Sealed Trees and the Perfect Subtree Property for Weakly Compact - - PowerPoint PPT Presentation

Sealed Trees and the Perfect Subtree Property for Weakly Compact Cardinals Sandra M¨ uller

Universit¨ at Wien

November 1, 2019 Joint with Yair Hayut

Rutgers MAMLS 2019

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 1

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Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 2

κ-Trees

Definition

Let κ be a regular cardinal. A tree T of height κ is called a normal κ-tree if each level of T has size <κ, every node splits, for every t ∈ T and α < κ above the height of t, there is some t′ of level α in T such that t <T t′, and for every limit ordinal α < κ and every branch up to α there is at most one least upper bound in T.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 3

The Branch Spectrum

Definition

Let κ be a regular cardinal. The Branch Spectrum of κ is the set Sκ = {|[T]| | T is a normal κ-tree}.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 4

The Branch Spectrum

Definition

Let κ be a regular cardinal. The Branch Spectrum of κ is the set Sκ = {|[T]| | T is a normal κ-tree}.

Examples

Sω = {2ℵ0}.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 4

The Branch Spectrum

Definition

Let κ be a regular cardinal. The Branch Spectrum of κ is the set Sκ = {|[T]| | T is a normal κ-tree}.

Examples

Sω = {2ℵ0}. For κ > ω, there are no κ-Kurepa trees iff max(Sκ) = κ.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 4

The Branch Spectrum

Definition

Let κ be a regular cardinal. The Branch Spectrum of κ is the set Sκ = {|[T]| | T is a normal κ-tree}.

Examples

Sω = {2ℵ0}. For κ > ω, there are no κ-Kurepa trees iff max(Sκ) = κ. For κ > ℵ1 of uncountable cofinalty, tree property holds at κ iff min(Sκ) = κ.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 4

Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound max(Sκ) = κ or κ+ / ∈ Sκ inaccessible cardinal

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 5

Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound max(Sκ) = κ or κ+ / ∈ Sκ inaccessible cardinal min(Sκ) = κ weakly compact cardinal

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 5

Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound max(Sκ) = κ or κ+ / ∈ Sκ inaccessible cardinal min(Sκ) = κ weakly compact cardinal κ+ / ∈ Sκ and min(Sκ) = κ ?

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 5

Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound max(Sκ) = κ or κ+ / ∈ Sκ inaccessible cardinal min(Sκ) = κ weakly compact cardinal κ+ / ∈ Sκ and min(Sκ) = κ ? The following gives an upper bound.

Proposition

Let κ be <µ-supercompact, where µ is strongly inaccessible. Then, there is a forcing extension in which κ is weakly compact, Sκ = {κ, κ++}. Proof idea: Consider Col(κ, <µ) × Add(κ, µ+).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 5

Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound max(Sκ) = κ or κ+ / ∈ Sκ inaccessible cardinal min(Sκ) = κ weakly compact cardinal κ+ / ∈ Sκ and min(Sκ) = κ ? The following gives an upper bound.

Proposition

Let κ be <µ-supercompact, where µ is strongly inaccessible. Then, there is a forcing extension in which κ is weakly compact, Sκ = {κ, κ++}. Proof idea: Consider Col(κ, <µ) × Add(κ, µ+).

Question

Is this optimal?

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 5

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Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 6

A First Lower Bound and Sealed Trees

If for some weakly compact cardinal κ, κ+ / ∈ Sκ then 0# exists:

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

A First Lower Bound and Sealed Trees

If for some weakly compact cardinal κ, κ+ / ∈ Sκ then 0# exists:

Fact (essentially Solovay)

If 0# does not exists then every weakly compact cardinal carries a tree with κ+ many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

A First Lower Bound and Sealed Trees

If for some weakly compact cardinal κ, κ+ / ∈ Sκ then 0# exists:

Fact (essentially Solovay)

If 0# does not exists then every weakly compact cardinal carries a tree with κ+ many branches. The tree actually has the following stronger property.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

A First Lower Bound and Sealed Trees

If for some weakly compact cardinal κ, κ+ / ∈ Sκ then 0# exists:

Fact (essentially Solovay)

If 0# does not exists then every weakly compact cardinal carries a tree with κ+ many branches. The tree actually has the following stronger property.

Definition

Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf(κ) > ω.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

A First Lower Bound and Sealed Trees

The tree actually has the following stronger property.

Definition

Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf(κ) > ω. Strongly sealed trees with κ many branches exist in ZFC: Take T ⊆ 2<κ to be the tree of all x such that {α ∈ dom(x) | x(α) = 1} is finite.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

A First Lower Bound and Sealed Trees

The tree actually has the following stronger property.

Definition

Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf(κ) > ω. Strongly sealed trees with κ many branches exist in ZFC: Take T ⊆ 2<κ to be the tree of all x such that {α ∈ dom(x) | x(α) = 1} is finite.

