{
  "aliases": [
    "recursive-definition",
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  "body": "\n## Summary\n\nA definition by recursion defines a function on the [[natural numbers|0MWKJ44V5A]] in two parts.\nIt gives the value at $0$, and a rule that gives the value at $n + 1$ from the value at $n$.\nThe recursion theorem says that such a definition always gives exactly one function.\nPrecisely, let $X$ be a set, let $a$ be an element of $X$ and let $g \\colon \\mathbb{N} \\times X \\to X$ be a function.\nThen there is exactly one function $f \\colon \\mathbb{N} \\to X$ with $f(0) = a$ and $f(n + 1) = g(n, f(n))$ for each $n$ in $\\mathbb{N}$.\n\n## Remarks\n\nThe factorial is an example: $0! = 1$ and $(n + 1)! = (n + 1) \\cdot n!$.\nThe [[addition|0W5DSDFQ4G]] and the [[multiplication|0JRC32D6K2]] of natural numbers have definitions by recursion.\nThe proof of the recursion theorem uses the [[principle of mathematical induction|0ASJVEFA9J]].\n\n## Explanations\n\n### L1 human\n\nA recursive definition tells you where to start and how to take one step.\nFor the factorial, you start with $0! = 1$, and each step multiplies by the next number.\nStep by step, this gives $1! = 1$, $2! = 2$, $3! = 6$ and $4! = 24$.\n",
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  "msc": [
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  "name": "Definition by recursion",
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        "what": "an inventory row",
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