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  "body": "\n## Definition\n\n*Projective determinacy* (PD) is the following statement in the language of set theory.\nFor every projective set $X \\subseteq \\N^{\\N}$, that is, every set in some level $\\boldsymbol{\\Sigma}^1_n$ of the [[projective hierarchy|8DZTNVTGY2]], the game $G(\\N, X)$ is [[determined|QXECDPAYTN]].\n\n## Remarks\n\nExample: every Borel set $X$ of the [[Baire space|P2WS16HKD7]] is projective, and its game is determined in ZFC by the Borel determinacy theorem.\nSo the Borel instances of PD are theorems of ZFC.\n\nNon-example: the axiom of choice gives a set $X \\subseteq \\N^{\\N}$ whose game is not determined.\nPD does not cover this set, and under PD such a set is not projective.\n\nFor each $n$, the statement that all $\\boldsymbol{\\Sigma}^1_n$ games are determined is a separate hypothesis, and PD is the conjunction of all of them.\nThe statement that all analytic games, the case $n = 1$, are determined follows from the existence of a measurable cardinal.\n",
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