Iteration hypotheses and the strong sealing of universally Baire - - PowerPoint PPT Presentation

Iteration hypotheses and the strong sealing of universally Baire sets

• W. Hugh Woodin

Harvard University

November 2018

Universally Baire sets

Definition (Feng-Magidor-Woodin) A set A ⊆ Rn is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → Rn; the preimage of A by π has the property of Baire in the space Ω.

Universally Baire sets

Definition (Feng-Magidor-Woodin) A set A ⊆ Rn is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → Rn; the preimage of A by π has the property of Baire in the space Ω. ◮ Universally Baire sets have the property of Baire

◮ Simply take Ω = Rn and π to be the identity.

Universally Baire sets

Definition (Feng-Magidor-Woodin) A set A ⊆ Rn is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → Rn; the preimage of A by π has the property of Baire in the space Ω. ◮ Universally Baire sets have the property of Baire

◮ Simply take Ω = Rn and π to be the identity.

◮ Universally Baire sets are Lebesgue measurable.

Universally Baire sets

Definition (Feng-Magidor-Woodin) A set A ⊆ Rn is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → Rn; the preimage of A by π has the property of Baire in the space Ω. ◮ Universally Baire sets have the property of Baire

◮ Simply take Ω = Rn and π to be the identity.

◮ Universally Baire sets are Lebesgue measurable. Theorem Assume V = L. Then every set A ⊆ R is the image of a universally Baire set by a continuous function F : R → R.

The universally Baire sets are the ultimate generalization

• f the projective sets

◮ in the context of large cardinals

Theorem Suppose that there is a proper class of Woodin cardinals and suppose A ⊆ R is universally Baire. ◮ Then every set B ∈ L(A, R) ∩ P(R) is universally Baire.

The universally Baire sets are the ultimate generalization

• f the projective sets

◮ in the context of large cardinals

Theorem Suppose that there is a proper class of Woodin cardinals and suppose A ⊆ R is universally Baire. ◮ Then every set B ∈ L(A, R) ∩ P(R) is universally Baire. Theorem Suppose that there is a proper class of Woodin cardinals. (1) (Martin-Steel) Suppose A ⊆ R is universally Baire.

◮ Then A is determined.

(2) (Steel) Suppose A ⊆ R is universally Baire.

◮ Then A has a universally Baire scale.

HODL(A,R) and measurable cardinals

Definition Suppose that A ⊆ R. Then HODL(A,R) is the class HOD as defined within L(A, R). ◮ The Axiom of Choice must hold in HODL(A,R)

◮ even if L(A, R) | = AD.

HODL(A,R) and measurable cardinals

Definition Suppose that A ⊆ R. Then HODL(A,R) is the class HOD as defined within L(A, R). ◮ The Axiom of Choice must hold in HODL(A,R)

◮ even if L(A, R) | = AD.

Theorem (Solovay:1967) Suppose that A ⊆ R and L(A, R) | = AD. ◮ Then ωV

1 is a measurable cardinal in HODL(A,R).

◮ Solovay’s theorem gave the first connection between the Axiom of Determinacy (AD) and large cardinal axioms.

The least measurable cardinal of HODL(A,R)

Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. ◮ Then ωV

1 is the least measurable cardinal in HODL(A,R).

The least measurable cardinal of HODL(A,R)

Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. ◮ Then ωV

1 is the least measurable cardinal in HODL(A,R).

◮ If stronger large cardinals exist in HODL(A,R), they can only

• ccur above ωV

1 .

The least measurable cardinal of HODL(A,R)

Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. ◮ Then ωV

1 is the least measurable cardinal in HODL(A,R).

◮ If stronger large cardinals exist in HODL(A,R), they can only

• ccur above ωV

1 .

Definition Suppose that A ⊆ R is universally Baire. Then ◮ ΘL(A,R) is the supremum of the ordinals α such that there exists a surjection, π : R → α, such that π ∈ L(A, R).

HODL(A,R) and Woodin cardinals

Lemma Suppose that A ⊆ R. Then: ◮ There are no measurable cardinals κ in HODL(A,R) such that κ ≥ ΘL(A,R).

HODL(A,R) and Woodin cardinals

Lemma Suppose that A ⊆ R. Then: ◮ There are no measurable cardinals κ in HODL(A,R) such that κ ≥ ΘL(A,R). Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. Then: ◮ ΘL(A,R) is a Woodin cardinal in HODL(A,R).

