Determinacy of Refinements to the Difference Hierarchy of Co-Analytic

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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 ω -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 ∃M(M T, M is iterable) =⇒ Det(ω -Π1 + Γ)

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -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 ∃M(M T, M is iterable) =⇒ Det(ω -Π1 + Γ)

1 T ⊆ ZFC, Γ ⊆ ∆1

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 2 / 24 0 Consider: T = KP + Σ1-Sep, Γ = Σ2.

Goal

Theorems of the form:

2 1 ∃M(M T, M is iterable) =⇒ Det(ω -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 ω -Π1 + Γ M¨unster,July 2015 2 / 24 Goal

Theorems of the form:

2 1 ∃M(M T, M is iterable) =⇒ Det(ω -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 ω -Π1 + Γ Münster,July 2015 2 / 24 Det ↔ 0† 2 1 2 1 1 (ω + 1)-Π1 ω -Π1 + ∆1 . . ω2-Π1 1 2 1 0 Classes we’re . ω -Π1 + Σ2 . interested in ω2-Π1 + Σ0 1 1 1 3-Π1 ω2-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 ω -Π1 + Γ Münster,July 2015 3 / 24 Det ↔ 0† 2 1 1 ω -Π1 + ∆1 . . ω2-Π1 + Σ0 Classes we’re 1 2 interested in 2 1 0 ω -Π1 + Σ1 2 1 ω -Π1 Refined difference hierarchy

Det ↔ 0]

Picture

2 1 (ω + 1)-Π1 1 ∆2 2 1 ω -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 ω -Π1 + Γ Münster,July 2015 3 / 24 2 1 1 ω -Π1 + ∆1 . . ω2-Π1 + Σ0 Classes we’re 1 2 interested in 2 1 0 ω -Π1 + Σ1 2 1 ω -Π1 Refined difference hierarchy

Picture

Det ↔ 0† 2 1 (ω + 1)-Π1 1 ∆2 2 1 ω -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 ω -Π1 + Γ M¨unster,July 2015 3 / 24 Classes we’re interested in

Picture

Det ↔ 0† 2 1 2 1 1 (ω + 1)-Π1 ω -Π1 + ∆1 1 . ∆2 . ω2-Π1 1 2 1 0 . ω -Π1 + Σ2 Π1 . 1 2 1 0 ω -Π1 + Σ1 1 3-Π1 ∆1 1 ω2-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 ω -Π1 + Γ M¨unster,July 2015 3 / 24 Picture

Det ↔ 0† 2 1 2 1 1 (ω + 1)-Π1 ω -Π1 + ∆1 1 . ∆2 . ω2-Π1 1 2 1 0 Classes we’re . ω -Π1 + Σ2 Π1 . interested in 1 2 1 0 ω -Π1 + Σ1 1 3-Π1 ∆1 1 ω2-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 ω -Π1 + Γ M¨unster,July 2015 3 / 24 I each Aβ ∈ Γ;

I Aα = ∅; and x ∈ A ↔ the least β such that x∈ / Aβ is odd

So (A0 \ A1) ∪ (A2 \ 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 | β 6 αi such that:

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 4 / 24 I Aα = ∅; and x ∈ A ↔ the least β such that x∈ / Aβ is odd

So (A0 \ A1) ∪ (A2 \ 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 | β 6 αi such that: I each Aβ ∈ Γ;

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 4 / 24 and

x ∈ A ↔ the least β such that x∈ / Aβ is odd

So (A0 \ A1) ∪ (A2 \ A3) ∪ · · · . Fact If α > 1 is a computable ordinal then

1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2

Diﬀerence Hierarchy

I Aα = ∅;

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 4 / 24 So (A0 \ A1) ∪ (A2 \ A3) ∪ · · · . Fact If α > 1 is a computable ordinal then

1 1 1 1 Π1 ( α-Π1 ( (α + 1)-Π1 ( ∆2

Diﬀerence Hierarchy

I Aα = ∅; and x ∈ A ↔ the least β such that x∈ / Aβ is odd

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -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

I Aα = ∅; and x ∈ A ↔ the least β such that x∈ / Aβ is odd

So (A0 \ A1) ∪ (A2 \ A3) ∪ · · · .

