Constructions from squares and diamonds (joint work with Assaf Rinot) Chris Lambie-Hanson Department of Mathematics Bar-Ilan University 6th European Set Theory Conference Budapest, Hungary 4 July 2017

I: A forcing axiom

Small approximations Fix a regular, uncountable cardinal κ.

Small approximations Fix a regular, uncountable cardinal κ. It is often desirable to construct objects of size κ+ using approximations of size < κ.

Small approximations Fix a regular, uncountable cardinal κ. It is often desirable to construct objects of size κ+ using approximations of size < κ. Example: Kurepa trees

Small approximations Fix a regular, uncountable cardinal κ. It is often desirable to construct objects of size κ+ using approximations of size < κ. Example: Kurepa trees Such constructions obviously cannot be carried out in a linear fashion and often require some additional set-theoretic hypotheses to ensure success.

Small approximations Fix a regular, uncountable cardinal κ. It is often desirable to construct objects of size κ+ using approximations of size < κ. Example: Kurepa trees Such constructions obviously cannot be carried out in a linear fashion and often require some additional set-theoretic hypotheses to ensure success. One such hypothesis is the existence of a (κ, 1)-morass, the definition of which is motivated by precisely such constructions.

Consequences of morass The existence of a (κ, 1)-morass implies the existence of:

Consequences of morass The existence of a (κ, 1)-morass implies the existence of: • a κ-Kurepa tree;

Consequences of morass The existence of a (κ, 1)-morass implies the existence of: • a κ-Kurepa tree; • a κ-thin tall superatomic Boolean algebra;

Consequences of morass The existence of a (κ, 1)-morass implies the existence of: • a κ-Kurepa tree; • a κ-thin tall superatomic Boolean algebra; • a (κ+, (κ+)∗)-Hausdorff gap in P(κ);

Consequences of morass The existence of a (κ, 1)-morass implies the existence of: • a κ-Kurepa tree; • a κ-thin tall superatomic Boolean algebra; • a (κ+, (κ+)∗)-Hausdorff gap in P(κ); • a B κ -sequence.

Consequences of morass The existence of a (κ, 1)-morass implies the existence of: • a κ-Kurepa tree; • a κ-thin tall superatomic Boolean algebra; • a (κ+, (κ+)∗)-Hausdorff gap in P(κ); • a B κ -sequence. Forcing axioms equivalent to the existence of morasses were developed by Velleman and by Shelah and Stanley and provide a streamlined approach to obtaining existence results such as those above.

Diamonds In a series of papers entitled “Models with second order properties,” Shelah et al. develop techniques for carrying out similar constructions assuming only a relative of ♦κ.

Diamonds In a series of papers entitled “Models with second order properties,” Shelah et al. develop techniques for carrying out similar constructions assuming only a relative of ♦κ. These ideas were later deployed by Foreman-Magidor-Shelah to prove the consistency, relative to the consistency of a huge cardinal, of the existence of an ultrafilter U on ω1 such that |ωω1 /U| = ω1 and by Foreman to prove the consistency, again relative to a huge cardinal, of the existence of an ℵ1-dense ideal on ℵ2.

A middle road We develop here an intermediate approach, a forcing axiom for such constructions that follows from the conjunction of B κ and ♦κ.

A middle road We develop here an intermediate approach, a forcing axiom for such constructions that follows from the conjunction of B κ and ♦κ. Definition B κ is the assertion that there is a sequence Cα | α ∈ Γ such that:

A middle road We develop here an intermediate approach, a forcing axiom for such constructions that follows from the conjunction of B κ and ♦κ. Definition B κ is the assertion that there is a sequence Cα | α ∈ Γ such that: 1 κ+ ∩ cof(κ) ⊆ Γ ⊆ acc(κ+);

A middle road We develop here an intermediate approach, a forcing axiom for such constructions that follows from the conjunction of B κ and ♦κ. Definition B κ is the assertion that there is a sequence Cα | α ∈ Γ such that: 1 κ+ ∩ cof(κ) ⊆ Γ ⊆ acc(κ+); 2 for all α ∈ Γ, Cα is club in α and otp(Cα) ≤ κ;

A middle road We develop here an intermediate approach, a forcing axiom for such constructions that follows from the conjunction of B κ and ♦κ. Definition B κ is the assertion that there is a sequence Cα | α ∈ Γ such that: 1 κ+ ∩ cof(κ) ⊆ Γ ⊆ acc(κ+); 2 for all α ∈ Γ, Cα is club in α and otp(Cα) ≤ κ; 3 for all β ∈ Γ and all α ∈ acc(Cβ), we have α ∈ Γ and Cα = Cβ ∩ α.

The forcing class Pκ We isolate a class Pκ of forcing notions. For every P ∈ Pκ, each p ∈ P is associated with a realm xp ∈ [κ+]<κ on which the condition p can specify information about the final desired structure.

The forcing class Pκ We isolate a class Pκ of forcing notions. For every P ∈ Pκ, each p ∈ P is associated with a realm xp ∈ [κ+]<κ on which the condition p can specify information about the final desired structure. If q ≤ p, then xq ⊇ xp.

The forcing class Pκ We isolate a class Pκ of forcing notions. For every P ∈ Pκ, each p ∈ P is associated with a realm xp ∈ [κ+]<κ on which the condition p can specify information about the final desired structure. If q ≤ p, then xq ⊇ xp. If π is a partial, continuous, order-preserving injection from a subset of κ+ to κ+, then π acts on {p ∈ P | xp ⊆ dom(π)} in such a way that xπ.p = π“xp.

