p-ideal dichotomy and a strong form of the souslin hypothesis · kurepa observed that the existence...

P-ideal dichotomy and a strong form of theSouslin Hypothesis

Borisa Kuzeljevic

Institute of Mathematics CAS

External meeting of the Institute of Mathematics, Ostrava

Borisa Kuzeljevic (IM CAS) PID and a strong form of SH September 20, 2017 1 / 14

Joint work with Stevo Todorcevic

Souslin’s problem

Souslin’s question

Is it possible to replace ’separability’ with ’countable chain condition’ inthe Cantor’s characterization of the real line?

Souslin hypothesis

Every dense, complete linearly ordered set satisfying the countable chaincondition is isomorphic to R.

• The counterexample is called a Souslin line.

• Kurepa observed that the existence of a Souslin line is equivalent tothe existence of a Souslin tree, i.e. a tree of height ω1 which containsno uncountable chain and no uncountable antichain.

Souslin’s problem

Souslin hypothesis

Souslin’s problem

Souslin hypothesis

Souslin’s problem

Souslin hypothesis

Strong form of the Souslin hypothesis

• Kurepa asked if there is a tree of height ω1 which contains nouncountable chain and no uncountable level.

• Yes (Aronszajn 1934).

• Every Souslin tree is Aronszajn.

• If there is an increasing function from an Aronszajn tree to therationals (or equivalently: it can be partioned into a countable unionof antichains), then it is certainly not Souslin.

• These trees are called special Aronzajn trees.

Aronszajn’s construction

• Fix an enumeration of the rationals {r1, r2, . . . }, and let ϕ(rn) =1

• For each β < ω1 consider collection Aβ of sequences 〈aξ : ξ < β〉which satisfy the following three conditions:

1 aξ’s are distinct and∑ξ<β

ϕ(aξ) <∞.

2 for every α < β < ω1, every ε > 0, and every e ∈ Aα, there is e′ ∈ Aβ

which is an extension of e, and moreover∑x∈e′

ϕ(x) <∑x∈e

ϕ(x) + ε.

3 Aβ is countable for each β < ω1.

• Let the ordering on⋃β<ω1

Aβ be end-extension.

• In this way, a special Aronszajn tree is defined.

Souslin hypothesis

Theorem (Solovay-Tennenbaum 1971)

MAω1 implies the Souslin Hypothesis

Theorem (Baumgartner-Malitz-Reinhardt 1970)

MAω1 implies that every Aronszajn tree is special.

Theorem (Todorcevic 1991)

Principle (K) implies that every Aronszajn tree is special.

Theorem (Abraham-Todorcevic 1997)

P-ideal dichotomy implies the Souslin hypothesis

Souslin hypothesis

P-ideal dichotomy

For every P-ideal I of countable subsets of some set S :

(1) either there is an uncountable X ⊆ S such that [X ]ω ⊆ I;

(2) or S =⋃

n<ω Xn where each Xn is orthogonal to I.

• We assume that every ideal on S conains all the finite subsets of S .

• I is a P-ideal if for every countable family A of elements of I, there isB ∈ I such that A \ B is finite for each A ∈ A.

• Y is orthogonal to I if there is no infinite set in I contained in Y .

P-ideal dichotomy

For every P-ideal I of countable subsets of some set S :

(1) either there is an uncountable X ⊆ S such that [X ]ω ⊆ I;

(2) or S =⋃

n<ω Xn where each Xn is orthogonal to I.

• We assume that every ideal on S conains all the finite subsets of S .

• I is a P-ideal if for every countable family A of elements of I, there isB ∈ I such that A \ B is finite for each A ∈ A.

• Y is orthogonal to I if there is no infinite set in I contained in Y .

P-ideal dichotomy is consistent with CH.

Theorem (Balcar-Jech-Pazak 2005)

P-ideal dichotomy implies that every complete weakly distributive algebraB satisfying the ccc supports a strictly positive continuous submeasure.

P-ideal dichotomy and MAω1 jointly imply that every nonseparable Banachspace contains a closed convex subset supported by all of its points.

Theorem (Viale 2008)

P-ideal dichotomy implies the Singular cardinal hypothesis.

Consistency of PID

PFA implies PID.

Proof:

Suppose that I is a P-ideal on a set S , which is not a countable union ofsets orthogonal to I . Then the poset P for obtaining an unocuntable setΓ⊆ S such that [Γ]ω ⊆ I is as follows: Elements of P are finite ∈-chains ofcountable elementary submodels, and the order is given by: Finite chain qis stronger then the finite chain p if and only if for every model M ∈ p,and every model N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a pointwhich does not belong to any set X ∈ P(S) ∩M which is orthogonal to I ,and bM ∈ I is a subset of S ∩M which almost contains every set in I ∩M.

Consistency of PID

PFA implies PID.

Proof:

Suppose that I is a P-ideal on a set S , which is not a countable union ofsets orthogonal to I . Then the poset P for obtaining an unocuntable setΓ⊆ S such that [Γ]ω ⊆ I is as follows:

Elements of P are finite ∈-chains ofcountable elementary submodels, and the order is given by: Finite chain qis stronger then the finite chain p if and only if for every model M ∈ p,and every model N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a pointwhich does not belong to any set X ∈ P(S) ∩M which is orthogonal to I ,and bM ∈ I is a subset of S ∩M which almost contains every set in I ∩M.

Consistency of PID

PFA implies PID.