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 4 / 24 Diﬀerence Hierarchy

I Aα = ∅; and x ∈ A ↔ the least β such that x∈ / Aβ is odd

So (A0 \ A1) ∪ (A2 \ 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 ω -Π1 + Γ M¨unster,July 2015 4 / 24 2 1 Let A ∈ ω -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 < ω , and if they both succeed then I wins if he gets into Aω2 .

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 ∈ α-Γ + Λ

if A ∈ (α + 1)-Γ, as witnessed by the sequence hAβ | β 6 α + 1i, but Aα ∈ Λ.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,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 ∈ α-Γ + Λ

if A ∈ (α + 1)-Γ, as witnessed by the sequence hAβ | β 6 α + 1i, but Aα ∈ Λ.

2 1 Let A ∈ ω -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 < ω , and if they both succeed then I wins if he gets into Aω2 .

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -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.

Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 6 / 24 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

Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

1 I Characterise membership in Π1 sets by well-orders

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 6 / 24 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

Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

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

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 6 / 24 I Using a winning strategy for the auxiliary game, a winning strategy for the original game is deﬁned

Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

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

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 6 / 24 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.

Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

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

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 6 / 24 Auxiliary Game

1 The proof follows Martin’s “integration” method for proving α-Π1 determinacy from indiscernibles. The ingredients of that proof are:

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

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 6 / 24 2 I The ordinal components ηi ∈ ℵω are partitioned so as to create ω many countable orderings. Each should witness that 2 x = ha0, a1, a2,...i ∈ Aβ for some β < ω .

I We say that the play is badly lost if one of these orderings witnesses that x∈ / Aβ. If the ﬁrst such mistake occurs with β even then it is badly lost for I, otherwise for II.

I II wins the auxiliary game if the play is not badly lost for either player; I wins if it is badly lost for II.

Details of the Auxiliary Game

I ha0, η0i ha2, η2i ...

II ha1, η1i ha3, η3i

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 7 / 24 I We say that the play is badly lost if one of these orderings witnesses that x∈ / Aβ. If the ﬁrst such mistake occurs with β even then it is badly lost for I, otherwise for II.

I II wins the auxiliary game if the play is not badly lost for either player; I wins if it is badly lost for II.

Details of the Auxiliary Game

I ha0, η0i ha2, η2i ...

II ha1, η1i ha3, η3i

2 I The ordinal components ηi ∈ ℵω are partitioned so as to create ω many countable orderings. Each should witness that 2 x = ha0, a1, a2,...i ∈ Aβ for some β < ω .

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 7 / 24 I II wins the auxiliary game if the play is not badly lost for either player; I wins if it is badly lost for II.

Details of the Auxiliary Game

I ha0, η0i ha2, η2i ...

II ha1, η1i ha3, η3i

2 I The ordinal components ηi ∈ ℵω are partitioned so as to create ω many countable orderings. Each should witness that 2 x = ha0, a1, a2,...i ∈ Aβ for some β < ω .

I We say that the play is badly lost if one of these orderings witnesses that x∈ / Aβ. If the ﬁrst such mistake occurs with β even then it is badly lost for I, otherwise for II.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 7 / 24 Details of the Auxiliary Game

I ha0, η0i ha2, η2i ...

II ha1, η1i ha3, η3i

2 I The ordinal components ηi ∈ ℵω are partitioned so as to create ω many countable orderings. Each should witness that 2 x = ha0, a1, a2,...i ∈ Aβ for some β < ω .

I We say that the play is badly lost if one of these orderings witnesses that x∈ / Aβ. If the ﬁrst such mistake occurs with β even then it is badly lost for I, otherwise for II.

I II wins the auxiliary game if the play is not badly lost for either player; I wins if it is badly lost for II.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 7 / 24 Now consider extending the proof to our situation: we don’t have a 2 1 2 1 ω -Π1 set, but a ω -Π1 + Γ set, so we modify the win condition to be: I wins if the play is badly lost for II or it is not badly lost for either player and x ∈ Aω2 .

Aω2 is an element of Γ, so this condition is no longer open; to ﬁnd a winning strategy we need to analyse the complexity of this condition. We will need a lightface condition, so the ﬁrst task is to work out what 0 ω “lightface Σn” should mean for a subset of (ω × ℵω) .