HODL(A,R) and Woodin cardinals

Lemma Suppose that A ⊆ R. Then: ◮ There are no measurable cardinals κ in HODL(A,R) such that κ ≥ ΘL(A,R). Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. Then: ◮ ΘL(A,R) is a Woodin cardinal in HODL(A,R). The Inner Model Test Question Suppose there is a proper class of Woodin cardinals. Suppose ϕ is a Σ2-sentence defining a large cardinal axiom. ◮ Can HODL(A,R) | = ϕ, for some universally Baire set A?

V = Ultimate-L versus the Ω Conjecture

Theorem Assume there is a proper class of strong cardinals and a proper class of Woodin cardinals. Then the following are equivalent.

• 1. V = Ultimate-L.
• 2. There is a universally Baire set A with infinitely many Woodin

cardinals in HODL(A,R) such that for all Σ2-sentences ϕ:

◮ V | = ϕ if and only if HODL(A,R)|ΘL(A,R) | = ϕ.

V = Ultimate-L versus the Ω Conjecture

Theorem Assume there is a proper class of strong cardinals and a proper class of Woodin cardinals. Then the following are equivalent.

• 1. V = Ultimate-L.
• 2. There is a universally Baire set A with infinitely many Woodin

cardinals in HODL(A,R) such that for all Σ2-sentences ϕ:

◮ V | = ϕ if and only if HODL(A,R)|ΘL(A,R) | = ϕ.

Theorem Assume there is a proper class of strong cardinals and a proper class of Woodin cardinals. Then the following are equivalent.

• 1. Ω Conjecture.
• 2. There is a universally Baire set A with infinitely many Woodin

cardinals in HODL(A,R) such that for all Σ2-sentences ϕ:

◮ V | =Ω ϕ if and only if HODL(A,R)|ΘL(A,R) | =Ω ϕ.

Some useful notation

Notation Suppose that there is a proper class of Woodin cardinals.

• 1. Γ∞ denotes the set of all A ⊆ R such that A is universally

Baire.

• 2. Suppose V [g] is a set-generic extension of V . Then

◮ Rg denotes RV [g]. ◮ Γ∞

g

denotes Γ∞)V [g].

The Sealing Theorem

Theorem (Sealing Theorem) Suppose that δ is supercompact and that there is a proper class of Woodin cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and Vδ+1 is countable in V [G]. Then the following hold. (1) Γ∞

G = P(RG) ∩ L(Γ∞ G , RG).

(2) Suppose that γ is a limit of Woodin cardinals in V and that G is V -generic for some partial P ∈ Vγ. Then (Γ∞)Vγ[G] = Γ∞

G .

The Sealing Theorem

Theorem (Sealing Theorem) Suppose that δ is supercompact and that there is a proper class of Woodin cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and Vδ+1 is countable in V [G]. Then the following hold. (1) Γ∞

G = P(RG) ∩ L(Γ∞ G , RG).

(2) Suppose that γ is a limit of Woodin cardinals in V and that G is V -generic for some partial P ∈ Vγ. Then (Γ∞)Vγ[G] = Γ∞

G .

(3) Γ∞

H = P(RH) ∩ L(Γ∞ H , RH).

(4) There is an elementary embedding j : L(Γ∞

G , RG) → L(Γ∞ H , RH)

such that for all A ∈ Γ∞

G , j(A) = (A)V [H], where (A)V [H] is

the interpretation of A in V [H].

The Projective Sealing Theorem

Theorem (Projective Sealing Theorem) Suppose that δ is a limit of strong cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and δ is countable in V [G]. ◮ Then V [G]ω+1 ≺ V [H]ω+1.

The Projective Sealing Theorem

Theorem (Projective Sealing Theorem) Suppose that δ is a limit of strong cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and δ is countable in V [G]. ◮ Then V [G]ω+1 ≺ V [H]ω+1. With stronger large cardinal assumptions one gets Vω+1 ≺ V [G]ω+1 ≺ V [H]ω+1.

The Projective Sealing Theorem

Theorem (Projective Sealing Theorem) Suppose that δ is a limit of strong cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and δ is countable in V [G]. ◮ Then V [G]ω+1 ≺ V [H]ω+1. With stronger large cardinal assumptions one gets Vω+1 ≺ V [G]ω+1 ≺ V [H]ω+1. ◮ This might seem to suggest the same might be true for the Sealing Theorem. In particular that:

◮ Some large cardinal hypothesis implies Γ∞ = P(R) ∩ L(Γ∞, R).

