A microscopic approach to Souslin trees constructions Forcing and - - PowerPoint PPT Presentation

A microscopic approach to Souslin trees constructions

Forcing and its Applications Retrospective Workshop The Fields Institute, Toronto, Canada 01-April-2015 Assaf Rinot Bar-Ilan University

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This is joint work with Ari M. Brodsky, and still in progress..

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ}

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1 ◮ CHλ asserts that 2λ = λ+

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1 ◮ CHλ asserts that 2λ = λ+ ◮ acc(C) = {α ∈ C | sup(C ∩ α) = α > 0}

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1 ◮ CHλ asserts that 2λ = λ+ ◮ acc(C) = {α ∈ C | sup(C ∩ α) = α > 0} ◮ nacc(C) = C \ acc(C)

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1 ◮ CHλ asserts that 2λ = λ+ ◮ acc(C) = {α ∈ C | sup(C ∩ α) = α > 0} ◮ nacc(C) = C \ acc(C) ◮ succσ(C) = {α ∈ C | otp(C ∩ α) = j + 1 for some j < σ}

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Some conventions

◮ E κ θ = {α < κ | cf(α) = θ} ◮ CH asserts that 2ℵ0 = ℵ1 ◮ CHλ asserts that 2λ = λ+ ◮ acc(C) = {α ∈ C | sup(C ∩ α) = α > 0} ◮ nacc(C) = C \ acc(C) ◮ succσ(C) = {α ∈ C | otp(C ∩ α) = j + 1 for some j < σ}

e.g., succ3(ω1 \ ω) = {ω + 1, ω + 2, ω + 3}.

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

◮ Tδ = {x ∈ T | otp(x↓, ⊳) = δ} is the δth-level of T ;

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

◮ Tδ = {x ∈ T | otp(x↓, ⊳) = δ} is the δth-level of T ; ◮ The height of T is min{δ ∈ Ord | Tδ = ∅};

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

◮ Tδ = {x ∈ T | otp(x↓, ⊳) = δ} is the δth-level of T ; ◮ The height of T is min{δ ∈ Ord | Tδ = ∅}; ◮ T is χ−complete if any ⊳-increasing sequence of length < χ

admits a bound;

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

◮ Tδ = {x ∈ T | otp(x↓, ⊳) = δ} is the δth-level of T ; ◮ The height of T is min{δ ∈ Ord | Tδ = ∅}; ◮ T is χ−complete if any ⊳-increasing sequence of length < χ

admits a bound;

◮ T is χ−slim if |Tα| = |α| whenever α ≥ χ.

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κ-trees

Definition

◮ A tree is a poset T = T, ⊳ in which x↓ := {y ∈ T | y ⊳ x}

is well-ordered for all x ∈ T;

◮ Tδ = {x ∈ T | otp(x↓, ⊳) = δ} is the δth-level of T ; ◮ The height of T is min{δ ∈ Ord | Tδ = ∅}; ◮ T is χ−complete if any ⊳-increasing sequence of length < χ

admits a bound;

◮ T is χ−slim if |Tα| = |α| whenever α ≥ χ.

Definition

◮ A κ-tree is a tree of height κ whose levels are of size < κ; ◮ A κ-Aronszajn tree is a κ-tree having no branches of size κ; ◮ A κ-Souslin tree is a κ-Aronszajn tree having no antichains of

size κ.

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The role of κ

Aronszajn and Souslin trees are useful objects that give rise to rich counterexamples in mathematics. The literature concerning these trees splits roughly into two: ◮ Papers that deal with the construction of Aronszajn/Souslin trees with some additional features. ◮ Papers that deal with the construction of the trees from weaker and weaker hypotheses, or consistency results concerning non-existence.

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The role of κ

Aronszajn and Souslin trees are useful objects that give rise to rich counterexamples in mathematics. The literature concerning these trees splits roughly into two: ◮ Papers that deal with the construction of Aronszajn/Souslin trees with some additional features. ◮ Papers that deal with the construction of the trees from weaker and weaker hypotheses, or consistency results concerning non-existence. We shall now dedicate a few minutes to review some known results, highlighting that the behavior of κ-Aronszajn and κ-Souslin trees depends heavily on the identity of κ.

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κ-Aronszajn trees

Theorem (K¨

• nig, 1927)

There exists no ℵ0-Aronszajn tree.

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κ-Aronszajn trees

Theorem (K¨

• nig, 1927)

There exists no ℵ0-Aronszajn tree.

Theorem (Aronszajn, 1935)

There exists an ℵ1-Aronszajn tree.

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κ-Aronszajn trees

Theorem (K¨

• nig, 1927)

There exists no ℵ0-Aronszajn tree.

Theorem (Specker, 1949. λ = ω is due to Aronszajn, 1935)

If λ is regular and λ<λ = λ, then there exists a λ+-Aronszajn tree.

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κ-Aronszajn trees

Theorem (K¨

• nig, 1927)

There exists no ℵ0-Aronszajn tree.

Theorem (Specker, 1949. λ = ω is due to Aronszajn, 1935)

If λ is regular and λ<λ = λ, then there exists a λ+-Aronszajn tree.