Proof:

Suppose that I is a P-ideal on a set S , which is not a countable union ofsets orthogonal to I . Then the poset P for obtaining an unocuntable setΓ⊆ S such that [Γ]ω ⊆ I is as follows: Elements of P are finite ∈-chains ofcountable elementary submodels, and the order is given by:

Finite chain qis stronger then the finite chain p if and only if for every model M ∈ p,and every model N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a pointwhich does not belong to any set X ∈ P(S) ∩M which is orthogonal to I ,and bM ∈ I is a subset of S ∩M which almost contains every set in I ∩M.

Consistency of PID

PFA implies PID.

Proof:

Suppose that I is a P-ideal on a set S , which is not a countable union ofsets orthogonal to I . Then the poset P for obtaining an unocuntable setΓ⊆ S such that [Γ]ω ⊆ I is as follows: Elements of P are finite ∈-chains ofcountable elementary submodels, and the order is given by: Finite chain qis stronger then the finite chain p if and only if for every model M ∈ p,and every model N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a pointwhich does not belong to any set X ∈ P(S) ∩M which is orthogonal to I ,and bM ∈ I is a subset of S ∩M which almost contains every set in I ∩M.

The Proper Forcing Axiom (PFA) has many important consequences onthe universe of set theory:

• all the consequences of PID;

• c = ℵ2;

• each directed system of cardinality at most ℵ1 is cofinally equivalentto one of: 1, ω, ω1, ω × ω1, [ω1]<ω;

• basis for uncountable linear orders has five elements;

• every automorphism of the Calkin algebra is inner:

• every Aronszajn tree is special.

Consistency of PFA

The consistency of PFA was proved by using countable support iteration ofproper posets (Baumgartner and Shelah late 70’s).

Recently, Neeman obtained a finite supports proof of the consistency ofthe PFA (in 2012).

The key novelty being the use of two-size ∈-chains of elementarysubmodels of H(θ) as side conditions.

We used his method to obtain the consistency of PID.

Consistency of PFA

Non-special Aronszajn tree

A tree T is called almost Souslin if it has no stationary antichains.

Theorem (K.-Todorcevic)

There is a forcing notion A which forces PID, and such that every almostSouslin tree T from V , remains almost Souslin in V A.

In the proof of one instance of PID, the original poset for introducing anuncountable set inside the ideal is changed in the following way:

Elements of P are strong finite ∈-chains of countable elementarysubmodels of H(θ), and the order is given by: Finite chain q is strongerthen the finite chain p if and only if for every model M ∈ p, and everymodel N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a point which doesnot belong to any set X ⊆ S from the Skolem closure of M which isorthogonal to I , and bM ∈ I is a subset of S ∩M which almost containsevery set in I ∩M.

Elements of P are strong finite ∈-chains of countable elementarysubmodels of H(θ), and the order is given by:

Finite chain q is strongerthen the finite chain p if and only if for every model M ∈ p, and everymodel N ∈ M ∩ (q \ p) we have xN ∈ bM , where xN is a point which doesnot belong to any set X ⊆ S from the Skolem closure of M which isorthogonal to I , and bM ∈ I is a subset of S ∩M which almost containsevery set in I ∩M.

The forcing A

• θ is a supercompact cardinal;

• F is a Laver function;

• <w is a well ordering of H(θ) such that for every x , y ∈ H(θ), if|trcl(x)| < |trcl(y)|, then x <w y ;

• Z = {α < θ : (H(α),∈,F � α,<w �H(α)) ≺ (H(θ),∈,F , <w )};• S = {M ∈ [H(θ)]ω : M ≺ (H(θ),∈,F , <w )};• T = {H(α) : α ∈ Z ∧ H(α)ω ⊆ H(α)};• Neeman calls the elements of S small nodes, and the elements of Ttransitive nodes, so we will follow his abbreviations.

The forcing A

Definition (The forcing A)

(1) Let A be the set of all pairs 〈s, p〉 such that:

1 s is a finite ∈-chain of models from S ∪ T , closed under intersections;2 p is a partial function on θ such that H(α) ∈ s for each α ∈ dom(p),

and that A∩H(α)“F (α) is a P-ideal on some ordinal γ which is not acountable union of sets orthogonal to F (α)”;

3 for every α ∈ dom(p), p(α) is a strong ∈-chain of countableelementary submodels of H(α+);

4 for every α ∈ dom(p), if M ∈ s/α and there are no transitive nodes ins between H(α) and M, then M ∩ H(α+) ∈ p(α).

(2) Ordering on A is defined as follows: 〈s∗, p∗〉 ≤ 〈s, p〉 iff s∗ ⊇ s and foreach α ∈ dom(p), p∗(α) ⊇ p(α), and moreover for every M ∈ p(α), ifN ∈ (p∗(α) ∩M) \ p(α), then 〈s∗ ∩ H(α), p∗ � α〉 A∩H(α)

˙xαN ∈ ˙bαM .

The forcing A

(1) Let A be the set of all pairs 〈s, p〉 such that:1 s is a finite ∈-chain of models from S ∪ T , closed under intersections;

2 p is a partial function on θ such that H(α) ∈ s for each α ∈ dom(p),and that A∩H(α)“F (α) is a P-ideal on some ordinal γ which is not acountable union of sets orthogonal to F (α)”;

˙xαN ∈ ˙bαM .

The forcing A

(1) Let A be the set of all pairs 〈s, p〉 such that:1 s is a finite ∈-chain of models from S ∪ T , closed under intersections;2 p is a partial function on θ such that H(α) ∈ s for each α ∈ dom(p),

˙xαN ∈ ˙bαM .

The forcing A

˙xαN ∈ ˙bαM .

The forcing A

˙xαN ∈ ˙bαM .

The forcing A

˙xαN ∈ ˙bαM .

p-ideal dichotomy and a strong form of the souslin hypothesis · kurepa observed that the existence...

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