The Projective Sealing Theorem

Theorem (Projective Sealing Theorem) Suppose that δ is a limit of strong cardinals. Suppose V [G] ⊂ V [H] are set-generic extensions of V and δ is countable in V [G]. ◮ Then V [G]ω+1 ≺ V [H]ω+1. With stronger large cardinal assumptions one gets Vω+1 ≺ V [G]ω+1 ≺ V [H]ω+1. ◮ This might seem to suggest the same might be true for the Sealing Theorem. In particular that:

◮ Some large cardinal hypothesis implies Γ∞ = P(R) ∩ L(Γ∞, R).

This would identify a large cardinal hypothesis for which the answer to the Inner Model Test Question is no.

Tower Sealing

Definition (Tower sealing) Suppose that there is a proper class of Woodin cardinals and that δ is a Woodin cardinal. ◮ Tower Sealing holds at δ is whenever G is V -generic for either the P<δ-stationary tower at δ or the Q<δ-stationary tower at δ, then j(Γ∞) = Γ∞

G

where j : V → M ⊂ V [G] is the generic elementary embedding given by G.

Tower Sealing

Definition (Tower sealing) Suppose that there is a proper class of Woodin cardinals and that δ is a Woodin cardinal. ◮ Tower Sealing holds at δ is whenever G is V -generic for either the P<δ-stationary tower at δ or the Q<δ-stationary tower at δ, then j(Γ∞) = Γ∞

G

where j : V → M ⊂ V [G] is the generic elementary embedding given by G. Lemma Suppose that δ is a Woodin limit of Woodin cardinals and that Tower Sealing holds at δ. ◮ Then Γ∞ = P(R) ∩ L(Γ∞, R).

The Strong Sealing Theorem

Theorem (Strong Sealing Theorem) Suppose that δ is an extendible cardinal. Then there is a proper class of κ such that: ◮ κ is a measurable Woodin cardinal and κ > δ. ◮ Suppose G is V -generic for some P ∈ Vκ and Vδ+1 is countable in V [G].

◮ Then Tower Sealing holds at κ in V [G].

The δ-cover and δ-approximation properties

Definition (Hamkins) Suppose N is a transitive class, N | = ZFC, and that δ is an uncountable regular cardinal of V .

• 1. N has the δ-cover property if for all σ ⊂ N, if |σ| < δ then

there exists τ ⊂ N such that:

◮ σ ⊂ τ, ◮ τ ∈ N, ◮ |τ| < δ.

The δ-cover and δ-approximation properties

Definition (Hamkins) Suppose N is a transitive class, N | = ZFC, and that δ is an uncountable regular cardinal of V .

• 1. N has the δ-cover property if for all σ ⊂ N, if |σ| < δ then

there exists τ ⊂ N such that:

◮ σ ⊂ τ, ◮ τ ∈ N, ◮ |τ| < δ.

• 2. N has the δ-approximation property if for all sets X ⊂ N,

the following are equivalent.

◮ X ∈ N. ◮ For all σ ∈ N if |σ| < δ then σ ∩ X ∈ N.

The Hamkins Uniqueness Theorem

Theorem (Hamkins) Suppose N0 and N1 both have the δ-approximation property and the δ-cover property. Suppose ◮ N0 ∩ H(δ+) = N1 ∩ H(δ+). Then: ◮ N0 = N1.

The Hamkins Uniqueness Theorem

Theorem (Hamkins) Suppose N0 and N1 both have the δ-approximation property and the δ-cover property. Suppose ◮ N0 ∩ H(δ+) = N1 ∩ H(δ+). Then: ◮ N0 = N1. Corollary Suppose N has the δ-approximation property and the δ-cover

• property. Let A = N ∩ H(δ+).

◮ Then N ∩ H(γ) is (uniformly) definable in H(γ) from A,

◮ for all strong limit cardinals γ > δ+.

◮ N is a Σ2-definable class from parameters.

Set Theoretic Geology

Definition (Hamkins) A transitive class N is a ground of V if ◮ N | = ZFC. ◮ There is a partial order P ∈ N and an N-generic filter G ⊆ P such that V = N[G].

◮ G is allowed to be trivial in which case N = V .

Set Theoretic Geology

Definition (Hamkins) A transitive class N is a ground of V if ◮ N | = ZFC. ◮ There is a partial order P ∈ N and an N-generic filter G ⊆ P such that V = N[G].

◮ G is allowed to be trivial in which case N = V .

