Previous Results
Gale and Stewart (1953) proved that if the payoff set is an open or closed subset of Aω then the Gale–Stewart game with that payoff set is always determined. Over the next twenty years, this was extended to slightly higher levels of the Borel hierarchy through ever more complicated proofs. This led to the question of whether the game must be determined whenever the payoff set is a Borel subset of Aω. It was known that, using the axiom of choice, it is possible to construct a subset of {0,1}ω that is not determined (Kechris 1995, p. 139).
Harvey Friedman (1971) proved that that any proof that all Borel subsets of Cantor space ({0,1}ω ) were determined would require repeated use of the axiom of replacement, an axiom not typically required to prove theorems about "small" objects such as Cantor space.
Read more about this topic: Borel Determinacy Theorem
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