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  "body": "\n## Statement\n\nLet $A$ be a nonempty set with the discrete topology, and give $A^{\\N}$ the product topology.\nLet $X \\subseteq A^{\\N}$ be a Borel set, that is, a member of the [[Borel sigma-algebra|0WQXY32Q8D]] of $A^{\\N}$.\nThen the game $G(A, X)$ is [[determined|QXECDPAYTN]]: one of the two players has a winning strategy.\n\n## Remarks\n\nThe theorem was proved by Donald A. Martin in [[ZFC|0HE5WDQ9FJ]].\nThe base case is the Gale–Stewart theorem: games with an open or a closed payoff set are determined.\nMartin's proof \"unravels\" a Borel game: it builds an auxiliary game on a larger set of moves in which the payoff set becomes clopen.\nWinning strategies of the auxiliary game then give winning strategies of the original game.\nThe induction runs along the levels of the Borel hierarchy, and each step uses power sets of the sets of moves of the previous step.\n\nThe hypothesis that $X$ is Borel cannot simply be dropped: with the axiom of choice one can build a set $X \\subseteq \\set{0, 1}^{\\N}$ that is not determined.\nHarvey Friedman showed that the theorem cannot be proved in second-order arithmetic, so the use of large sets in the proof is unavoidable.\n\n## Explanations\n\n### L1 human\n\nImagine an infinite game where two players take turns choosing numbers, and the winner is decided by whether the infinite sequence of moves lands in a set $X$.\nSuppose $X$ can be built from simple sets by countable unions, countable intersections and complements.\nThen one of the two players has a strategy that always wins.\n\n### L2 human\n\nThe theorem is a basic tool of descriptive set theory.\nFrom it one gets, for instance, Wadge's lemma for Borel subsets $B$ and $C$ of the Baire space $\\N^{\\N}$.\nIt says that $B = f^{-1}(C)$ for some continuous map $f$, or $C = g^{-1}(\\N^{\\N} \\setminus B)$ for some continuous map $g$.\nGoing beyond Borel sets needs more than ZFC: determinacy of analytic games follows from a measurable cardinal, and projective determinacy follows from infinitely many Woodin cardinals.\nThe theorem is also a famous example of a statement about countable objects whose proof requires uncountably many iterations of the power set operation.\n",
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