Lemma (Hamkins) Suppose N is a ground of V . Then for all sufficiently large regular cardinals δ: ◮ N has the δ-approximation property. ◮ N has the δ-cover property. Corollary The grounds of V are Σ2-definable classes from parameters.

The sealing hypotheses

Definition (Sealing Hypothesis) There exists a proper class of Woodin cardinals and there exists a ground N of V and there exists δ such that ◮ δ is a supercompact cardinal in N, ◮ N ∩ Vδ+1 is countable in V .

The sealing hypotheses

Definition (Sealing Hypothesis) There exists a proper class of Woodin cardinals and there exists a ground N of V and there exists δ such that ◮ δ is a supercompact cardinal in N, ◮ N ∩ Vδ+1 is countable in V . Definition (Strong Sealing Hypothesis) There exists a ground N of V and there exists δ such that ◮ δ is an extendible cardinal in N, ◮ N ∩ Vδ+1 is countable in V . ◮ By the definability of grounds, these are each first order hypotheses.

Extenders

Notation Suppose E is an extender. Then ◮ κE = CRT(E) and κ∗

E = jE(κE),

◮ ρ(E) = sup{α Vα ⊂ Ult0(V , E)}, ◮ ιE = sup{γ νE ⊆ sup(jE[γ])}, where jE : V → ME ∼ = Ult0(V , E) is the ultrapower embedding and νE = sup{ξ + 1 ξ = jE(f )(s) for all s ∈ [ξ]<ω, f ∈ V }.

Iteration trees

Definition: Iteration trees An iteration tree, T , on V of length η is a tree order <T on η with minimum element 0 and which is a suborder of the standard order, together with a sequence Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α such that the following hold.

• 1. M0 = V ,
• 2. jγ,α : Mγ → Mα for all γ <T α < η,

and ...

definition continued

• 3. Suppose that α + 1 < η. Then α + 1 has an immediate

predecessor, α∗, in the tree order <T and:

◮ Eα ∈ Mα and

◮ Mα | = “ Eα is an extender which is not ω-huge”;

◮ If α∗ < α then ιEα + 1 ≤ min{ρ(Eβ) α∗ ≤ β < α}; ◮ Mα+1 = Ult0(Mα∗, Eα) and jα∗,α+1 : Mα∗ → Mα+1 is the associated embedding.

and ...

definition continued

• 3. Suppose that α + 1 < η. Then α + 1 has an immediate

predecessor, α∗, in the tree order <T and:

◮ Eα ∈ Mα and

◮ Mα | = “ Eα is an extender which is not ω-huge”;

◮ If α∗ < α then ιEα + 1 ≤ min{ρ(Eβ) α∗ ≤ β < α}; ◮ Mα+1 = Ult0(Mα∗, Eα) and jα∗,α+1 : Mα∗ → Mα+1 is the associated embedding.

• 4. If 0 < β < η is a limit ordinal then the set of α such that

α <T β is cofinal in β and

◮ Mβ = limα<T β Mα

relative to the embeddings; jα,β.

(+θ) iteration trees

Definition Suppose that that T is an iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. Suppose that θ ∈ Ord. ◮ Then the iteration tree, T , is a (+θ)-iteration tree if for all α + 1 < η, sup{ιEβ α + 1 ≤ β and β∗ ≤ α} + θ ≤ ρ(Eα) where for each β + 1 < η, β∗ is the T predecessor of β + 1.

Strongly closed iteration trees

Definition Suppose that T is an iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. The iteration tree T is strongly closed if ◮ T is a (+1)-iteration tree and for all α + 1 < η: ◮ Eα is LTH(Eα)-strong in Mα. ◮ LTH(Eα) is strongly inaccessible in Mα.

Strongly closed iteration trees

Definition Suppose that T is an iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. The iteration tree T is strongly closed if ◮ T is a (+1)-iteration tree and for all α + 1 < η: ◮ Eα is LTH(Eα)-strong in Mα. ◮ LTH(Eα) is strongly inaccessible in Mα. ◮ Strongly closed iteration trees are (+ω)-iterations trees and much more.

The successor theorem

Theorem Suppose that T = Mα, Eβ, jγ,α : α ≤ η, β + 1 ≤ η, γ <T α. is a countable strongly closed iteration tree on V of length (η + 1) and that for all limit ǫ ≤ η, T |ǫ has at most one cofinal wellfounded branch. Suppose that E ∈ Mη is an extender and that α < η is such that ιE + 1 < LTH(Eξ) for all α ≤ ξ < η. ◮ Then Ult0(Mα, E) is wellfounded.

