\n2. $\\ulcorner \\cdot \\urcorner$ is computable;\n3. the set of codes $\\set{\\ulcorner w \\urcorner : w \\in \\Sigma^*}$ is a [[computable set|PPGD5KW8JC]], and the inverse map from codes to strings is computable.\n\nFor a formal language, a Gödel numbering of its [[formulas|HYJ10AF6RT]] or proofs is the restriction of a Gödel numbering of its strings of symbols.\n\n## Remarks\n\nExample: give the symbols injective codes $c(s) \\ge 1$, and let $p_1 = 2, p_2 = 3, p_3 = 5, \\ldots$ be the primes.\nSet $\\ulcorner s_1 s_2 \\cdots s_n \\urcorner = p_1^{c(s_1)} p_2^{c(s_2)} \\cdots p_n^{c(s_n)}$.\nBy unique factorization the map is injective, and the exponents of a number can be found by trial division, so all three clauses hold.\n\nNon-example: the map that sends each string to its length is computable but fails clause 1, since different strings of the same length get the same number.\n\nThe code of a formula $\\varphi$ is written $\\ulcorner \\varphi \\urcorner$, and in a theory of arithmetic its numeral is also written $\\ulcorner \\varphi \\urcorner$.\nAny two Gödel numberings are related by a computable bijection between their sets of codes.\n",
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