Theorem (Magidor-Shelah, 1996)

Modulo large cardinals, it is consistent with GCH, that for some singular cardinal λ, there exists no λ+-Aronszajn tree.

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κ-Aronszajn trees

Theorem (K¨

• nig, 1927)

There exists no ℵ0-Aronszajn tree.

Theorem (Specker, 1949. λ = ω is due to Aronszajn, 1935)

If λ is regular and λ<λ = λ, then there exists a λ+-Aronszajn tree.

Theorem (Magidor-Shelah, 1996)

Modulo large cardinals, it is consistent with GCH, that for some singular cardinal λ, there exists no λ+-Aronszajn tree.

Theorem (Erd˝

• s-Taski, 1943)

If κ is weakly compact, then there exists no κ-Aronszajn tree.

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λ+-Souslin trees

Definition (Jensen, 1972)

For S ⊆ κ, ♦(S) asserts the existence of a sequence Aα | α ∈ S such that {α ∈ S | A ∩ α = Aα} is stationary for all A ⊆ κ.

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λ+-Souslin trees

Definition (Jensen, 1972)

For S ⊆ κ, ♦(S) asserts the existence of a sequence Aα | α ∈ S such that {α ∈ S | A ∩ α = Aα} is stationary for all A ⊆ κ.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

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λ+-Souslin trees

Definition (Jensen, 1972)

For S ⊆ κ, ♦(S) asserts the existence of a sequence Aα | α ∈ S such that {α ∈ S | A ∩ α = Aα} is stationary for all A ⊆ κ.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree. This gives a method to construct Souslin tree at the level of successor of regulars. How to handle successor of singulars?

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λ+-Souslin trees

Definition (Jensen, 1972)

For S ⊆ κ, ♦(S) asserts the existence of a sequence Aα | α ∈ S such that {α ∈ S | A ∩ α = Aα} is stationary for all A ⊆ κ.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

Definition (Jensen, 1972)

λ(S) asserts the existence of a sequence Cδ | δ < λ+ such that for all limit δ < λ+:

◮ Cδ is a club in δ of order-type ≤ λ; ◮ if β ∈ acc(Cδ), then β ∈ S and Cδ ∩ β = Cβ.

Write λ for λ(∅).

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λ+-Souslin trees

Definition (Jensen, 1972)

For S ⊆ κ, ♦(S) asserts the existence of a sequence Aα | α ∈ S such that {α ∈ S | A ∩ α = Aα} is stationary for all A ⊆ κ.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

Definition (Jensen, 1972)

λ(S) asserts the existence of a sequence Cδ | δ < λ+ such that for all limit δ < λ+:

◮ Cδ is a club in δ of order-type ≤ λ; ◮ if β ∈ acc(Cδ), then β ∈ S and Cδ ∩ β = Cβ.

Theorem (Jensen, 1972)

If there exists S ⊆ λ+ for which λ(S) + ♦(S) holds, then there exists a λ+-Souslin tree.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

Note

◮ A special λ+-tree is λ+-Aronszajn; ◮ A λ+-Souslin tree is non-special.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

Note

◮ A special λ+-tree is λ+-Aronszajn; ◮ A λ+-Souslin tree is non-special.

Remark

Aronszajn’s and Specker’s constructions from λ<λ = λ may be steered to yield a special λ+-tree.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

Definition

A λ+-tree is specializable if it is special in some extended universe

• f ZFC with the same cardinal structure.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

Definition

A λ+-tree is specializable if it is special in some extended universe

• f ZFC with the same cardinal structure.

Theorem (Baumgartner-Mailtz-Reinhardt, 1970)

An ℵ1-tree is Aronszajn iff it is specializable.

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Special and specializable λ+-trees

Definition

A λ+-tree is special if it is the union of λ many antichains.

Definition

A λ+-tree is specializable if it is special in some extended universe

• f ZFC with the same cardinal structure.

Theorem (Baumgartner-Mailtz-Reinhardt, 1970)

An ℵ1-tree is Aronszajn iff it is specializable.

Theorem (implicit in David, 1990)

If V = L, then for every regular λ, the canonical λ-complete λ+-Souslin tree constructed using fine structure, is specializable.

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Non-specializable λ+-Souslin trees

Theorem (Baumgartner, 1970’s, building on Laver)

ℵ1 entails a non-specializable ℵ2-Aronszajn tree.

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Non-specializable λ+-Souslin trees

Theorem (Baumgartner, 1980’s, improving Devlin)

GCH +ℵ1 entails a non-specializable ℵ2-Souslin tree.

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Non-specializable λ+-Souslin trees

Theorem (Baumgartner, 1980’s, improving Devlin)

GCH +ℵ1 entails a non-specializable ℵ2-Souslin tree.

Theorem (Cummings, 1997)

ℵ1 ≤ λ<λ = λ + ♦ λ entails a non-specializable λ-complete λ+-Souslin tree.

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Non-specializable λ+-Souslin trees

Theorem (Baumgartner, 1980’s, improving Devlin)

GCH +ℵ1 entails a non-specializable ℵ2-Souslin tree.

Theorem (Cummings, 1997)

ℵ1 ≤ λ<λ = λ + ♦ λ entails a non-specializable λ-complete λ+-Souslin tree.

