{
  "aliases": [
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  "body": "\n## Statement\n\nLet $R$ be a [[well-founded relation|0230ECX1Q5]] on a set $A$ and let $P(x)$ be a [[predicate|0T3DPBG981]] on $A$.\nAssume that, for each $x \\in A$, if $P(y)$ holds for each $y \\in A$ with $y \\mathrel{R} x$, then $P(x)$ holds.\nThen $P(x)$ holds for each $x \\in A$.\n\n## Remarks\n\nProof: if the set of $x$ with $\\neg P(x)$ is nonempty, it has an $R$-minimal element $m$. Then $P(y)$ holds for each $y \\mathrel{R} m$, so $P(m)$ holds, a contradiction.\nFor $\u003c$ on $\\mathbb{N}$, this is strong induction.\nFor [[ordinal numbers|0RDJK7EQYY]], one often splits the step into three cases: zero, successor ordinals and limit ordinals.\n\n## Explanations\n\n### L1 human\n\nTransfinite induction extends induction on $\\mathbb{N}$ to every well-founded order.\nYou show: if the statement holds for everything below $x$, then it holds for $x$.\nSince you cannot go down forever, the statement holds everywhere.\n",
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  "name": "Transfinite induction",
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