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  "body": "\n## Definition\n\nThe *axiom of determinacy* (AD) is the following statement in the language of set theory.\nFor every set $X \\subseteq \\N^{\\N}$ of sequences of natural numbers, the game $G(\\N, X)$ is [[determined|QXECDPAYTN]].\nHere, in $G(\\N, X)$, players I and II alternately choose natural numbers $a_0, a_1, a_2, \\dots$, and player I wins if $(a_0, a_1, a_2, \\dots) \\in X$.\n\n## Remarks\n\nExample: for each Borel set $X$ of the [[Baire space|P2WS16HKD7]] $\\N^{\\N}$, the instance of AD for $X$ holds.\nIt is even a theorem of ZFC, by the Borel determinacy theorem.\n\nNon-example: AD is false in [[ZFC|0HE5WDQ9FJ]].\nThe [[axiom of choice|0HTGB13FE2]] gives a well-ordering of all strategies.\nA diagonal argument along it builds a set $X$ that defeats every strategy of each player.\n\nAD is therefore studied over ZF without choice.\nIt still implies the axiom of countable choice for sets of reals.\n",
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