Theorem (Cummings, 1997)

If λ is a singular cardinal of countable cofinality, λ + CHλ and µℵ1 < λ for all µ < λ, then there exists a non-specializable λ+-Souslin tree.

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Non-specializable λ+-Souslin trees

Theorem (Baumgartner, 1980’s, improving Devlin)

GCH +ℵ1 entails a non-specializable ℵ2-Souslin tree.

Theorem (Cummings, 1997)

ℵ1 ≤ λ<λ = λ + ♦ λ entails a non-specializable λ-complete λ+-Souslin tree.

Theorem (Cummings, 1997)

If λ is a singular cardinal of countable cofinality, λ + CHλ and µℵ1 < λ for all µ < λ, then there exists a non-specializable λ+-Souslin tree.

Theorem (Cummings, 1997)

If λ is a singular cardinal of uncountable cofinality, λ + CHλ and µℵ0 < λ for all µ < λ, then there exists a non-specializable λ+-Souslin tree.

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To sum up

The construction of λ+-Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular.

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To sum up

The construction of λ+-Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality.

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To sum up

The construction of λ+-Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality.

Question

Do one really have to come up with such a long list of variations each time that a fundamental construction is discovered?

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To sum up

The construction of λ+-Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality.

Question

Do one really have to come up with such a long list of variations each time that a fundamental construction is discovered? Isn’t there any automatic translation between the different cardinals?

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An idea

Find a proxy!

• 1. Introduce a family of combinatorial principles from which the

constructions can be carried out uniformly;

• 2. Prove that this operational principle is a consequence of the

“usual” hypotheses.

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An idea

Find a proxy!

• 1. Introduce a family of combinatorial principles from which the

constructions can be carried out uniformly;

• 2. Prove that this operational principle is a consequence of the

“usual” hypotheses. This part is done only once, and then will be utilized each time that a new construction is discovered.

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The proxy principle

Goal

The proxy principle will allow to translate constructions from one cardinal to another, to calibrate the hypotheses needed to carry a construction, and will capture all known ♦-based constructions.

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The proxy principle

Goal

The proxy principle will allow to translate constructions from one cardinal to another, to calibrate the hypotheses needed to carry a construction, and will capture all known ♦-based constructions.

Definition

P(κ, µ, R, θ, S, ν, σ, ̟) asserts that ♦(κ) holds, and so is the corresponding P−(κ, µ, R, θ, S, ν, σ, ̟).

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The proxy principle

Definition

P(κ, µ, R, θ, S, ν, σ, ̟) asserts that ♦(κ) holds, and so is the corresponding P−(κ, µ, R, θ, S, ν, σ, ̟).

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The proxy principle

Definition

P(κ, µ, R, θ, S, ν, σ, ̟) asserts that ♦(κ) holds, and so is the corresponding P−(κ, µ, R, θ, S, ν, σ, ̟).

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ;

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ;

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ R is a binary relation over P(κ);

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ R is a binary relation over P(κ);

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ R is a binary relation over P(κ); ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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The proxy principle

Example of a binary relation R

⊑, where D ⊑ C iff ∃β such that D = C ∩ β.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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The proxy principle

Example of a binary relation R

⊑χ, where D ⊑χ C iff D ⊑ C or otp(C) < χ.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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The proxy principle

Example of a binary relation R

⊑∗, where D ⊑∗ C iff ∃α < sup(D) with D \ α ⊑ C \ α.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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The proxy principle

Example of a binary relation R

χ⊑∗, where Dχ⊑∗C iff cf(sup(D)) < χ or D ⊑∗ C.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S,

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S, there exists δ ∈ S

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The proxy principle

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S, there exists δ ∈ S with |Cδ| < ν such that:

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The proxy principle

Recall

succσ(C) = {α ∈ C | otp(C ∩ α) = j + 1 for some j < σ}.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S, there exists δ ∈ S with |Cδ| < ν such that:

◮ ∀i < min{δ, θ}∀C ∈ Cδ sup{β ∈ C | succσ(C \ β) ⊆ Ai} = δ; 12 / 32

The proxy principle

Recall

succ̟(C) = {α ∈ C | otp(C ∩ α) = j + 1 for some j < ̟}.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subsets of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S, there exists δ ∈ S with |Cδ| < ν such that:

◮ ∀i < min{δ, θ}∀C ∈ Cδ sup{β ∈ C | succσ(C \ β) ⊆ Ai} = δ; ◮ ∀i < min{δ, θ} sup

C∈Cδ{β ∈ C | succ̟(C \ β) ⊆ Ai} = δ,

unless ̟ = 0.

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Default values

Don’t worry, we have some default values! Whenever omitted, let θ = 1, S = {κ}, ν = 2, σ = 1, ̟ = 0.

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Default values

Don’t worry, we have some default values! Whenever omitted, let θ = 1, S = {κ}, ν = 2, σ = 1, ̟ = 0.

