Lower Bounds for the Perfect Subtree Property at Weakly Compact - - PowerPoint PPT Presentation

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Lower Bounds for the Perfect Subtree Property at Weakly Compact Cardinals Sandra M uller Universit at Wien September 23, 2019 Joint with Yair Hayut 15th International Luminy Workshop in Set Theory Sandra M uller (Universit at

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Lower Bounds for the Perfect Subtree Property at Weakly Compact Cardinals Sandra M¨ uller

Universit¨ at Wien

September 23, 2019 Joint with Yair Hayut

15th International Luminy Workshop in Set Theory

Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 1

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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, each level has at least one split, for every limit ordinal α < κ and every branch up to α there is at most one least upper bound in T, and 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′.

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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}.

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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 = {ℵ0, 2ℵ0}.

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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 = {ℵ0, 2ℵ0}. For κ > ω, if there are no κ-Kurepa trees, then κ+ / ∈ Sκ.

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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 = {ℵ0, 2ℵ0}. For κ > ω, if there are no κ-Kurepa trees, then κ+ / ∈ Sκ. For κ > ℵ1, if the tree property holds at κ, then min(Sκ) = κ.

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Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound κ+ / ∈ Sκ inaccessible cardinal

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Upper Bounds

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

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Upper Bounds

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

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Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound 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+).

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Upper Bounds

Let κ > ℵ1. Branch Spectrum Upper bound 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?

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A First Lower Bound and Sealed Trees

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

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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.

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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.

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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.

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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.

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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?

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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.

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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.

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Construction of the Tree

Let T(Kκ+) be the following tree:

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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.

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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.

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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.

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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.

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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).

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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.

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A Sealed Tree in K

Theorem (Hayut, M.)

Let us assume that there is no inner model with a Woodin cardinal. Then for every regular 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 .

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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.

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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.

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The κ-Perfect Subtree Property

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

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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) Perfect Subtree Property at Weakly Compacts September 23, 2019 11

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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.

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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?

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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+).

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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

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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.

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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 + ADR.

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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 + 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).

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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 + 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).

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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 + 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 ).

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S P “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

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