Axiom of determinacy: Difference between revisions
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Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. |
Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. |
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By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy |
By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. |
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Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). |
Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). |
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