Definition

P−(κ, µ, R, θ, S, ν, σ, ̟) asserts the existence of Cδ | δ < κ s.t.:

◮ for every limit δ < κ, Cδ is a collection of club subset of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every sequence Ai | i < θ of cofinal subsets of κ, and

every S ∈ S, there exists δ ∈ S with |Cδ| < ν such that:

◮ ∀i < min{δ, θ}∀C ∈ Cδ sup{β ∈ C | succσ(C \ β) ⊆ Ai} = δ; ◮ ∀i < min{δ, θ} sup

C∈Cδ{β ∈ C | succ̟(C \ β) ⊆ Ai} = δ,

unless ̟ = 0.

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Default values

Don’t worry, we have some default values! Whenever omitted, let θ = 1, S = {κ}, ν = 2, σ = 1, ̟ = 0.

Special case

P−(κ, µ, R) asserts the existence of Cδ | δ < κ such that:

◮ for every limit δ < κ, Cδ is a collection of club subset of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R;

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Default values

Don’t worry, we have some default values! Whenever omitted, let θ = 1, S = {κ}, ν = 2, σ = 1, ̟ = 0.

Special case

P−(κ, µ, R) asserts the existence of Cδ | δ < κ such that:

◮ for every limit δ < κ, Cδ is a collection of club subset of δ; ◮ 0 < |Cδ| < µ for all δ < κ; ◮ if C ∈ Cδ and β ∈ acc(C), then ∃D ∈ Cβ with (D, C) ∈ R; ◮ for every cofinal A ⊆ κ, there exists δ < κ with Cδ = {Cδ},

such that sup(nacc(Cδ) ∩ A) = δ.

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A Souslin tree from the weakest principle

Let κ denote a regular uncountable cardinal.

Proposition

P(κ, κ, ⊑∗, 1, {κ}, κ) entails a κ-Souslin tree.

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A Souslin tree from the weakest principle

Let κ denote a regular uncountable cardinal.

Proposition

P(κ, κ, ⊑∗, 1, {κ}, κ) entails a κ-Souslin tree.

Proposition

P(κ, κ, ⊑∗, 1, {E κ

≥χ}, κ) entails a χ-complete κ-Souslin tree,

provided |α|<χ < κ for all α < κ.

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A Souslin tree from the weakest principle

Let κ denote a regular uncountable cardinal.

Proposition

P(κ, κ, ⊑∗, 1, {κ}, κ) entails a κ-Souslin tree.

Proposition

P(κ, κ, χ⊑∗, 1, {E κ

≥χ}, κ) entails a χ-complete κ-Souslin tree,

provided |α|<χ < κ for all α < κ.

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Sanity check #1

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

15 / 32

Sanity check #1

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

Theorem

♦(E λ+

λ ) entails P(λ+, 2, λ⊑, {E λ+ λ }).

15 / 32

Sanity check #1

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

Theorem

♦(E λ+

λ ) entails P(λ+, 2, λ⊑, {E λ+ λ }).

Corollary

If λ<λ = λ and ♦(E λ+

λ ) holds, then there exists a λ-complete

λ+-Souslin tree.

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Sanity check #2

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If there exists S ⊆ λ+ for which λ(S) + ♦(S) holds, then there exists a λ+-Souslin tree.

16 / 32

Sanity check #2

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If there exists S ⊆ λ+ for which λ(S) + ♦(S) holds, then there exists a λ+-Souslin tree.

Theorem

λ + CHλ entails P(λ+, 2, ⊑, {E λ+

≥θ | θ < λ}).

16 / 32

Sanity check #2

Let λ denote an uncountable cardinal.

Theorem (Jensen, 1972)

If there exists S ⊆ λ+ for which λ(S) + ♦(S) holds, then there exists a λ+-Souslin tree.

Theorem

λ + CHλ entails P(λ+, 2, ⊑, {E λ+

≥θ | θ < λ}).

Corollary

If λ + CHλ holds, then for every χ < λ with λ<χ = λ, there exists a χ-complete λ+-Souslin tree.

16 / 32

Sanity check #3

Let λ denote an uncountable cardinal.

Theorem (Gregory, 1976)

If λ<λ = λ, 2λ = λ+ and exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ+-Souslin tree.

17 / 32

Sanity check #3

Let λ denote an uncountable cardinal.

Theorem (Gregory, 1976)

If λ<λ = λ, 2λ = λ+ and exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then P(λ+, 2, λ⊑∗, {E λ+ λ }) holds.

17 / 32

Sanity check #3

Let λ denote an uncountable cardinal.

Theorem (Gregory, 1976)

If λ<λ = λ, 2λ = λ+ and exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then P(λ+, 2, λ⊑∗, {E λ+ λ }) holds.

Corollary (Kojman-Shelah, 1993)

If λ<λ = λ, 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ-complete λ+-Souslin tree.

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Sanity check #4

Let λ denote an uncountable cardinal.

Theorem (Shelah, 1984)

If 2ℵ0 = ℵ1, NSℵ1 is saturated, then there exists an ℵ2-Souslin tree.

18 / 32

Sanity check #4

Let λ denote an uncountable cardinal.

Theorem (Shelah, 1984)

If 2ℵ0 = ℵ1, NSℵ1 is saturated, then there exists an ℵ2-Souslin tree.

Theorem

If 2ℵ1 = ℵ2, NSℵ1 is saturated, then P(ℵ2, 2, ℵ1⊑∗, {E ℵ2

ℵ1 }) holds.