The branch theorem

Theorem Suppose that T = Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. is a countable strongly closed iteration tree on V of limit length and that for all limit ǫ ≤ η, T |ǫ has at most one cofinal wellfounded branch. ◮ Then T has a cofinal wellfounded branch.

Three special cases: the suitable families

Definition Suppose that T is an iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. The iteration tree T is weakly suitable if T is a strongly closed iteration tree with κEγ ≤ ιEγ < κ∗

Eγ ≤ LTH(Eα)

for all γ + 1 < η, and such that for all β + 1 <T α < α + 1 < η: ◮ κ∗

Eβ ≤ κEα.

◮ Either κ∗

Eβ = κEα or LTH(Eβ) < κα.

Suitable and Strongly Suitable iteration trees

Definition Suppose that T is a weakly suitable iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. Then ◮ The iteration tree T is suitable if for all α + 1 < η, if α∗ < α then ιEα < min{κ∗

Eǫ

α∗ ≤ ǫ < α}, where α∗ is the ≤T -predecessor of α + 1.

Suitable and Strongly Suitable iteration trees

Definition Suppose that T is a weakly suitable iteration tree on V with associated sequence, Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α. Then ◮ The iteration tree T is suitable if for all α + 1 < η, if α∗ < α then ιEα < min{κ∗

Eǫ

α∗ ≤ ǫ < α}, where α∗ is the ≤T -predecessor of α + 1. ◮ The iteration T is strongly suitable if T is suitable and

LTH(Eα) ≤ κ∗

Eα

for all α + 1 < η.

The unique branch hypotheses

Definition (Strong-UBH) Suppose that T is a countable weakly suitable iteration tree on V

• f limit length.

◮ Then T has at most one cofinal maximal wellfounded branch.

The unique branch hypotheses

Definition (Strong-UBH) Suppose that T is a countable weakly suitable iteration tree on V

• f limit length.

◮ Then T has at most one cofinal maximal wellfounded branch. Definition (Suitable-UBH) Suppose that T is a countable suitable iteration tree on V of limit length. ◮ Then T has at most one cofinal maximal wellfounded branch.

The unique branch hypotheses

Definition (Strong-UBH) Suppose that T is a countable weakly suitable iteration tree on V

• f limit length.

◮ Then T has at most one cofinal maximal wellfounded branch. Definition (Suitable-UBH) Suppose that T is a countable suitable iteration tree on V of limit length. ◮ Then T has at most one cofinal maximal wellfounded branch. Definition (Weak-UBH) Suppose that T is a countable strongly suitable iteration tree on V

• f limit length.

◮ Then T has at most one cofinal maximal wellfounded branch.

The first branch counterexample

Theorem Suppose that there is a supercompact cardinal. Then there exist an extender E such that νE =

• 22κME

where κ = κE and ME = Ult0(V , E), and a strongly suitable iteration tree T = Mα, Eβ, jγ,α : α < η, β + 1 < ω, γ <T α

• n ME of length ω such that:

◮ κEα > κ∗

E for all α < ω,

◮ T has two wellfounded branches.

The second branch counterexample

Theorem Suppose that there is a supercompact cardinal. Then there exist an extender E such that νE =

• 22κME

where κ = κE and ME = Ult0(V , E), and a strongly suitable iteration tree T = Mα, Eβ, jγ,α : α < η, β + 1 < ω, γ <T α

• n ME of length ω2 such that:

◮ κEα > κ∗

E for all α < ω2,

◮ T has only one cofinal branch and that branch is not wellfounded.

N-strong extenders

◮ An extender E is a suitable extender if

◮ ιE < κ∗

E ≤ LTH(E) = ρ(E)

◮ LTH(E) is strongly inaccessible.

N-strong extenders

◮ An extender E is a suitable extender if

◮ ιE < κ∗

E ≤ LTH(E) = ρ(E)

◮ LTH(E) is strongly inaccessible.

Definition Suppose that N is a proper class and that N+ is the theory of (V , N) with parameters from V .

• 1. A suitable extender E is N-strong if

jE(N) ∩ VLTH(E) = N ∩ VLTH(E).

N-strong extenders

◮ An extender E is a suitable extender if

◮ ιE < κ∗

E ≤ LTH(E) = ρ(E)

◮ LTH(E) is strongly inaccessible.

Definition Suppose that N is a proper class and that N+ is the theory of (V , N) with parameters from V .

