Basically a measurable cardinal is more than sufficient. A weaker basic principle — the existence of 0# is ample to verify coanalytic determinacy, and a little bit a lot more: The specific result is that the existence of 0# is such as the determinacy of all levels of the primary difference hierarchy underneath the ω2 degree, i.e. ω·n-Π11 determinacy For each n displaystyle n
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From projective determinacy it follows that, For each all-natural quantity n, You will find a transitive internal product that satisfies that there are n Woodin cardinals.
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The evidence that this sort of game titles are determined is rather basic: Player I just plays not to lose; that may be, player I plays to make certain that player II does not have a winning strategy just after I's move.
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sets is set. See reverse mathematics for other relations amongst determinacy and subsystems of 2nd-purchase arithmetic.
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Existence of ω1 Woodin cardinals implies that For each and every countable ordinal α, all online games on integers of length α and projective payoff are identified. Around speaking, α Woodin cardinals corresponds to determinacy of game titles on reals of size α (with a simple payoff established). Assuming a limit of Woodin cardinals κ with o(κ)=κ++ and ω Woodin cardinals above κ, online games of variable countable length where the sport ends when its length is admissible relative to the road of play and with projective payoff are established.