18 / 32

Sanity check #4

Let λ denote an uncountable cardinal.

Theorem (Shelah, 1984)

If 2ℵ0 = ℵ1, NSℵ1 is saturated, then there exists an ℵ2-Souslin tree.

Theorem

If 2ℵ1 = ℵ2, NSℵ1 is saturated, then P(ℵ2, 2, ℵ1⊑∗, {E ℵ2

ℵ1 }) holds.

Corollary

If 2ℵ0 = ℵ1, NSℵ1 is saturated, then there exists an ℵ1-complete ℵ2-Souslin tree.

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And so on..

19 / 32

And so on..

Okay, so you seem to found a way to redirect all ♦-based constructions of Souslin trees through a single construction. You haven’t yet shown me anything new!

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κ-trees which cohere modulo finite

Definition

A subtree T of <κκ is said to be coherent if for all δ < κ:

◮ if x, y ∈ Tδ, then {α < δ | x(α) = y(α)} is finite; ◮ if x, y ∈ δκ, and {α < δ | x(α) = y(α)} is finite,

then x ∈ Tδ iff y ∈ Tδ.

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κ-trees which cohere modulo finite

Definition

A subtree T of <κκ is said to be coherent if for all δ < κ:

◮ if x, y ∈ Tδ, then {α < δ | x(α) = y(α)} is finite; ◮ if x, y ∈ δκ, and {α < δ | x(α) = y(α)} is finite,

then x ∈ Tδ iff y ∈ Tδ.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a coherent ℵ1-Souslin tree.

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κ-trees which cohere modulo finite

Definition

A subtree T of <κκ is said to be coherent if for all δ < κ:

◮ if x, y ∈ Tδ, then {α < δ | x(α) = y(α)} is finite; ◮ if x, y ∈ δκ, and {α < δ | x(α) = y(α)} is finite,

then x ∈ Tδ iff y ∈ Tδ.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a coherent ℵ1-Souslin tree.

Theorem (Veliˇ ckovi´ c, 1986)

♦ ℵ1 entails a coherent ℵ2-Souslin tree.

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κ-trees which cohere modulo finite

In my talk at “Young Set Theory 2011” workshop, I asked about the consistency of a coherent λ+-Souslin tree for λ singular.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a coherent ℵ1-Souslin tree.

Theorem (Veliˇ ckovi´ c, 1986)

♦ ℵ1 entails a coherent ℵ2-Souslin tree.

20 / 32

κ-trees which cohere modulo finite

In my talk at “Young Set Theory 2011” workshop, I asked about the consistency of a coherent λ+-Souslin tree for λ singular.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a coherent ℵ1-Souslin tree.

Theorem (Veliˇ ckovi´ c, 1986)

♦ ℵ1 entails a coherent ℵ2-Souslin tree.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

20 / 32

κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1)

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a coherent ℵ1-Souslin tree.

Theorem (Veliˇ ckovi´ c, 1986)

♦ ℵ1 entails a coherent ℵ2-Souslin tree.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

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κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1) ◮ ♦ λ entails P(λ+, 2, ⊑, λ+)

Theorem (Veliˇ ckovi´ c, 1986)

♦ ℵ1 entails a coherent ℵ2-Souslin tree.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

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κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1) ◮ ♦ λ entails P(λ+, 2, ⊑, λ+) ◮ λ + CHλ entails P(λ+, 2, ⊑, λ+) for λ singular

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

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κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1) ◮ ♦ λ entails P(λ+, 2, ⊑, λ+) ◮ λ + CHλ entails P(λ+, 2, ⊑, λ+) for λ singular

Corollary

If λ + CHλ holds for λ singular, then there exists a coherent λ+-Souslin tree.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

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κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1) ◮ ♦ λ entails P(λ+, 2, ⊑, λ+) ◮ λ + CHλ entails P(λ+, 2, ⊑, λ+) for λ singular ◮ V = L entails P(κ, 2, ⊑, κ) for all regular uncountable κ

which is not weakly compact

Corollary

If λ + CHλ holds for λ singular, then there exists a coherent λ+-Souslin tree.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

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κ-trees which cohere modulo finite

Theorem

◮ ♦(ℵ1) entails P(ℵ1, 2, ⊑, ℵ1) ◮ ♦ λ entails P(λ+, 2, ⊑, λ+) ◮ λ + CHλ entails P(λ+, 2, ⊑, λ+) for λ singular ◮ V = L entails P(κ, 2, ⊑, κ) for all regular uncountable κ

which is not weakly compact

Corollary

If λ + CHλ holds for λ singular, then there exists a coherent λ+-Souslin tree.

Corollary

If V = L, then any regular uncountable κ is not weakly compact iff there exists a coherent κ-Souslin tree.