• 1. A suitable extender E is N-strong if

jE(N) ∩ VLTH(E) = N ∩ VLTH(E).

• 2. A suitable extender E is completely-N-strong if

◮ E is N+-strong ◮ (VLTH(E), N ∩ VLTH(E)) ≺ (Vκ, N).

Strongly N-closed iteration trees

Definition Suppose that N is a proper class, N+ is the theory of (V , N) with parameters from V , and that T = Mα, Eβ, jγ,α : α < η, β + 1 < η, γ <T α, is a weakly suitable iteration tree on V . ◮ Then T is strongly N-closed if for all α + 1 < η, there exists E ∈ Mα such that

◮ E is a suitable extender in Mα. ◮ E is completely-j0,α(N)-strong in Mα. ◮ LTH(Eα) < LTH(E) and Eα = E|LTH(Eα). ◮ κ∗

Eα = κ∗ E

The N unique branch hypotheses

◮ Suppose that N+ is the theory of (V , N) with parameters. Definition (Strong-UBH(N) where N is a proper class) Suppose that T is a weakly suitable iteration tree on V and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch.

The N unique branch hypotheses

◮ Suppose that N+ is the theory of (V , N) with parameters. Definition (Strong-UBH(N) where N is a proper class) Suppose that T is a weakly suitable iteration tree on V and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch. Definition (Suitable-UBH(N) where N is a proper class) Suppose that T is a suitable iteration tree on V and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch.

The N unique branch hypotheses

◮ Suppose that N+ is the theory of (V , N) with parameters. Definition (Strong-UBH(N) where N is a proper class) Suppose that T is a weakly suitable iteration tree on V and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch. Definition (Suitable-UBH(N) where N is a proper class) Suppose that T is a suitable iteration tree on V and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch. Definition (Strong-UBH(N:ω) where N is a proper class) Suppose that T is a weakly suitable iteration tree on V of length ω and that T is strongly N-closed. ◮ Then T as at most one cofinal wellfounded branch.

Inner model theory predictions

Prediction 1 Suppose that δ is a huge cardinal. ◮ Strong-UBH(N) fails at δ for all N ⊂ Vδ.

Inner model theory predictions

Prediction 1 Suppose that δ is a huge cardinal. ◮ Strong-UBH(N) fails at δ for all N ⊂ Vδ. Prediction 2 Suppose that δ is a huge cardinal. ◮ Strong-UBH(N:ω) holds at δ for some N ⊂ Vδ.

Inner model theory predictions

Prediction 1 Suppose that δ is a huge cardinal. ◮ Strong-UBH(N) fails at δ for all N ⊂ Vδ. Prediction 2 Suppose that δ is a huge cardinal. ◮ Strong-UBH(N:ω) holds at δ for some N ⊂ Vδ. Prediction 3 Suppose that δ is a huge cardinal. ◮ Suitable-UBH(N) holds at δ for some N ⊂ Vδ. ◮ This also predicts that if there is a proper class of huge cardinals then the Ω Conjecture holds.

The cofinality of the universally Baire sets

Notation Suppose there is a proper class of Woodin cardinals. Then δ(Γ∞) = sup{ΘL(A,R) A ∈ Γ∞}.

The cofinality of the universally Baire sets

Notation Suppose there is a proper class of Woodin cardinals. Then δ(Γ∞) = sup{ΘL(A,R) A ∈ Γ∞}. ◮ Assume the Sealing Hypothesis or just: Γ∞ = P(R ∩ L(Γ∞, R). Then δ(Γ∞) = ΘL(Γ∞,R).

The cofinality of the universally Baire sets

Notation Suppose there is a proper class of Woodin cardinals. Then δ(Γ∞) = sup{ΘL(A,R) A ∈ Γ∞}. ◮ Assume the Sealing Hypothesis or just: Γ∞ = P(R ∩ L(Γ∞, R). Then δ(Γ∞) = ΘL(Γ∞,R). The cofinality question Suppose there is a proper class of Woodin cardinals. ◮ What is cof(δ(Γ∞))?

The cofinality of the universally Baire sets

Notation Suppose there is a proper class of Woodin cardinals. Then δ(Γ∞) = sup{ΘL(A,R) A ∈ Γ∞}. ◮ Assume the Sealing Hypothesis or just: Γ∞ = P(R ∩ L(Γ∞, R). Then δ(Γ∞) = ΘL(Γ∞,R). The cofinality question Suppose there is a proper class of Woodin cardinals. ◮ What is cof(δ(Γ∞))? Lemma Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

◮ Then cof(δ(Γ∞)) is ω1, ω2, or ω3.