Complexity

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 8 / 24 Aω2 is an element of Γ, so this condition is no longer open; to find a winning strategy we need to analyse the complexity of this condition. We will need a lightface condition, so the first task is to work out what 0 ω “lightface Σn” should mean for a subset of (ω × ℵω) .

Complexity

“Being badly lost” is an open condition because a play is badly lost iﬀ there is an initial position where the orderings for one player are wrong. Altogether this means that the above auxiliary game is open, and so it is determined. Now consider extending the proof to our situation: we don’t have a 2 1 2 1 ω -Π1 set, but a ω -Π1 + Γ set, so we modify the win condition to be: I wins if the play is badly lost for II or it is not badly lost for either player and x ∈ Aω2 .

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 8 / 24 We will need a lightface condition, so the ﬁrst task is to work out what 0 ω “lightface Σn” should mean for a subset of (ω × ℵω) .

Complexity

Aω2 is an element of Γ, so this condition is no longer open; to ﬁnd a winning strategy we need to analyse the complexity of this condition.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 8 / 24 Complexity

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 8 / 24 One can also deﬁne lightface open subsets of (κω)n × κm × ωj in the obvious way.

If we replace Lκ with hLκ[~c], ∈,~ci for some countable set of ordinals ~c, then we can make the same deﬁnition to get the lightface in ~c open sets.

Eﬀective Descriptive Set Theory on κω

Deﬁnition ω Call a subset R of κ generalised lightface open if there is a Σ1(Lκ) set X ⊆ κ<ω such that:

x ∈ R ⇐⇒ ∃p ∈ X(p ⊆ x)

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 9 / 24 If we replace Lκ with hLκ[~c], ∈,~ci for some countable set of ordinals ~c, then we can make the same deﬁnition to get the lightface in ~c open sets.

Eﬀective Descriptive Set Theory on κω

Deﬁnition ω Call a subset R of κ generalised lightface open if there is a Σ1(Lκ) set X ⊆ κ<ω such that:

x ∈ R ⇐⇒ ∃p ∈ X(p ⊆ x)

One can also deﬁne lightface open subsets of (κω)n × κm × ωj in the obvious way.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 9 / 24 Eﬀective Descriptive Set Theory on κω

Deﬁnition ω Call a subset R of κ generalised lightface open if there is a Σ1(Lκ) set X ⊆ κ<ω such that:

x ∈ R ⇐⇒ ∃p ∈ X(p ⊆ x)

One can also deﬁne lightface open subsets of (κω)n × κm × ωj in the obvious way.

If we replace Lκ with hLκ[~c], ∈,~ci for some countable set of ordinals ~c, then we can make the same deﬁnition to get the lightface in ~c open sets.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 9 / 24 Definition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iff there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then:

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open;

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 10 / 24 0 0 I P is Πn iff ¬P is Σn; 0 0 0 I P is ∆n iff it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn;

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 The Generalised Lightface Borel Hierarchy

Let P be a relation on κω, then: Deﬁnition 0 I P is called Σ1 if P is generalised lightface open; 0 0 ω I P is Σn+1 iﬀ there is a Πn predicate R ⊆ κ × ω such that

x ∈ P ⇐⇒ ∃a ∈ ω(R(x, a));

0 0 I P is Πn iﬀ ¬P is Σn; 0 0 0 I P is ∆n iﬀ it is Σn and Πn.

Note that we go up by ω unions, not κ unions.

0 If we replace “lightface” with “lightface in ~c” then we get the Σn(~c) hierarchy on κω.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 10 / 24 1 I The idea is that Σ1 relations are Π1 over any admissible containing Lκ. The leftmost path through the the corresponding tree is then a deﬁnable element.

I This allows us to reduce the complexities of properties in the determinacy arguments, and hence prove determinacy of the auxiliary games in weak models.

Recursion Theory

I We can prove the analogue of the Kleene Basis theorem in this ω 1 context: If X ⊆ κ is Σ1 and non-empty, it has an element deﬁnable over any admissible set M with Lκ ∈ M.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 11 / 24 I This allows us to reduce the complexities of properties in the determinacy arguments, and hence prove determinacy of the auxiliary games in weak models.

Recursion Theory

I We can prove the analogue of the Kleene Basis theorem in this ω 1 context: If X ⊆ κ is Σ1 and non-empty, it has an element deﬁnable over any admissible set M with Lκ ∈ M.