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A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a free ℵ1-Souslin tree.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a free ℵ1-Souslin tree. Jensen construct the levels of the tree by recursion, where the nodes of limit level α are obtained by forcing with finite conditions

• ver some countable elementary submodel that knows about the

diamond sequence and the tree constructed so far.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Theorem (Jensen, 1970’s)

♦(ℵ1) entails a free ℵ1-Souslin tree. Jensen construct the levels of the tree by recursion, where the nodes of limit level α are obtained by forcing with finite conditions

• ver some countable elementary submodel that knows about the

diamond sequence and the tree constructed so far. Genericity entails freeness.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Observation

CH +♦(E ℵ2

ℵ1 ) entails a free ℵ2-Souslin tree.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Observation

CH +♦(E ℵ2

ℵ1 ) entails a free ℵ2-Souslin tree.

Construct a ℵ1-complete tree by recursion, where the nodes of level α of uncountable cofinality are obtained by forcing with countable conditions over some ℵ1-sized elementary submodel that knows about anything relevant.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Observation

CH +♦(E ℵ2

ℵ1 ) entails a free ℵ2-Souslin tree.

Construct a ℵ1-complete tree by recursion, where the nodes of level α of uncountable cofinality are obtained by forcing with countable conditions over some ℵ1-sized elementary submodel that knows about anything relevant. The model is of size ℵ1 to accompany all relevant dense sets.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes ti | i < n from a fixed level δ < κ, the product tree of the upper cones

i<n ti ↑ is again κ-Souslin.

Observation

CH +♦(E ℵ2

ℵ1 ) entails a free ℵ2-Souslin tree.

Construct a ℵ1-complete tree by recursion, where the nodes of level α of uncountable cofinality are obtained by forcing with countable conditions over some ℵ1-sized elementary submodel that knows about anything relevant. The model is of size ℵ1 to accompany all relevant dense sets. The ℵ1-completeness of the tree and the countable conditions are necessary for the existence of a generic over the ℵ1-sized model.

21 / 32

A concept of “being productive” for Souslin trees

Question

How about free λ+-Souslin tree for λ singular?

Observation

CH +♦(E ℵ2

ℵ1 ) entails a free ℵ2-Souslin tree.

Construct a ℵ1-complete tree by recursion, where the nodes of level α of uncountable cofinality are obtained by forcing with countable conditions over some ℵ1-sized elementary submodel that knows about anything relevant. The model is of size ℵ1 to accompany all relevant dense sets. The ℵ1-completeness of the tree and the countable conditions are necessary for the existence of a generic over the ℵ1-sized model.

21 / 32

A concept of “being productive” for Souslin trees

Question

How about free λ+-Souslin tree for λ singular? Freeness requires that the generic meet λ many dense sets, but the tree cannot be λ-complete, and there cannot be a generic for the relevant poset over a model of size λ.

21 / 32

A concept of “being productive” for Souslin trees

Question

How about free λ+-Souslin tree for λ singular? Freeness requires that the generic meet λ many dense sets, but the tree cannot be λ-complete, and there cannot be a generic for the relevant poset over a model of size λ. But, there is another way:

Theorem

P(κ, µ, ⊑, κ) entails a µ-slim, free κ-Souslin tree.

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A concept of “being productive” for Souslin trees

Theorem

P(κ, µ, ⊑, κ) entails a µ-slim, free κ-Souslin tree.

Corollary

If λ + CHλ holds for λ singular, then there exists a free λ+-Souslin tree.

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A concept of “being productive” for Souslin trees

Corollary

If λ + CHλ holds for λ singular, then there exists a free λ+-Souslin tree.

Corollary

If V = L, then any regular uncountable κ is not weakly compact iff there exists a free κ-Souslin tree.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is χ-free, if for every nonzero ν < χ and any sequence of distinct nodes ti | i < ν from a fixed level δ < κ, the product tree of the upper cones

i<ν ti ↑ is again κ-Souslin.

21 / 32

A concept of “being productive” for Souslin trees

Definition

A κ-Souslin tree T is χ-free, if for every nonzero ν < χ and any sequence of distinct nodes ti | i < ν from a fixed level δ < κ, the product tree of the upper cones

i<ν ti ↑ is again κ-Souslin.

From GCH-type assumption, we can also construct χ-free trees for uncountable χ. For instance:

Corollary

If λ + CHλ holds for λ singular, then there exists a logλ(λ+)-free λ+-Souslin tree.

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Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree.

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Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree. The same is true for ℵ2-Souslin trees.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree. The same is true for ℵ2-Souslin trees. So, one would expect that the fact that free is weaker than coherent be reflected in the hypothesis needed to construct such.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree. The same is true for ℵ2-Souslin trees. So, one would expect that the fact that free is weaker than coherent be reflected in the hypothesis needed to construct such.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree. The same is true for ℵ2-Souslin trees. So, one would expect that the fact that free is weaker than coherent be reflected in the hypothesis needed to construct such.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

Theorem

P(κ, κ, ⊑, κ) entails a free κ-Souslin tree.

22 / 32

Calibrating

• P. Larson proved that any coherent ℵ1-Souslin tree contains a

regularly emebedded free ℵ1-Souslin tree. In particular, a coherent ℵ1-Souslin tree entails a free one. Larson and Zapletal (independently) proved that the following is consistent: there exists a free ℵ1-Souslin tree, but no coherent ℵ1-Souslin tree. The same is true for ℵ2-Souslin trees. So, one would expect that the fact that free is weaker than coherent be reflected in the hypothesis needed to construct such.

