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  "body": "\n## Statement\n\nThe central result of the subject is: **Theorem** (Martin–Harrington, 1978). The following are equivalent over ZFC:\n\nAll lightface analytic games are determined.\n\n0${}^{\\#}$ exists.\n\nDonald A. Martin proved that if 0${}^{\\#}$ exists then all lightface analytic games are determined. The key tool is the sequence of Silver indiscernibles encoded by 0${}^{\\#}$. These indiscernibles provide a uniform way to build winning strategies for lightface analytic games: the computable tree defining the payoff set can be analyzed using the indiscernibles, and the resulting structure allows one to determine which player wins and to explicitly construct a winning strategy.\n\nLeo Harrington proved the converse in 1978: if all lightface analytic games are determined, then 0${}^{\\#}$ exists. This direction is considerably harder. Harrington's proof uses the theory of admissible sets and admissible ordinals (in particular, Barwise compactness) to show that determinacy of a certain lightface analytic game forces the existence of a model of set theory with a non-trivial elementary embedding *j* : *L* → *L*, which is equivalent to the existence of 0${}^{\\#}$. A later forcing-free proof of this direction was given by Sami (1999).\n",
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