1 I The idea is that Σ1 relations are Π1 over any admissible containing Lκ. The leftmost path through the the corresponding tree is then a deﬁnable element.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 11 / 24 Recursion Theory

I We can prove the analogue of the Kleene Basis theorem in this ω 1 context: If X ⊆ κ is Σ1 and non-empty, it has an element deﬁnable over any admissible set M with Lκ ∈ M.

1 I The idea is that Σ1 relations are Π1 over any admissible containing Lκ. The leftmost path through the the corresponding tree is then a deﬁnable element.

I This allows us to reduce the complexities of properties in the determinacy arguments, and hence prove determinacy of the auxiliary games in weak models.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 11 / 24 We can then prove that the auxiliary game is determined using 0 arguments analogous to those used to establish ordinary Σn determinacy. Example 2 1 0 ∗ 0 If A ∈ ω -Π1 + Σ2 then the auxiliary winning set A is Σb2 and, if M is a transitive model of KP + Σ1-Sep containing hℵii then there is a ∗ Σ1-deﬁnable winning strategy for A in M.

Determinacy of the Auxiliary Game

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 12 / 24 Example 2 1 0 ∗ 0 If A ∈ ω -Π1 + Σ2 then the auxiliary winning set A is Σb2 and, if M is a transitive model of KP + Σ1-Sep containing hℵii then there is a ∗ Σ1-deﬁnable winning strategy for A in M.

Determinacy of the Auxiliary Game

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 12 / 24 Determinacy of the Auxiliary Game

ω 0 0 A basic fact is that if A ⊆ ω is Σn, it is also Σn in this sense, considered as a subset of each κω. A quick calculation then shows that, 2 1 0 if the main game is ω -Π1 + Σn for n > 1 then the auxiliary game is 0 ω 0 Σn(hℵi | i < ωi) on (ω × ℵω) , a pointclass we abbreviate to Σbn. We can then prove that the auxiliary game is determined using 0 arguments analogous to those used to establish ordinary Σn determinacy. Example 2 1 0 ∗ 0 If A ∈ ω -Π1 + Σ2 then the auxiliary winning set A is Σb2 and, if M is a transitive model of KP + Σ1-Sep containing hℵii then there is a ∗ Σ1-deﬁnable winning strategy for A in M.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 12 / 24 The kind of indiscernibility we use is as follows: Deﬁnition

A closed-unbounded class of ordinals C is a class of Σn generating indiscernibles for the theory T if, letting AT [~c] be the least transitive model of the theory T (in the language including a predicate for ~c) containing the sequence ~c,

~ AT [~c] ≡Σn AT [d]

So, taking T = KP + Σ1-Sep, n = 1, this principle would imply that the 0 winning strategy for the Σb2 auxiliary game behaves the same when deﬁned over any of these models.

Generating Indiscernibles

Having shown that the auxiliary game is determined, we need a form of indiscernibility to perform the “integration” part of Martin’s method.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 13 / 24 letting AT [~c] be the least transitive model of the theory T (in the language including a predicate for ~c) containing the sequence ~c,

~ AT [~c] ≡Σn AT [d]

So, taking T = KP + Σ1-Sep, n = 1, this principle would imply that the 0 winning strategy for the Σb2 auxiliary game behaves the same when deﬁned over any of these models.

Generating Indiscernibles

Having shown that the auxiliary game is determined, we need a form of indiscernibility to perform the “integration” part of Martin’s method. The kind of indiscernibility we use is as follows: Deﬁnition

A closed-unbounded class of ordinals C is a class of Σn generating indiscernibles for the theory T if,

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 13 / 24 So, taking T = KP + Σ1-Sep, n = 1, this principle would imply that the 0 winning strategy for the Σb2 auxiliary game behaves the same when deﬁned over any of these models.

Generating Indiscernibles

~ AT [~c] ≡Σn AT [d]

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 13 / 24 Generating Indiscernibles

~ AT [~c] ≡Σn AT [d]

So, taking T = KP + Σ1-Sep, n = 1, this principle would imply that the 0 winning strategy for the Σb2 auxiliary game behaves the same when deﬁned over any of these models.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 13 / 24 Deﬁnition

Let Mλ be the λth iterate in the iteration of M by F, with measurable κλ, and let P be the Prikry forcing for Fλ:

<ω P = {hp, Xi | p ∈ [κλ] , X ∈ Fλ ∩ Mλ} hp, Xi 6 hq, Yi ↔ q is an initial segment of p ∧ X ∪ (p \ q) ⊆ Y

Theorem (L.S.)