Question

How about strongly sealed κ-trees with at least κ+ many branches?

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 7

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Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 8

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Construction of the Tree

Let T(Kκ+) be the following tree:

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Construction of the Tree

Let T(Kκ+) be the following tree: Nodes: ¯ M, ¯ x, where ¯ M = trcl(HullKκ+(ρ ∪ {x})) for some ρ < κ, x ∈ Kκ+ ∩ κ2 and x collapses to ¯ x.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Construction of the Tree

Let T(Kκ+) be the following tree: Nodes: ¯ M, ¯ x, where ¯ M = trcl(HullKκ+(ρ ∪ {x})) for some ρ < κ, x ∈ Kκ+ ∩ κ2 and x collapses to ¯ x. Tree order: M0, x0 ≤ M1, x1 if there is some ordinal ρ such that M0 = trcl(HullM1(ρ ∪ {x1})) and x1 collapses to x0.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Construction of the Tree

Let T(Kκ+) be the following tree: Nodes: ¯ M, ¯ x, where ¯ M = trcl(HullKκ+(ρ ∪ {x})) for some ρ < κ, x ∈ Kκ+ ∩ κ2 and x collapses to ¯ x. Tree order: M0, x0 ≤ M1, x1 if there is some ordinal ρ such that M0 = trcl(HullM1(ρ ∪ {x1})) and x1 collapses to x0.

Claim

T(Kκ+) is a tree of height κ with at least (κ+)K many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K. Argue that each branch in V is in fact already in K, so |[T]| = (κ+)K (use maximality and universality of K).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable. Compare Rb and KV :

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable. Compare Rb and KV :

KV wins by universality.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable. Compare Rb and KV :

KV wins by universality. Rb does not move by maximality.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable. Compare Rb and KV :

KV wins by universality. Rb does not move by maximality.

Let K∞ be the iterate of K with Rb K∞, so b ∈ K∞.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Every branch in V is already in K

Claim

Each branch through T in V is already in K, so |[T]| = (κ+)K. Let b be a branch through T and Rb the direct limit of the models on the branch. ρω(Rb) = κ. Rb is countably iterable. Compare Rb and KV :

KV wins by universality. Rb does not move by maximality.

Let K∞ be the iterate of K with Rb K∞, so b ∈ K∞. A more careful analysis yields b ∈ K.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K. Argue that each branch in V is in fact already in K, so |[T]| = (κ+)K (use maximality and universality of K).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K. Argue that each branch in V is in fact already in K, so |[T]| = (κ+)K (use maximality and universality of K). Use forcing absoluteness to see that T is strongly sealed.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every inaccessible cardinal κ, there is a strongly sealed κ-tree with exactly (κ+)K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ+ many branches. Proof idea: Construct a κ-tree T in K with |[T]| ≥ (κ+)K. Argue that each branch in V is in fact already in K, so |[T]| = (κ+)K (use maximality and universality of K). Use forcing absoluteness to see that T is strongly sealed. Use covering to obtain (κ+)K = (κ+)V .

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 9

Sealed Trees in the context of Woodin cardinals

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 10

Sealed Trees in the context of Woodin cardinals

Observation

Strongly sealed κ-trees with κ+ many branches cannot exist in the context

• f a Woodin cardinal δ > κ.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 10

Sealed Trees in the context of Woodin cardinals

Observation

Strongly sealed κ-trees with κ+ many branches cannot exist in the context

• f a Woodin cardinal δ > κ.

Why? Woodin’s stationary tower forcing with critical point κ+ will introduce new branches to any κ-tree T, while preserving the regularity of κ, as well as many large cardinal properties of κ.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 10

The κ-Perfect Subtree Property

The following lemma yields a canonical weakening of being strongly sealed.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 11

The κ-Perfect Subtree Property

The following lemma yields a canonical weakening of being strongly sealed.

Lemma (Folklore)

Let κ be a cardinal. The following are equivalent for a tree T of height κ:

1 T has a perfect subtree. 2 Every set forcing that adds a fresh subset to κ also adds a branch to

T.

3 There is a κ-closed forcing that adds a branch to T. Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 11

The κ-Perfect Subtree Property

The following lemma yields a canonical weakening of being strongly sealed.

Lemma (Folklore)

Let κ be a cardinal. The following are equivalent for a tree T of height κ:

1 T has a perfect subtree. 2 Every set forcing that adds a fresh subset to κ also adds a branch to

T.

3 There is a κ-closed forcing that adds a branch to T.

Definition

Let κ be an uncountable cardinal. The Perfect Subtree Property (PSP) for κ is the statement that every κ-tree with at least κ+ many branches has a perfect subtree.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 11

The κ-Perfect Subtree Property

Definition

Let κ be an uncountable cardinal. The Perfect Subtree Property (PSP) for κ is the statement that every κ-tree with at least κ+ many branches has a perfect subtree. What is the consistency strength of this statement?

