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  "body": "\n## Summary\n\nThe axiom of dependent choice, DC, allows a sequence of countably many choices.\nEach choice may depend on the choices made before.\nIt is weaker than the full [[axiom of choice|0HTGB13FE2]] and stronger than the axiom of countable choice.\nIt is the form of choice that most of real analysis needs, and the [[precise statement|TPXEFSBQYB]] uses a binary relation with no dead ends.\n\n## Explanations\n\n### L1 human\n\nImagine a game where from every position there is at least one legal move.\nDependent choice says you can then play forever: there is an infinite sequence of positions, each one a legal move from the one before.\nFor a finite number of moves this needs no axiom, but an infinite run needs infinitely many choices made one after another.\nFor example, let $R$ be a relation on the real numbers such that each $x$ is related to at least one $y \u003e x$.\nThen DC gives a sequence $x_0 \u003c x_1 \u003c x_2 \u003c \\cdots$ in which each term is related to the next.\nWhen there is no explicit rule that picks $y$ from $x$, the axiom is what makes the sequence exist.\n",
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