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Determinacy of Refinements to the Difference Hierarchy of Co-Analytic

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Determinacy of Refinements to the Difference Hierarchy of Co-Analytic Determinacy of Refinements to the Difference Hierarchy of Co-analytic sets Chris Le Sueur http://www.maths.bris.ac.uk/~cl7907/ University of Bristol 3rd M¨unsterConference in Inner Model Theory, the Core Model Induction and Hod Mice 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 1 / 24 1 T ⊆ ZFC, Γ ⊆ ∆1 For the above to hold we would like to have: T ` Det(Γ) 0 Consider: T = KP + Σ1-Sep, Γ = Σ2. Goal Theorems of the form: 2 1 9M(M T, M is iterable) =) Det(! -Π1 + Γ) 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 2 / 24 For the above to hold we would like to have: T ` Det(Γ) 0 Consider: T = KP + Σ1-Sep, Γ = Σ2. Goal Theorems of the form: 2 1 9M(M T, M is iterable) =) Det(! -Π1 + Γ) 1 T ⊆ ZFC, Γ ⊆ ∆1 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 2 / 24 0 Consider: T = KP + Σ1-Sep, Γ = Σ2. Goal Theorems of the form: 2 1 9M(M T, M is iterable) =) Det(! -Π1 + Γ) 1 T ⊆ ZFC, Γ ⊆ ∆1 For the above to hold we would like to have: T ` Det(Γ) 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 2 / 24 Goal Theorems of the form: 2 1 9M(M T, M is iterable) =) Det(! -Π1 + Γ) 1 T ⊆ ZFC, Γ ⊆ ∆1 For the above to hold we would like to have: T ` Det(Γ) 0 Consider: T = KP + Σ1-Sep, Γ = Σ2. 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 2 / 24 Det $ 0y 2 1 2 1 1 (! + 1)-Π1 ! -Π1 + ∆1 . !2-Π1 1 2 1 0 Classes we're . ! -Π1 + Σ2 . interested in !2-Π1 + Σ0 1 1 1 3-Π1 !2-Π1 1 1 2-Π1 Refined difference hierarchy 1 Π1 Det $ 0] 1 Difference hierarchy on Π1 Picture 1 ∆2 1 Π1 1 ∆1 . 0 Σ2 0 Σ1 Borel, Projective hierarchy 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 3 / 24 Det $ 0y 2 1 1 ! -Π1 + ∆1 . !2-Π1 + Σ0 Classes we're 1 2 interested in 2 1 0 ! -Π1 + Σ1 2 1 ! -Π1 Refined difference hierarchy Det $ 0] Picture 2 1 (! + 1)-Π1 1 ∆2 2 1 ! -Π1 1 . Π1 . 1 3-Π1 ∆1 1 . 1 . 2-Π1 0 1 Σ2 Π1 0 Π1 Σ1 Difference hierarchy on 1 Borel, Projective hierarchy 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 3 / 24 2 1 1 ! -Π1 + ∆1 . !2-Π1 + Σ0 Classes we're 1 2 interested in 2 1 0 ! -Π1 + Σ1 2 1 ! -Π1 Refined difference hierarchy Picture Det $ 0y 2 1 (! + 1)-Π1 1 ∆2 2 1 ! -Π1 1 . Π1 . 1 3-Π1 ∆1 1 . 1 . 2-Π1 0 1 Σ2 Π1 Det $ 0] 0 Π1 Σ1 Difference hierarchy on 1 Borel, Projective hierarchy 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 3 / 24 Classes we're interested in Picture Det $ 0y 2 1 2 1 1 (! + 1)-Π1 ! -Π1 + ∆1 1 . ∆2 . !2-Π1 1 2 1 0 . ! -Π1 + Σ2 Π1 . 1 2 1 0 ! -Π1 + Σ1 1 3-Π1 ∆1 1 !2-Π1 . 1 1 . 2-Π1 Refined difference hierarchy 0 1 Σ2 Π1 Det $ 0] 0 Π1 Σ1 Difference hierarchy on 1 Borel, Projective hierarchy 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 3 / 24 Picture Det $ 0y 2 1 2 1 1 (! + 1)-Π1 ! -Π1 + ∆1 1 . ∆2 . !2-Π1 1 2 1 0 Classes we're . ! -Π1 + Σ2 Π1 . interested in 1 2 1 0 ! -Π1 + Σ1 1 3-Π1 ∆1 1 !2-Π1 . 1 1 . 2-Π1 Refined difference hierarchy 0 1 Σ2 Π1 Det $ 0] 0 Π1 Σ1 Difference hierarchy on 1 Borel, Projective hierarchy 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 3 / 24 I each Aβ 2 Γ; I Aα = ?; and x 2 A $ the least β such that x2 = Aβ is odd So (A0 n A1) [ (A2 n A3) [··· . Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 I Aα = ?; and x 2 A $ the least β such that x2 = Aβ is odd So (A0 n A1) [ (A2 n A3) [··· . Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: I each Aβ 2 Γ; 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 and x 2 A $ the least β such that x2 = Aβ is odd So (A0 n A1) [ (A2 n A3) [··· . Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: I each Aβ 2 Γ; I Aα = ?; 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 So (A0 n A1) [ (A2 n A3) [··· . Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: I each Aβ 2 Γ; I Aα = ?; and x 2 A $ the least β such that x2 = Aβ is odd 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: I each Aβ 2 Γ; I Aα = ?; and x 2 A $ the least β such that x2 = Aβ is odd So (A0 n A1) [ (A2 n A3) [··· . 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 Difference Hierarchy Definition 1 Let Γ be a pointclass closed under countable intersections (e.g. Π1), α be a countable ordinal. We say a set A is α-Γ if there is a sequence hAβ j β 6 αi such that: I each Aβ 2 Γ; I Aα = ?; and x 2 A $ the least β such that x2 = Aβ is odd So (A0 n A1) [ (A2 n A3) [··· . Fact If α > 1 is a computable ordinal then 1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 4 / 24 2 1 Let A 2 ! -Π1 + Γ. In order to win the game for A, both players are 2 trying not to be the first one to go out of an Aβ for β < ! , and if they both succeed then I wins if he gets into A!2 . Refining the Difference Hierarchy We can refine the difference hierarchy by restricting the final set in the sequence. Definition For Λ ⊆ Γ, we say A 2 α-Γ + Λ if A 2 (α + 1)-Γ, as witnessed by the sequence hAβ j β 6 α + 1i, but Aα 2 Λ. 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 5 / 24 Refining the Difference Hierarchy We can refine the difference hierarchy by restricting the final set in the sequence. Definition For Λ ⊆ Γ, we say A 2 α-Γ + Λ if A 2 (α + 1)-Γ, as witnessed by the sequence hAβ j β 6 α + 1i, but Aα 2 Λ. 2 1 Let A 2 ! -Π1 + Γ. In order to win the game for A, both players are 2 trying not to be the first one to go out of an Aβ for β < ! , and if they both succeed then I wins if he gets into A!2 . 2 1 Chris Le Sueur (University of Bristol) Determinacy of ! -Π1 + Γ M¨unster,July 2015 5 / 24 1 I Characterise membership in Π1 sets by well-orders I Define an auxiliary game in which the players must confirm that they played into certain sets by exhibiting those wellorders I The auxiliary game is constructed so as to be determined I Using a winning strategy for the auxiliary game, a winning strategy for the original game is defined I In moving from the auxiliary strategy to that for the original game, the players must \imagine" the auxiliary moves being played by their opponent; indiscernibility ensures that this is possible.
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