Theorem

P(κ, 2, ⊑, κ) entails a coherent κ-Souslin tree.

Theorem

P(κ, κ, ⊑, κ) entails a free κ-Souslin tree.

Theorem

P(κ, κ, ⊑∗, 1) entails a κ-Souslin tree.

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Specializable Souslin trees

23 / 32

Specializable Souslin trees

Recall (implicit in David, 1990)

If V = L, then for every regular λ, there exists a λ+-Souslin tree which is specializable.

24 / 32

Specializable Souslin trees

Recall (implicit in David, 1990)

If V = L, then for every regular λ, there exists a λ+-Souslin tree which is specializable.

Proposition

If λ<λ = λ, then any coherent λ-splitting λ+-Souslin tree is specializable.

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Specializable Souslin trees

Recall (implicit in David, 1990)

If V = L, then for every regular λ, there exists a λ+-Souslin tree which is specializable.

Proposition

If λ<λ = λ, then any coherent λ-splitting λ+-Souslin tree is specializable.

Corollary

If λ<λ = λ, P(λ+, 2, ⊑, λ+) entails a specializable λ+-Souslin tree.

24 / 32

Specializable Souslin trees

Theorem

If λ<λ = λ, P(λ+, λ+, λ⊑∗, 1, {E λ+

λ }, λ+, 1, 1) entails a

λ-complete, specializable λ+-Souslin tree.

Corollary

If λ<λ = λ, P(λ+, 2, ⊑, λ+) entails a specializable λ+-Souslin tree.

24 / 32

Specializable Souslin trees

Theorem

If λ<λ = λ, P(λ+, λ+, λ⊑∗, 1, {E λ+

λ }, λ+, 1, 1) entails a

λ-complete, specializable λ+-Souslin tree.

Recall

If 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then P(λ+, 2, λ⊑∗, {E λ+ λ }) holds.

24 / 32

Specializable Souslin trees

Theorem

If λ<λ = λ, P(λ+, λ+, λ⊑∗, 1, {E λ+

λ }, λ+, 1, 1) entails a

λ-complete, specializable λ+-Souslin tree.

Recall

If 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then P(λ+, 2, λ⊑∗, {E λ+ λ }) holds.

Corollary

If λ<λ = λ, 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ-complete, specializable

λ+-Souslin tree.

24 / 32

Specializable Souslin trees

Recall (implicit in David, 1990)

If V = L, then for every regular λ, there exists a λ+-Souslin tree which is specializable.

Recall (Gregory, 1976)

If λ<λ = λ, 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If λ<λ = λ, 2λ = λ+ and there exists a nonreflecting stationary subset of E λ+

<λ, then there exists a λ-complete, specializable

λ+-Souslin tree.

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non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {λ+}, 2, ω) entails a non-Specializable λ+-Souslin tree.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ. This covers the Baumgartner and Cummings constructions from GCH +λ and ♦ λ.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ. This covers the Baumgartner and Cummings constructions from GCH +λ and ♦ λ. In addition, the relation ⊑χ is weak enough to make P(λ+, 2, ⊑χ, 1, {λ+}, 2, ω) consistent with large cardinals that refute ∗

λ.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ. This covers the Baumgartner and Cummings constructions from GCH +λ and ♦ λ. In addition, the relation ⊑χ is weak enough to make P(λ+, 2, ⊑χ, 1, {λ+}, 2, ω) consistent with large cardinals that refute ∗

λ. Thereby, covering a seemingly unrelated scenario of

Shelah and Ben-David.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ. This covers the Baumgartner and Cummings constructions from GCH +λ and ♦ λ. In addition, the relation ⊑χ is weak enough to make P(λ+, 2, ⊑χ, 1, {λ+}, 2, ω) consistent with large cardinals that refute ∗

λ. Thereby, covering a seemingly unrelated scenario of

Shelah and Ben-David.

A model of “all Aronszajn trees are nonspecial”

It is consistent that κ is supercompact, λ = κ+ω, and there exists a non-Specializable λ+-Souslin tree.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ.

Theorem

P(λ+, 2, ⊑χ, λ+, {λ+}, 2, ω) entails a free, non-Specializable λ+-Souslin tree.

25 / 32

non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, 1, {E λ+

≥κ}, 2, ω) entails a non-Specializable λ+-Souslin

tree, which is κ-complete, provided that λ<κ = λ.

Theorem

P(λ+, 2, ⊑χ, λ+, {E λ+

≥κ}, 2, ω) entails a free, non-Specializable

λ+-Souslin tree, which is κ-complete, provided that λ<κ = λ.

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non-Specializable Souslin trees

Let χ < λ denote infinite cardinals.

Theorem

P(λ+, 2, ⊑χ, λ+, {E λ+

≥κ}, 2, ω) entails a free, non-Specializable

λ+-Souslin tree, which is κ-complete, provided that λ<κ = λ.

A model of “all Aronszajn trees are nonspecial”

It is consistent that κ is supercompact, λ = κ+ω, and there exists a free, non-Specializable λ+-Souslin tree.

