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  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nThen each [[set|0SJ3RX9TXA]] $A$ has a [[well-order|0F1D6PK9RZ]].\n\n## Remarks\n\nAnother name is Zermelo's theorem.\nOver ZF, it is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].\nA well-order of $\\mathbb{R}$ is hard or impossible to visualize, because its construction needs the axiom of choice.\nWith a well-order, each set allows proofs by [[transfinite induction|0ZHQMQ2C1N]].\n\n## History\n\nErnst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that each set has an order with a first element, a next element, and so on, possibly beyond infinity.\nFor $\\mathbb{N}$ the usual order works.\nFor $\\mathbb{R}$ such an order exists, but it is very hard to picture.\n",
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