If M KP + Σn-Sep + V = L[F] then the class of iteration points is a class of generating indiscernibles for KP + Σn-Sep

This requires us to show that, although P is a class forcing over Mλ, KP + Σn-Sep holds in the generic extension.

Generating Indiscernibles

We obtain indiscernibles by starting with a mouse M T.(M must satisfy T in the language with its M-ultraﬁlter F as a predicate)

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 14 / 24 Theorem (L.S.)

If M KP + Σn-Sep + V = L[F] then the class of iteration points is a class of generating indiscernibles for KP + Σn-Sep

This requires us to show that, although P is a class forcing over Mλ, KP + Σn-Sep holds in the generic extension.

Generating Indiscernibles

We obtain indiscernibles by starting with a mouse M T.(M must satisfy T in the language with its M-ultraﬁlter F as a predicate) Deﬁnition

Let Mλ be the λth iterate in the iteration of M by F, with measurable κλ, and let P be the Prikry forcing for Fλ:

<ω P = {hp, Xi | p ∈ [κλ] , X ∈ Fλ ∩ Mλ} hp, Xi 6 hq, Yi ↔ q is an initial segment of p ∧ X ∪ (p \ q) ⊆ Y

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 14 / 24 This requires us to show that, although P is a class forcing over Mλ, KP + Σn-Sep holds in the generic extension.

Generating Indiscernibles

We obtain indiscernibles by starting with a mouse M T.(M must satisfy T in the language with its M-ultraﬁlter F as a predicate) Deﬁnition

Let Mλ be the λth iterate in the iteration of M by F, with measurable κλ, and let P be the Prikry forcing for Fλ:

<ω P = {hp, Xi | p ∈ [κλ] , X ∈ Fλ ∩ Mλ} hp, Xi 6 hq, Yi ↔ q is an initial segment of p ∧ X ∪ (p \ q) ⊆ Y

Theorem (L.S.)

If M KP + Σn-Sep + V = L[F] then the class of iteration points is a class of generating indiscernibles for KP + Σn-Sep

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 14 / 24 Generating Indiscernibles

We obtain indiscernibles by starting with a mouse M T.(M must satisfy T in the language with its M-ultraﬁlter F as a predicate) Deﬁnition

Let Mλ be the λth iterate in the iteration of M by F, with measurable κλ, and let P be the Prikry forcing for Fλ:

<ω P = {hp, Xi | p ∈ [κλ] , X ∈ Fλ ∩ Mλ} hp, Xi 6 hq, Yi ↔ q is an initial segment of p ∧ X ∪ (p \ q) ⊆ Y

Theorem (L.S.)

If M KP + Σn-Sep + V = L[F] then the class of iteration points is a class of generating indiscernibles for KP + Σn-Sep

This requires us to show that, although P is a class forcing over Mλ, KP + Σn-Sep holds in the generic extension. 2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 14 / 24 ˙ ˙ I Deﬁne a set of “names,” which can be thought of as Lθ[~c] where ~c is a constant symbol for a Prikry-generic sequence. We give each ˙ name a rank according to where it appears in Lθ[~c].

I After we ﬁx a generic ~c ⊆ κλ, the interpretation of any name in ˙ Lθ[~c] is just the corresponding set in Lθ[~c].

I Deﬁne a ranked language LP which in addition to everything from α L{∈} contains ranked variables vi for α < θ and all the names from ˙ Lθ[~c]. A sentence of LP is ranked if all variables in it are ranked.

Ramiﬁed Forcing Language

I We need to show that the forcing behaves nicely, but don’t want to show it’s pre-tame. Let θ = On ∩ Mλ.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 15 / 24 I After we ﬁx a generic ~c ⊆ κλ, the interpretation of any name in ˙ Lθ[~c] is just the corresponding set in Lθ[~c].

Ramiﬁed Forcing Language

I We need to show that the forcing behaves nicely, but don’t want to show it’s pre-tame. Let θ = On ∩ Mλ.

