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  "body": "\n## Statement\n\nLet $\\kappa$ be an uncountable [[regular cardinal|ZCE333FAWC]].\nLet $S \\subseteq \\kappa$ be [[stationary|6FDYJVG1CK]] with $0 \\notin S$.\nLet $f \\colon S \\to \\kappa$ be a [[function|0XHDTE5YMZ]] that is regressive, that is, $f(\\alpha) \u003c \\alpha$ for each $\\alpha \\in S$.\nThen there is $\\gamma \u003c \\kappa$ such that the set $\\set{\\alpha \\in S : f(\\alpha) = \\gamma}$ is stationary in $\\kappa$.\n\n## Remarks\n\nThe proof is by contradiction.\nIf each fibre $f^{-1}(\\gamma)$ is nonstationary, pick a [[club set|GWS5QXH7WW]] $C_\\gamma$ disjoint from it.\nThe diagonal intersection $D = \\set{\\alpha \u003c \\kappa : \\alpha \\in \\bigcap_{\\gamma \u003c \\alpha} C_\\gamma}$ is club, so it meets $S$ in some ordinal $\\alpha$.\nThen $\\gamma = f(\\alpha) \u003c \\alpha$ gives $\\alpha \\in C_\\gamma$, which contradicts $f(\\alpha) = \\gamma$.\nExample: on the limit ordinals below $\\omega_1$, any function with $f(\\alpha) \u003c \\alpha$ is constant on a stationary set.\nStationarity is necessary: on the successor ordinals, $f(\\beta + 1) = \\beta$ is regressive and injective, so all its fibres have one element.\nThe lemma is also called the pressing-down lemma.\n\n## Explanations\n\n### L1 human\n\nSuppose that to each point of a large set of countable ordinals you assign a smaller ordinal.\nFodor's lemma says that you must then assign the same value to a large set of points.\nYou cannot keep pushing every point down to a different place.\n\"Large\" here means stationary, the right notion of size for these sets.\n\n### L2 human\n\nFodor's lemma is the basic tool for working with stationary sets.\nIt is the infinite analogue of the pigeonhole principle for regressive functions, and it relies on closure of the club filter under diagonal intersections.\nIt is used in Solovay's splitting theorem for stationary sets and in Silver's theorem on singular cardinals.\n",
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