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 11

The κ-Perfect Subtree Property

Definition

Let κ be an uncountable cardinal. The Perfect Subtree Property (PSP) for κ is the statement that every κ-tree with at least κ+ many branches has a perfect subtree. What is the consistency strength of this statement?

Proposition

Let κ be < µ-supercompact, where µ is strongly inaccessible. Then, there is a forcing extension in which κ is weakly compact, Sκ = {κ, κ++} and the Perfect Subtree Property holds at κ. Proof idea: Consider Col(κ, <µ) × Add(κ, µ+).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 11

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Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 12

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR there is no κ-tree with exactly κ+ many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR there is no κ-tree with exactly κ+ many branches. Then there is a non-domestic mouse. In particular, there is a model of ZF + DC + ADR.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR there is no κ-tree with exactly κ+ many branches. Then there is a non-domestic mouse. In particular, there is a model of ZF + DC + ADR. Proof idea: Consider the tree T(S) for S = S(κ) the stack of mice on Kc||κ (cf. Andretta-Neeman-Steel and Jensen-Schimmerling-Schindler-Steel).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR there is no κ-tree with exactly κ+ many branches. Then there is a non-domestic mouse. In particular, there is a model of ZF + DC + ADR. Proof idea: Consider the tree T(S) for S = S(κ) the stack of mice on Kc||κ (cf. Andretta-Neeman-Steel and Jensen-Schimmerling-Schindler-Steel). T(S) has exactly (κ+)V many branches (using covering as in JSSS).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

A Non-domestic Mouse from the κ-PSP

Theorem (Hayut, M.)

Let κ be a weakly compact cardinal and let us assume that the Perfect Subtree Property holds at κ OR there is no κ-tree with exactly κ+ many branches. Then there is a non-domestic mouse. In particular, there is a model of ZF + DC + ADR. Proof idea: Consider the tree T(S) for S = S(κ) the stack of mice on Kc||κ (cf. Andretta-Neeman-Steel and Jensen-Schimmerling-Schindler-Steel). T(S) has exactly (κ+)V many branches (using covering as in JSSS). T(S) does not have a perfect subtree (argue that set of branches does not change in an Add(κ, 1)-generic extension of V ).

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 13

How sealed is the tree?

Definition (Neeman, Steel)

Let κ be a regular cardinal. A normal tree T of height κ is sealed if the set

• f branches of T cannot be modified by set forcing P satisfying the

following properties:

1 P × P does not collapse κ, 2 P × P preserves cf(κ) > ω, and 3 P does not add any new sets of reals. Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 14

How sealed is the tree?

Definition (Neeman, Steel)

Let κ be a regular cardinal. A normal tree T of height κ is sealed if the set

• f branches of T cannot be modified by set forcing P satisfying the

following properties:

1 P × P does not collapse κ, 2 P × P preserves cf(κ) > ω, and 3 P does not add any new sets of reals.

Theorem

Let κ be a weakly compact cardinal and assume that there is no non-domestic premouse. Then there is a sealed κ-tree with exactly κ+ many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 14

Looking beyond

Definition (Neeman, Steel)

A cardinal κ is Π1

1-Woodin, if for every A ⊆ Vκ, and for every

Π1

1-statement Φ for which Vκ, ∈, A |

= Φ, there is a <κ-A-strong cardinal µ such that Vµ, ∈, A ∩ Vµ | = Φ. Note: The least Π1

1-Woodin cardinal is below the least Shelah cardinal.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 15

Looking beyond

Definition (Neeman, Steel)

A cardinal κ is Π1

1-Woodin, if for every A ⊆ Vκ, and for every

Π1

1-statement Φ for which Vκ, ∈, A |

= Φ, there is a <κ-A-strong cardinal µ such that Vµ, ∈, A ∩ Vµ | = Φ. Note: The least Π1

1-Woodin cardinal is below the least Shelah cardinal.

Theorem

Let κ be Π1

1-Woodin and assume SBHκ. Then either κ is

Π1

1-κ+-subcompact in an inner model or there is a sealed κ-tree on κ with

κ+ many branches.

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 15

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Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 16

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Conjecture

Let κ be a weakly compact cardinal with the Perfect Subtree Property. Then there is an inner model with a pair of cardinals λ < µ such that λ is <µ-supercompact and µ is inaccessible.

https://arxiv.org/abs/1910.05159

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 17

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n d a n d S e a l e d T r e e s B e l

• w

W

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i n c a r d i

• n

a l s A N

• n
• d
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e s t i c M

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s e “Never say there is nothing beautiful in the world anymore. There is always something to make you wonder in the shape of a tree, the trembling of a leaf.”

Albert Schweitzer

https://arxiv.org/abs/1910.05159

Sandra M¨ uller (Universit¨ at Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 18