˙ ˙ I Deﬁne a set of “names,” which can be thought of as Lθ[~c] where ~c is a constant symbol for a Prikry-generic sequence. We give each ˙ name a rank according to where it appears in Lθ[~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 15 / 24 I Deﬁne a ranked language LP which in addition to everything from α L{∈} contains ranked variables vi for α < θ and all the names from ˙ Lθ[~c]. A sentence of LP is ranked if all variables in it are ranked.

Ramiﬁed Forcing Language

I We need to show that the forcing behaves nicely, but don’t want to show it’s pre-tame. Let θ = On ∩ Mλ.

I After we ﬁx a generic ~c ⊆ κλ, the interpretation of any name in ˙ Lθ[~c] is just the corresponding set in Lθ[~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 15 / 24 Ramiﬁed Forcing Language

I We need to show that the forcing behaves nicely, but don’t want to show it’s pre-tame. Let θ = On ∩ Mλ.

I After we ﬁx a generic ~c ⊆ κλ, the interpretation of any name in ˙ Lθ[~c] is just the corresponding set in Lθ[~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 15 / 24 ∗ ˙ ˙ 1 p x ∈ y iﬀ x ∈ Lα[~c], y ∈ Lβ[~c] and: 1 α = β = 0 and either x ∈ y ∈ κ or x ∈ p ∧ y = ~c˙; or γ γ ∗ 2 α < β, y = {z | ϕ(z )} and p ϕ(x); or else ˙ 3 α > β and ∃z ∈ Lγ[~c] for some γ, either β > γ or β = γ = 0 and ∗ p z = x ∧ z ∈ y

Ramiﬁed Forcing

We can now deﬁne the (weak) forcing relation.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 16 / 24 Ramiﬁed Forcing

We can now deﬁne the (weak) forcing relation.

∗ ˙ ˙ 1 p x ∈ y iﬀ x ∈ Lα[~c], y ∈ Lβ[~c] and: 1 α = β = 0 and either x ∈ y ∈ κ or x ∈ p ∧ y = ~c˙; or γ γ ∗ 2 α < β, y = {z | ϕ(z )} and p ϕ(x); or else ˙ 3 α > β and ∃z ∈ Lγ[~c] for some γ, either β > γ or β = γ = 0 and ∗ p z = x ∧ z ∈ y

We can now deﬁne the (weak) forcing relation.

∗ ∗ α 2 p x = y iff p ∀z (z ∈ x ↔ z ∈ y) for α the maximum of the ranks of x and y. ∗ ∗ ∗ 3 p ϕ ∧ ψ iff p ϕ and p ψ. ∗ ∗ 4 p ¬ϕ iff ∀q ∈ P(q 6 p =⇒ q 1 ϕ). ∗ α α ˙ ∗ 5 p ∃x (ϕ(x )) iff there is some t ∈ Lα[~c] such that p ϕ(t). ∗ S ˙ ∗ 6 p ∃x(ϕ(x)) iff there is some t ∈ α Lα[~c] such that p ϕ(t). 2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 16 / 24 I This is proved first for ∆1 formulæ simultaneously with the Prikry lemma, and relies on the fact that we don’t quantify over P in the ∗ definition of for atomic formulæ. I This slide was a bit empty, so here’s a picture of mice playing a game:

Forcing Theorems

I This forcing relation is deﬁnable, and in fact if ϕ is a Σn sentence ∗ Mλ of the forcing language, then p ϕ is Σn .

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 17 / 24 I This slide was a bit empty, so here’s a picture of mice playing a game:

Forcing Theorems

I This forcing relation is definable, and in fact if ϕ is a Σn sentence ∗ Mλ of the forcing language, then p ϕ is Σn . I This is proved first for ∆1 formulæ simultaneously with the Prikry lemma, and relies on the fact that we don’t quantify over P in the ∗ definition of for atomic formulæ.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 17 / 24 Forcing Theorems

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 17 / 24 Deﬁnition

In this case, the generic extension Mλ[G] is Lθ[~c], where ~c = S{p | hp, Xi ∈ G}.

From now on we denote M[G] as Mλ[~c] as above. Note that, as in modern forcing, the generic extension is the class of names interpreted by the generic.

