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  "body": "\n## Statement\n\nLet $(P, \\le)$ be a nonempty [[partially ordered set|0QV2JS9RJ3]] in which every [[chain|0S8FE7ZMFM]] has a [[supremum|0R3H5PANRQ]].\nLet $f \\colon P \\to P$ be a map with $x \\le f(x)$ for every $x \\in P$.\nThen $f$ has a fixed point: there is $x \\in P$ with $f(x) = x$.\n\n## Remarks\n\nThe map $f$ need not be monotone; the hypothesis $x \\le f(x)$ says that $f$ is *inflationary*.\nProof idea: fix $a \\in P$ and let $T$ be the smallest subset of $P$ that contains $a$, is closed under $f$, and contains the supremum of each of its nonempty chains.\nA careful induction shows that $T$ is a chain, so it has a supremum $s$, and $s \\in T$.\nThen $f(s) \\in T$, so $f(s) \\le s \\le f(s)$ and $f(s) = s$.\nThe proof uses no form of the axiom of choice.\nCombined with the axiom of choice, the theorem shows that such a poset has a maximal element.\nOtherwise a choice of some $f(x) \u003e x$ for each $x$ would be an inflationary map with no fixed point.\nThis is a standard step in the proof of [[Zorn's lemma|04RXTQA5ZA]] from the axiom of choice.\nThe hypothesis $x \\le f(x)$ is needed: on the chain $\\set{0, 1}$, where every chain has a supremum, the map swapping $0$ and $1$ has no fixed point.\nThe completeness hypothesis is needed too: on $\\N$, the inflationary map $n \\mapsto n + 1$ has no fixed point, and the chain $\\N$ has no supremum.\n\n## Explanations\n\n### L1 human\n\nSuppose a rule never moves any point downward, and every chain of points has a top.\nThen some point is not moved at all.\nThe idea is to start anywhere and keep applying the rule, jumping to the top of the chain built so far whenever the steps run out.\nThis climb cannot go on forever, so it stops at a fixed point.\n\n### L2 human\n\nThe theorem is a choice-free fixed-point principle for chain-complete posets.\nIt is a standard route to [[Zorn's lemma|04RXTQA5ZA]] from the axiom of choice, and it is used to build objects by transfinite recursion without naming ordinals.\nUnlike the [[Knaster–Tarski theorem|RHXP3HA64M]], it does not need monotonicity, but it says nothing about least fixed points.\n",
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