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 ⊆ R n is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → R n ; 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 ⊆ R n is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → R n ; 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 = R n and π to be the identity.

Universally Baire sets Definition (Feng-Magidor-Woodin) A set A ⊆ R n is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → R n ; 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 = R n and π to be the identity. ◮ Universally Baire sets are Lebesgue measurable.

Universally Baire sets Definition (Feng-Magidor-Woodin) A set A ⊆ R n is universally Baire if: ◮ For all topological spaces Ω ◮ For all continuous functions π : Ω → R n ; 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 = R n 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 of 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 of 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.

HOD L ( A , R ) and measurable cardinals Definition Suppose that A ⊆ R . Then HOD L ( A , R ) is the class HOD as defined within L ( A , R ). ◮ The Axiom of Choice must hold in HOD L ( A , R ) ◮ even if L ( A , R ) | = AD .

HOD L ( A , R ) and measurable cardinals Definition Suppose that A ⊆ R . Then HOD L ( A , R ) is the class HOD as defined within L ( A , R ). ◮ The Axiom of Choice must hold in HOD L ( A , R ) ◮ even if L ( A , R ) | = AD . Theorem (Solovay:1967) Suppose that A ⊆ R and L ( A , R ) | = AD . ◮ Then ω V 1 is a measurable cardinal in HOD L ( A , R ) . ◮ Solovay’s theorem gave the first connection between the Axiom of Determinacy ( AD ) and large cardinal axioms.

The least measurable cardinal of HOD L ( A , R ) Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. 1 is the least measurable cardinal in HOD L ( A , R ) . ◮ Then ω V

The least measurable cardinal of HOD L ( A , R ) Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. 1 is the least measurable cardinal in HOD L ( A , R ) . ◮ Then ω V ◮ If stronger large cardinals exist in HOD L ( A , R ) , they can only occur above ω V 1 .

The least measurable cardinal of HOD L ( A , R ) Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. 1 is the least measurable cardinal in HOD L ( A , R ) . ◮ Then ω V ◮ If stronger large cardinals exist in HOD L ( A , R ) , they can only occur above ω V 1 . Definition Suppose that A ⊆ R is universally Baire. Then ◮ Θ L ( A , R ) is the supremum of the ordinals α such that there exists a surjection, π : R → α , such that π ∈ L ( A , R ).

HOD L ( A , R ) and Woodin cardinals Lemma Suppose that A ⊆ R . Then: ◮ There are no measurable cardinals κ in HOD L ( A , R ) such that κ ≥ Θ L ( A , R ) .

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

HOD L ( A , R ) and Woodin cardinals Lemma Suppose that A ⊆ R . Then: ◮ There are no measurable cardinals κ in HOD L ( A , R ) such that κ ≥ Θ L ( A , R ) . Theorem Suppose that there is a proper class of Woodin cardinals and that A is universally Baire. Then: ◮ Θ L ( A , R ) is a Woodin cardinal in HOD L ( A , R ) . The Inner Model Test Question Suppose there is a proper class of Woodin cardinals. Suppose ϕ is a Σ 2 -sentence defining a large cardinal axiom. ◮ Can HOD L ( 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 HOD L ( A , R ) such that for all Σ 2 -sentences ϕ : = ϕ if and only if HOD L ( A , R ) | Θ L ( A , R ) | ◮ V | = 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 HOD L ( A , R ) such that for all Σ 2 -sentences ϕ : = ϕ if and only if HOD L ( A , R ) | Θ L ( A , R ) | ◮ V | = 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 HOD L ( A , R ) such that for all Σ 2 -sentences ϕ : = Ω ϕ if and only if HOD L ( A , R ) | Θ L ( A , R ) | ◮ V | = 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 ◮ R g denotes R V [ g ] . ◮ Γ ∞ denotes Γ ∞ ) V [ g ] . 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 V δ +1 is countable in V [ G ] . Then the following hold. (1) Γ ∞ G = P ( R G ) ∩ L (Γ ∞ G , R G ) . (2) Suppose that γ is a limit of Woodin cardinals in V and that G is V -generic for some partial P ∈ V γ . Then (Γ ∞ ) V γ [ 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 V δ +1 is countable in V [ G ] . Then the following hold. (1) Γ ∞ G = P ( R G ) ∩ L (Γ ∞ G , R G ) . (2) Suppose that γ is a limit of Woodin cardinals in V and that G is V -generic for some partial P ∈ V γ . Then (Γ ∞ ) V γ [ G ] = Γ ∞ G . (3) Γ ∞ H = P ( R H ) ∩ L (Γ ∞ H , R H ) . (4) There is an elementary embedding j : L (Γ ∞ G , R G ) → L (Γ ∞ H , R H ) G , j ( A ) = ( A ) V [ H ] , where ( A ) V [ H ] is such that for all A ∈ Γ ∞ 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 V ω +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 V ω +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 ).