Forcing Theorems

The deﬁnition of genericity is odd due to the weak setting: Deﬁnition

Mλ G ⊆ P is Mλ-generic if it both meets all Σn dense subclasses of P and decides every Σn sentence of LP.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 18 / 24 From now on we denote M[G] as Mλ[~c] as above. Note that, as in modern forcing, the generic extension is the class of names interpreted by the generic.

Forcing Theorems

The deﬁnition of genericity is odd due to the weak setting: Deﬁnition

Mλ G ⊆ P is Mλ-generic if it both meets all Σn dense subclasses of P and decides every Σn sentence of LP.

Deﬁnition

In this case, the generic extension Mλ[G] is Lθ[~c], where ~c = S{p | hp, Xi ∈ G}.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 18 / 24 Forcing Theorems

The deﬁnition of genericity is odd due to the weak setting: Deﬁnition

Mλ G ⊆ P is Mλ-generic if it both meets all Σn dense subclasses of P and decides every Σn sentence of LP.

Deﬁnition

In this case, the generic extension Mλ[G] is Lθ[~c], where ~c = S{p | hp, Xi ∈ G}.

From now on we denote M[G] as Mλ[~c] as above. Note that, as in modern forcing, the generic extension is the class of names interpreted by the generic.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 18 / 24 In fact more is true. By the Prikry property, we have the following: Theorem (L.S.)

For any p = {c0,..., cl} and y an arbitrary constant Σn-deﬁnable in Mλ[~c] (without indiscernible parameters above cl). Suppose ψ is Πn−1. Then: Mλ[~c] ∃zψ(z, y) ⇔ Mλ ∃Yhp, Yi ∃zψ(z, y)

Now we can show that Mλ[~c] KP + Σn-Sep.

Forcing Theorems

Theorem (L.S.) The Truth Lemma holds for any such generic, and we can always ﬁnd one by taking the Prikry sequence ~c a countable sequence of critical points coﬁnal in κλ.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 19 / 24 Now we can show that Mλ[~c] KP + Σn-Sep.

Forcing Theorems

Theorem (L.S.) The Truth Lemma holds for any such generic, and we can always ﬁnd one by taking the Prikry sequence ~c a countable sequence of critical points coﬁnal in κλ.

In fact more is true. By the Prikry property, we have the following: Theorem (L.S.)

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 19 / 24 Forcing Theorems

Theorem (L.S.) The Truth Lemma holds for any such generic, and we can always ﬁnd one by taking the Prikry sequence ~c a countable sequence of critical points coﬁnal in κλ.

In fact more is true. By the Prikry property, we have the following: Theorem (L.S.)

Now we can show that Mλ[~c] KP + Σn-Sep.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 19 / 24 I I.e. Mλ[~c] = AT [~c].

I We thus have the ingredients required to mimic Martin’s proof: deﬁnable winning strategies for the auxiliary game, and suitable indiscernibles.

Indiscernibility

I Now it’s easy to see that all Mλ[~c] models are Σn elementary equivalent, minimal and models of KP + Σn-Sep with ~c ∈ Mλ[~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 20 / 24 I We thus have the ingredients required to mimic Martin’s proof: deﬁnable winning strategies for the auxiliary game, and suitable indiscernibles.

Indiscernibility

I Now it’s easy to see that all Mλ[~c] models are Σn elementary equivalent, minimal and models of KP + Σn-Sep with ~c ∈ Mλ[~c].

I I.e. Mλ[~c] = AT [~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 20 / 24 Indiscernibility

I Now it’s easy to see that all Mλ[~c] models are Σn elementary equivalent, minimal and models of KP + Σn-Sep with ~c ∈ Mλ[~c].

I I.e. Mλ[~c] = AT [~c].

I We thus have the ingredients required to mimic Martin’s proof: deﬁnable winning strategies for the auxiliary game, and suitable indiscernibles.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 20 / 24 I They “imagine” their opponent has played indiscernibles ~c.

I They then move according to the auxiliary strategy’s output, as computed by any model Mλ[~c] = AT [~c].

I This strategy is winning in V because any counterexample would be a real existing by Shoenﬁeld absoluteness in a suitable Mλ[~c].

Fitting it Together

I To win the original game, the player must be able to ignore the components of the auxiliary game that are not played in the original.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 21 / 24 I They then move according to the auxiliary strategy’s output, as computed by any model Mλ[~c] = AT [~c].