The forcing class Pκ We isolate a class Pκ of forcing notions. For every P ∈ Pκ, each p ∈ P is associated with a realm xp ∈ [κ+]<κ on which the condition p can specify information about the final desired structure. If q ≤ p, then xq ⊇ xp. If π is a partial, continuous, order-preserving injection from a subset of κ+ to κ+, then π acts on {p ∈ P | xp ⊆ dom(π)} in such a way that xπ.p = π“xp. Definition A subset D ⊆ P is sharply dense if: • for all p ∈ P, there is q ≤ p such that q ∈ D and xq = cl(xp);

The forcing class Pκ We isolate a class Pκ of forcing notions. For every P ∈ Pκ, each p ∈ P is associated with a realm xp ∈ [κ+]<κ on which the condition p can specify information about the final desired structure. If q ≤ p, then xq ⊇ xp. If π is a partial, continuous, order-preserving injection from a subset of κ+ to κ+, then π acts on {p ∈ P | xp ⊆ dom(π)} in such a way that xπ.p = π“xp. Definition A subset D ⊆ P is sharply dense if: • for all p ∈ P, there is q ≤ p such that q ∈ D and xq = cl(xp); • for all p ∈ P and all π with xp ⊆ dom(π), we have p ∈ D iff π.p ∈ D.

The forcing axiom Definition Suppose that P ∈ Pκ, D is a sharply dense subset of P, and G is a filter over P.

The forcing axiom Definition Suppose that P ∈ Pκ, D is a sharply dense subset of P, and G is a filter over P. We say G meets D cofinally if, for all y ∈ [κ+]<κ, there is p ∈ G ∩ D such that y ⊆ xp.

The forcing axiom Definition Suppose that P ∈ Pκ, D is a sharply dense subset of P, and G is a filter over P. We say G meets D cofinally if, for all y ∈ [κ+]<κ, there is p ∈ G ∩ D such that y ⊆ xp. Theorem Suppose that B κ and ♦κ hold. Suppose also that P ∈ Pκ and D = {Dη | η < κ} is a collection of sharply dense subsets of P.

The forcing axiom Definition Suppose that P ∈ Pκ, D is a sharply dense subset of P, and G is a filter over P. We say G meets D cofinally if, for all y ∈ [κ+]<κ, there is p ∈ G ∩ D such that y ⊆ xp. Theorem Suppose that B κ and ♦κ hold. Suppose also that P ∈ Pκ and D = {Dη | η < κ} is a collection of sharply dense subsets of P. Then there is a filter G over P such that, for all η < κ, G meets Dη cofinally.

II: Super-Souslin trees

Super-Souslin trees Definition Let λ be an infinite cardinal, and let (T,

Souslin trees and morasses Lemma (Shelah) If (T,

Souslin trees and morasses Lemma (Shelah) If (T,

Souslin trees and morasses Lemma (Shelah) If (T,

Souslin trees and morasses Lemma (Shelah) If (T,

Souslin trees and morasses Lemma (Shelah) If (T,

An upper bound Theorem (Laver-Shelah) Suppose λ < µ are infinite cardinals, with µ weakly compact. Then there is a forcing extension by a λ+-directed closed forcing in which:

An upper bound Theorem (Laver-Shelah) Suppose λ < µ are infinite cardinals, with µ weakly compact. Then there is a forcing extension by a λ+-directed closed forcing in which: • µ = λ++; • 2λ = λ+; • SHλ++ holds.

A lower bound Using the forcing axiom from Section I, we obtain the following theorem.

A lower bound Using the forcing axiom from Section I, we obtain the following theorem. Theorem Suppose λ is an infinite cardinal and B λ+ and ♦λ+ both hold. Then there is a λ++-super-Souslin tree.

A lower bound Using the forcing axiom from Section I, we obtain the following theorem. Theorem Suppose λ is an infinite cardinal and B λ+ and ♦λ+ both hold. Then there is a λ++-super-Souslin tree. Remark In 2010, in private communication, Foreman told Rinot that one can construct an ℵ2-Souslin tree from the conjunction of ω1 and ♦ω1 . This work was never published, and no details of the construction were provided. We thank him for pointing us in this direction.

A lower bound The failure of B κ implies that κ+ is Mahlo in L.

A lower bound The failure of B κ implies that κ+ is Mahlo in L. Moreover, by a recent result of Shelah, for uncountable λ, ♦λ+ is equivalent to 2λ = λ+.

A lower bound The failure of B κ implies that κ+ is Mahlo in L. Moreover, by a recent result of Shelah, for uncountable λ, ♦λ+ is equivalent to 2λ = λ+. We therefore obtain the following improvement of the Shelah-Stanley result for uncountable λ.

A lower bound The failure of B κ implies that κ+ is Mahlo in L. Moreover, by a recent result of Shelah, for uncountable λ, ♦λ+ is equivalent to 2λ = λ+. We therefore obtain the following improvement of the Shelah-Stanley result for uncountable λ. Corollary If λ is an uncountable cardinal and 2λ = λ+ and SHλ++ both hold, then λ++ is Mahlo in L.

Photo credits: Alexandre Jacques

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