Theorem Assume V = Ultimate-L. Then ◮ cof(δ(Γ∞)) = ω1 ◮ Suppose V [G] is a generic extension of V such that ωV [G]

1

= ωV

1 .

◮ Then V [G] | = cof(δ(Γ∞)) = ω1.

Theorem Assume V = Ultimate-L. Then ◮ cof(δ(Γ∞)) = ω1 ◮ Suppose V [G] is a generic extension of V such that ωV [G]

1

= ωV

1 .

◮ Then V [G] | = cof(δ(Γ∞)) = ω1.

Lemma Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals. Then the following are equivalent.

◮ cof(δ(Γ∞)) = ω3. ◮ δ(Γ∞) = ω3.

Theorem Assume V = Ultimate-L. Then ◮ cof(δ(Γ∞)) = ω1 ◮ Suppose V [G] is a generic extension of V such that ωV [G]

1

= ωV

1 .

◮ Then V [G] | = cof(δ(Γ∞)) = ω1.

Lemma Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals. Then the following are equivalent.

◮ cof(δ(Γ∞)) = ω3. ◮ δ(Γ∞) = ω3. ◮ Suppose that δ is supercompact and V [G] is a δ-cc extension in which δ = ω2. Then δ(Γ∞) < ω3 in V [G].

Martin’s Maximum and the universally Baire sets

Theorem Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

Suppose that A ⊂ R. Then the following are equivalent. ◮ A and R\A have scales which are ω1-universally Baire. ◮ A is universally Baire.

Martin’s Maximum and the universally Baire sets

Theorem Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

Suppose that A ⊂ R. Then the following are equivalent. ◮ A and R\A have scales which are ω1-universally Baire. ◮ A is universally Baire. Corollary Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

◮ Then Γ∞ is definable in H(c+).

Martin’s Maximum and the universally Baire sets

Theorem Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

Suppose that A ⊂ R. Then the following are equivalent. ◮ A and R\A have scales which are ω1-universally Baire. ◮ A is universally Baire. Corollary Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals.

◮ Then Γ∞ is definable in H(c+). Suppose Martin’s Maximum holds and that there is a proper class

• f Woodin cardinals. Then:

◮ The cofinality question is a first order question about H(c+).

The influence of cof(UB)

Theorem (Schindler, Woodin) Assume there is a proper class of Woodin cardinals and: ◮ Martin’s Maximum++. ◮ cof(δ(Γ∞)) > ω1. ◮ There exists a Woodin cardinal δ and a set E ⊂ Vδ

• f strongly closed extenders such that:

◮ E witnesses that δ is a Woodin cardinal. ◮ Every non-overlapping iteration tree on (V , E) has at most one cofinal wellfounded branch.

Then Axiom(∗) holds.

Martin’s Maximum++ and covering

Theorem Suppose there is a proper class of Woodin cardinals, MM++ holds, and that X ⊂ δ(Γ∞) is a bounded set with |Y | = ω1. Then there exists X ⊂ Y ⊂ δ(Γ∞) such that Y ∈ L(A, R) for some A ∈ Γ∞ such that |Y |L(A,R) = ω1.

More notation

Notation Suppose δ is an uncountable regular cardinal. Then U(Ord<ω: δ) denotes the class of all δ-complete ultrafilters U such that η<ω ∈ U for some ordinal η.

More notation

Notation Suppose δ is an uncountable regular cardinal. Then U(Ord<ω: δ) denotes the class of all δ-complete ultrafilters U such that η<ω ∈ U for some ordinal η. Notation Suppose σ ⊂ U(Ord<ω: δ) for some δ and that σ is countable. Suppose A ⊆ ωω. Then A <W σ if for all surjections f : ω → σ, A and R\A are each Wadge reducible to the set σf = {x ∈ ωω | f (x(i)) : i < ω is a wellfounded tower}.

Martin’s Maximum++ and the Strong Sealing Hypothesis

Theorem Suppose that the Strong Sealing Hypothesis holds and that MM++ holds. Suppose that F : ω1 → Γ∞. Then for all δ ∈ Ord, there exists H : ω1 → U(Ord<ω: δ) and a closed unbounded set C ⊂ ω1 such that for all γ ∈ C, F(γ) <W H[γ]. ◮ The conclusion is just the version of Tower Sealing for the nonstationary ideal on ω1.