I This strategy is winning in V because any counterexample would be a real existing by Shoenﬁeld absoluteness in a suitable Mλ[~c].

Fitting it Together

I To win the original game, the player must be able to ignore the components of the auxiliary game that are not played in the original.

I They “imagine” their opponent has played indiscernibles ~c.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 21 / 24 I This strategy is winning in V because any counterexample would be a real existing by Shoenﬁeld absoluteness in a suitable Mλ[~c].

Fitting it Together

I To win the original game, the player must be able to ignore the components of the auxiliary game that are not played in the original.

I They “imagine” their opponent has played indiscernibles ~c.

I They then move according to the auxiliary strategy’s output, as computed by any model Mλ[~c] = AT [~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 21 / 24 Fitting it Together

I To win the original game, the player must be able to ignore the components of the auxiliary game that are not played in the original.

I They “imagine” their opponent has played indiscernibles ~c.

I They then move according to the auxiliary strategy’s output, as computed by any model Mλ[~c] = AT [~c].

I This strategy is winning in V because any counterexample would be a real existing by Shoenﬁeld absoluteness in a suitable Mλ[~c].

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 21 / 24 T Γ 0 “ ‘cleverness’ + ∃ a ‘clever mouse’ ” Σ1 0 KP + Σ1-Sep Σ2 0 KP + Σ2-Sep Σ3 0 KP + Σn+1-Sep n-Π3 − α 0 CK ZFC + P (κ) exists Σ1+α+3 (α < ω1 ) 1 ZFC ∆1.

Results

Theorem (L.S.) The implication:

2 1 ∃M(M T, M is iterable) =⇒ Det(ω -Π1 + Γ) holds for the following values of T and Γ:

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 22 / 24 Results

Theorem (L.S.) The implication:

2 1 ∃M(M T, M is iterable) =⇒ Det(ω -Π1 + Γ) holds for the following values of T and Γ:

T Γ 0 “ ‘cleverness’ + ∃ a ‘clever mouse’ ” Σ1 0 KP + Σ1-Sep Σ2 0 KP + Σ2-Sep Σ3 0 KP + Σn+1-Sep n-Π3 − α 0 CK ZFC + P (κ) exists Σ1+α+3 (α < ω1 ) 1 ZFC ∆1.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 22 / 24 2 Are there reverse implications, or at least limitations?

3 Does the generalised lightface hierarchy generate interesting eﬀective descriptive set theory?

4 Is the specialised forcing useful for anything else?

Open Questions

1 What other combinations of T, Γ can we ﬁnd proofs of?

C. M. Le Sueur. Determinacy of refinements to the difference hierarchy of co-analytic sets. submitted. 2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 23 / 24 3 Does the generalised lightface hierarchy generate interesting effective descriptive set theory?

4 Is the specialised forcing useful for anything else?

Open Questions

1 What other combinations of T, Γ can we ﬁnd proofs of?

2 Are there reverse implications, or at least limitations?

C. M. Le Sueur. Determinacy of refinements to the difference hierarchy of co-analytic sets. submitted. 2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ Münster,July 2015 23 / 24 4 Is the specialised forcing useful for anything else?

Open Questions

1 What other combinations of T, Γ can we ﬁnd proofs of?

2 Are there reverse implications, or at least limitations?

3 Does the generalised lightface hierarchy generate interesting eﬀective descriptive set theory?

1 What other combinations of T, Γ can we ﬁnd proofs of?

2 Are there reverse implications, or at least limitations?

3 Does the generalised lightface hierarchy generate interesting eﬀective descriptive set theory?

4 Is the specialised forcing useful for anything else?

Deﬁnition

M M Let M be a mouse, Qκ the Q-structure of M at κ and θ = On ∩ Qκ . <ω M is clever if, for any Σ1 formula ϕ(x, y) and parameter p ∈ [κ] ,

M M {ξ < κ | Qκ ϕ(ξ, p)} ∈ F =⇒ ~F κ M ∃τ < θ {ξ < τ | Jτ ϕ(ξ, p)} ∈ F ∩ Qκ

This implies Rowbottom’s theorem holds for partitions Σ1 deﬁnable over M.

2 1 Chris Le Sueur (University of Bristol) Determinacy of ω -Π1 + Γ M¨unster,July 2015 24 / 24