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Some more

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Generalizing Gregory’s theorem to singular cardinals

Recall (Gregory, 1976)

If λ<λ = λ, CHλ and there exists a nonreflecting stationary subset

• f E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If 2<λ = λ, CHλ +∗

λ and exists a nonreflecting stationary subset

• f E λ+

=cf(λ), then P(λ+, λ+, ⊑) holds.

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Generalizing Gregory’s theorem to singular cardinals

Recall (Gregory, 1976)

If λ<λ = λ, CHλ and there exists a nonreflecting stationary subset

• f E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If 2<λ = λ, CHλ +∗

λ and exists a nonreflecting stationary subset

• f E λ+

=cf(λ), then P(λ+, λ+, ⊑) holds.

Theorem (Cummings-Foreman-Magidor, 2001)

After Prikry forcing over a supercompact cardinal λ, ∗

λ holds, yet,

any stationary subset of E λ+

=cf(λ) reflects.

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Generalizing Gregory’s theorem to singular cardinals

Recall (Gregory, 1976)

If λ<λ = λ, CHλ and there exists a nonreflecting stationary subset

• f E λ+

<λ, then there exists a λ+-Souslin tree.

Theorem

If 2<λ = λ, CHλ +∗

λ and exists a nonreflecting stationary subset

• f E λ+

=cf(λ), then P(λ+, λ+, ⊑) holds.

Theorem (Cummings-Foreman-Magidor, 2001)

After Prikry forcing over a supercompact cardinal λ, ∗

λ holds, yet,

any stationary subset of E λ+

=cf(λ) reflects.

Theorem

After Prikry forcing over a measurable cardinal λ satisfying CHλ, P(λ+, λ+, ⊑, λ+) holds.

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Generalizing Gregory’s theorem to singular cardinals

Theorem

If 2<λ = λ, CHλ +∗

λ and exists a nonreflecting stationary subset

• f E λ+

=cf(λ), then P(λ+, λ+, ⊑) holds.

Theorem

After Prikry forcing over a measurable cardinal λ satisfying CHλ, P(λ+, λ+, ⊑, λ+) holds.

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Generalizing Gregory’s theorem to singular cardinals

The derived trees

◮ P(λ+, λ+, ⊑) entails a rigid λ+-Souslin tree; ◮ P(λ+, λ+, ⊑, λ+) entails a free λ+-Souslin tree; ◮ P(λ+, λ+, ⊑, λ+) entails an homogeneous λ+-Souslin tree.

Theorem

If 2<λ = λ, CHλ +∗

λ and exists a nonreflecting stationary subset

• f E λ+

=cf(λ), then P(λ+, λ+, ⊑) holds.

Theorem

After Prikry forcing over a measurable cardinal λ satisfying CHλ, P(λ+, λ+, ⊑, λ+) holds.

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More results

Let λ<λ = λ denote a regular uncountable cardinal.

◮ If CHλ, then adding a single λ-Cohen set entails

P(λ+, λ+, ⊑, λ+, {E λ+

λ }), and hence

free/homogeneous/specializable trees.

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More results

Let λ<λ = λ denote a regular uncountable cardinal.

◮ If CHλ, then adding a single λ-Cohen set entails

P(λ+, λ+, ⊑, λ+, {E λ+

λ }), and hence

free/homogeneous/specializable trees.

◮ If λ + CHλ, then a single λ-Cohen set entails

P(λ+, 2, ⊑, λ+, {E λ+

λ }, 2, ω), and hence

free/coherent/specializable/non-specializable trees.

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More results

Let λ<λ = λ denote a regular uncountable cardinal.

◮ If CHλ, then adding a single λ-Cohen set entails

P(λ+, λ+, ⊑, λ+, {E λ+

λ }), and hence

free/homogeneous/specializable trees.

◮ If λ + CHλ, then a single λ-Cohen set entails

P(λ+, 2, ⊑, λ+, {E λ+

λ }, 2, ω), and hence

free/coherent/specializable/non-specializable trees.

◮ If λ + ♦∗(λ+), then there exists a (free) λ+-Souslin tree T,

whose ω-reduced power tree ωT/U is λ+-Kurepa for any nonprincipal ultrafilter U over ω.

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The microscopic approach

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Diamond for Hκ

Recall that P(κ, · · · ) asserts that ♦(κ) + P−(κ, · · · ) holds.

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Diamond for Hκ

Recall that P(κ, · · · ) asserts that ♦(κ) + P−(κ, · · · ) holds.

Proposition

For κ regular uncountable, ♦(κ) iff ♦(Hκ).

Definition

♦(Hκ) asserts the existence of ϕ0 : κ → Hκ and ϕ1 : κ → Hκ as

• follows. For every a ∈ Hκ, A ⊆ Hκ, and p ∈ Hκ++, there exists an

elementary submodel M ≺ Hκ++ such that:

◮ p ∈ M; ◮ M ∩ κ ∈ κ; ◮ ϕ0(M ∩ κ) = a; ◮ ϕ1(M ∩ κ) = M ∩ A.

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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Diamond for Hκ

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A construction ´ a la microscopic approach

#include <NormalTree.h> #include <SealAntichain.h> //#include <Specialize.h> #include <SealAutomorphism.h> //#include <SealProductTree.h>

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