Determinacy in L(ℝ)

A two-player game is said to be determined if one of the two players has a winning strategy. Determinacy for a pointclass Γ is the assertion that all perfect information two-player games of length ω on natural numbers, with payoff in Γ, are determined. Determinacy may fail for games with payoff sets constructed using the axiom of choice, but turns out in contrast to be a key property of definable sets, from which many other properties and a rich structure theory can be derived. This paper presents proofs of determinacy for definable sets, starting with determinacy for the pointclass of analytic sets, continuing through projective sets, and ending with all sets in L(ℝ). Proofs of determinacy for these pointclasses involve large cardinal axioms at the level of measurable cardinals and Woodin cardinals. The paper presents some of the theory of these large cardinals, including the relevant definitions and all results that are needed for the determinacy proofs.

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