Sealed Trees and the Perfect Subtree Property for Weakly ...smueller/Slides/201911... · For >!,...

Sealed Trees and the Perfect Subtree Property forWeakly Compact Cardinals

Sandra Muller

Universitat Wien

November 1, 2019

Joint with Yair Hayut

Rutgers MAMLS 2019

Sandra Muller (Universitat Wien) Sealed Trees and PSP for Weakly Compacts November 1, 2019 1

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

κ-Trees

Definition

Let κ be a regular cardinal. A tree T of height κ is called a normal κ-treeif

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′ oflevel α in T such that t <T t

′, and

for every limit ordinal α < κ and every branch up to α there is atmost one least upper bound in T .

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 iffmin(Sκ) = κ.

The Branch Spectrum

Definition

Examples

Sω = {2ℵ0}.

For κ > ω, there are no κ-Kurepa trees iff max(Sκ) = κ.

The Branch Spectrum

Definition

Examples

The Branch Spectrum

Definition

Examples

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, thereis 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?

Upper Bounds

Let κ > ℵ1.

Proposition

Question

Is this optimal?

Upper Bounds

Let κ > ℵ1.

Proposition

Question

Is this optimal?

Upper Bounds

Let κ > ℵ1.

Proposition

Question

Is this optimal?

Upper Bounds

Let κ > ℵ1.

Proposition

Question

Is this optimal?

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

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 treewith 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 sealedif the set of branches of T cannot be modified by set forcing that forcescf(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?

Definition

Question

Definition

Question

Definition

Question

Definition

Question

Definition

Question

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Thenfor every inaccessible cardinal κ, there is a strongly sealed κ-tree withexactly (κ+)

Kmany 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 .

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

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 thatM0 = trcl(HullM1(ρ ∪ {x1})) and x1 collapses to x0.

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

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

Every branch in V is already in K

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 thebranch.

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 EK∞, so b ∈ K∞.

A more careful analysis yields b ∈ K.

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

KV wins by universality.

Rb does not move by maximality.

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

ρω(Rb) = κ.

Compare Rb and KV :

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

A Sealed Tree in K

Theorem (Hayut, M.)

Proof idea:

Sealed Trees in the context of Woodin cardinals

Observation

Strongly sealed κ-trees with κ+ many branches cannot exist in the contextof a Woodin cardinal δ > κ.

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

Observation

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 toT .

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) forκ is the statement that every κ-tree with at least κ+ many branches has aperfect subtree.

What is the consistency strength of this statement?

Proposition

Let κ be < µ-supercompact, where µ is strongly inaccessible. Then, thereis a forcing extension in which κ is weakly compact, Sκ = {κ, κ++} andthe Perfect Subtree Property holds at κ.

Lemma (Folklore)

Definition

Proposition

Lemma (Folklore)

Definition

Proposition

Definition

Proposition

Definition

Proposition

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

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 ofZF + 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 branchesdoes not change in an Add(κ, 1)-generic extension of V ).

Theorem (Hayut, M.)

Proof idea:

Theorem (Hayut, M.)

Proof idea:

Theorem (Hayut, M.)

Proof idea:

Theorem (Hayut, M.)

Proof idea:

Theorem (Hayut, M.)

Proof idea:

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 setof branches of T cannot be modified by set forcing P satisfying thefollowing 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 nonon-domestic premouse. Then there is a sealed κ-tree with exactly κ+

many branches.

How sealed is the tree?

Let κ be a regular cardinal. A normal tree T of height κ is sealed if the setof branches of T cannot be modified by set forcing P satisfying thefollowing 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 nonon-domestic premouse. Then there is a sealed κ-tree with exactly κ+

many branches.

Looking beyond

A cardinal κ is Π11-Woodin, if for every A ⊆ Vκ, and for every

Π11-statement Φ for which 〈Vκ,∈, A〉 |= Φ, there is a <κ-A-strong cardinal

µ such that 〈Vµ,∈, A ∩ Vµ〉 |= Φ.

Note: The least Π11-Woodin cardinal is below the least Shelah cardinal.

Theorem

Let κ be Π11-Woodin and assume SBHκ. Then either κ is

Π11-κ

+-subcompact in an inner model or there is a sealed κ-tree on κ withκ+ many branches.

Looking beyond

A cardinal κ is Π11-Woodin, if for every A ⊆ Vκ, and for every

Π11-statement Φ for which 〈Vκ,∈, A〉 |= Φ, there is a <κ-A-strong cardinal

µ such that 〈Vµ,∈, A ∩ Vµ〉 |= Φ.

Note: The least Π11-Woodin cardinal is below the least Shelah cardinal.

Theorem

Let κ be Π11-Woodin and assume SBHκ. Then either κ is

Π11-κ

+-subcompact in an inner model or there is a sealed κ-tree on κ withκ+ many branches.

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

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

κ-Trees and Upper

Bounds

A First Lower Bound

and Sealed Trees

BelowWoodin cardi-

A Non-domestic

“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

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