The UB Cofinality Conjecture

Definition (UB Cofinality Conjecture) Suppose that there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that MM++ holds. ◮ Then provably cof(δ(Γ∞)) > ω1.

The UB Cofinality Conjecture

Definition (UB Cofinality Conjecture) Suppose that there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that MM++ holds. ◮ Then provably cof(δ(Γ∞)) > ω1. Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that MM++ holds. Suppose that ◮ Suitable-UBH(N) holds at δ for some N ⊆ Vδ, for some huge cardinal δ.

The UB Cofinality Conjecture

Definition (UB Cofinality Conjecture) Suppose that there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that MM++ holds. ◮ Then provably cof(δ(Γ∞)) > ω1. Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that MM++ holds. Suppose that ◮ Suitable-UBH(N) holds at δ for some N ⊆ Vδ, for some huge cardinal δ. ◮ Then Axiom (∗) holds.

Forcing sets to be universally Baire?

◮ Suppose the Strong Sealing Hypothesis holds.

◮ Then Tower Sealing holds at δ for a proper class of Woodin cardinals δ.

◮ But it is not clear if Tower Sealing must hold at all Woodin cardinals.

Forcing sets to be universally Baire?

◮ Suppose the Strong Sealing Hypothesis holds.

◮ Then Tower Sealing holds at δ for a proper class of Woodin cardinals δ.

◮ But it is not clear if Tower Sealing must hold at all Woodin cardinals.

Theorem Suppose that the Strong Sealing Hypothesis holds, δ is a Woodin cardinal, and that Tower Sealing fails at δ for the P<δ tower. ◮ Then there exist generic extensions V [G] ⊂ V [H]

• f V such that

◮ V [G]δ = V [H]δ ◮ δ is a Woodin cardinal in V [H] ◮ Γ∞

G = Γ∞ H .

Forcing a “new” universally Baire set

Lemma Suppose there is a proper class of Woodin cardinals and: ◮ δ(Γ∞) = ω2 ◮ CH holds. Suppose G is V -generic for Namba forcing. Then: ◮ RG = RV

◮ and so AG = A for all A ∈ Γ∞.

◮ {A A ∈ Γ∞} is not Wadge cofinal in Γ∞

G .

◮ The hypothesis of the lemma is consistent from the existence

• f a Woodin cardinal which is a limit of Woodin cardinals.

The key consequence of the UB Cofinality Conjecture

Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and there is a supercompact cardinal. Then: ◮ There is a semiproper partial order P such that if G ⊂ P is V -generic then {AG A ∈ Γ∞} is not Wadge cofinal in Γ∞

G .

The key consequence of the UB Cofinality Conjecture

Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and there is a supercompact cardinal. Then: ◮ There is a semiproper partial order P such that if G ⊂ P is V -generic then {AG A ∈ Γ∞} is not Wadge cofinal in Γ∞

G .

◮ Proving P exists is essentially equivalent to the UB Cofinality Conjecture.

The key consequence of the UB Cofinality Conjecture

Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and there is a supercompact cardinal. Then: ◮ There is a semiproper partial order P such that if G ⊂ P is V -generic then {AG A ∈ Γ∞} is not Wadge cofinal in Γ∞

G .

◮ Proving P exists is essentially equivalent to the UB Cofinality Conjecture. ◮ Proving P exists from just the existence of even a proper class

• f I0 cardinals (without the Strong Sealing Hypothesis) would:

◮ Refute the Ultimate-L Conjecture. ◮ Negatively answer the Inner Model Test Question.

Semiproper forcing and the Ω Conjecture

Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that κ is a supercompact cardinal. Suppose that for some huge cardinal δ > κ: ◮ Suitable-UBH(N) holds at δ for some N ⊆ Vδ.

Semiproper forcing and the Ω Conjecture

Theorem (UB Cofinality Conjecture) Suppose there is a proper class of huge cardinals, ◮ the Strong Sealing Hypothesis holds, and that κ is a supercompact cardinal. Suppose that for some huge cardinal δ > κ: ◮ Suitable-UBH(N) holds at δ for some N ⊆ Vδ. Suppose ϕ is a Σ2-sentence. ◮ Then the following are equivalent.

◮ ϕ is Ω-consistent. ◮ There is a partial order P such that V P | = ϕ. ◮ There is a semiproper partial order P such that V P | = ϕ.

◮ The equivalence of the first two sentences is just the Ω Conjecture.