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  "body": "\n## Statement\n\nLet $X$ be a [[set|0SJ3RX9TXA]] of nonempty sets.\nThen there is a [[choice function|0YXEQWRGP0]] on $X$, that is, a function $c \\colon X \\to \\bigcup X$ such that $c(S) \\in S$ for each $S \\in X$.\n\n## Remarks\n\nAn equivalent form: for each [[family|05Q6G4EH2G]] $(A_i)_{i \\in I}$ of nonempty sets, the product $\\prod_{i \\in I} A_i$ is nonempty.\nOver ZF, it is equivalent to [[Zorn's lemma|04RXTQA5ZA]], to the [[well-ordering theorem|0N109NG5ZX]] and to the [[comparability of cardinals|0YBHSJMN3J]].\nIt is independent of ZF, if ZF is consistent.\nKurt Gödel showed in 1938 that ZF does not refute it, and Paul Cohen showed in 1963 that ZF does not prove it.\n\n## History\n\nErnst Zermelo formulated the axiom in 1904 for his proof of the [[well-ordering theorem|0N109NG5ZX]].\n\n## Explanations\n\n### L1 human\n\nFor finitely many nonempty boxes, you can pick one object from each box, one box at a time.\nFor infinitely many boxes without a rule to pick, the axiom of choice says that a selection still exists.\nFrom infinitely many pairs of shoes, you can take the left shoe of each pair.\nFrom infinitely many pairs of identical socks, there is no such rule, and you need the axiom.\n",
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