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  "body": "\n## Statement\n\nThe utility of the *β* function comes from the following result (Gödel 1931, *Hilfssatz* 1, p. 192-193), which is the purpose of the *β* function in Gödel’s incompleteness proof. This result is explained in more detail than in Gödel’s proof in (Mendelson 1997:186) and (Smith 2013:113-118).\n\n***β* function lemma**. For any sequence of natural numbers (*k*\u003csub\u003e0\u003c/sub\u003e, *k*\u003csub\u003e1\u003c/sub\u003e, ..., *k*\u003csub\u003e*n*\u003c/sub\u003e), there are natural numbers *b* and *c* such that, for every natural number *0* ≤ *i* ≤ *n*, *β*(*b*, *c*, *i*) = *k*\u003csub\u003e*i*\u003c/sub\u003e.\n\nAs an example, the sequence (2, 0, 2, 1) can be encoded by *b* = 3412752 and *c* = 24, since\n\n$$\\begin{array}{lllll} \\operatorname{rem}(3412752, (1+(0+1) \\times 24)) \u0026 = \u0026 \\operatorname{rem}(3412752, 25) \u0026 = \u0026 2, \\\\ \\operatorname{rem}(3412752, (1+(1+1) \\times 24)) \u0026 = \u0026 \\operatorname{rem}(3412752, 49) \u0026 = \u0026 0, \\\\ \\operatorname{rem}(3412752, (1+(2+1) \\times 24)) \u0026 = \u0026 \\operatorname{rem}(3412752, 73) \u0026 = \u0026 2, \\text{ and} \\\\ \\operatorname{rem}(3412752, (1+(3+1) \\times 24)) \u0026 = \u0026 \\operatorname{rem}(3412752, 97) \u0026 = \u0026 1. \\\\ \\end{array}$$\n\nThe proof of the *β* function lemma makes use of the [[Chinese remainder theorem|0FNN9B4W64]].\n",
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