# Constructions from squares and diamonds

July 15, 2017

## Transcript

1. ### 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

4. ### Small approximations Fix a regular, uncountable cardinal κ. It is

often desirable to construct objects of size κ+ using approximations of size < κ.
5. ### 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
6. ### 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.
7. ### 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.
8. ### Consequences of morass The existence of a (κ, 1)-morass implies

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

the existence of: • a κ-Kurepa tree;
10. ### 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;
11. ### 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);
12. ### 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.
13. ### 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.
14. ### 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.
15. ### 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.
16. ### 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.
17. ### 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:
18. ### 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+);
19. ### 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;
20. ### 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.
21. ### 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.
22. ### 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.
23. ### 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.
24. ### 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);
25. ### 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.
26. ### The forcing axiom Deﬁnition Suppose that P ∈ Pκ, D

is a sharply dense subset of P, and G is a ﬁlter over P.
27. ### 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.
28. ### 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.
29. ### 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.

31. ### Super-Souslin trees Deﬁnition Let λ be an inﬁnite cardinal, and

let (T, <T ) be a normal, splitting λ++-tree. • For α < λ++, Tα denotes the α-th level of T, and Tλ α denotes the set of injective functions a : λ → Tα. • Tλ = α<λ++ Tα. • If a, b ∈ Tλ, we say a <T b if, for all i < λ, a(i) <T b(t). • [Tλ]2 = {(a, b) | a, b ∈ Tλ and a <T b}. • T is a λ++-super-Souslin tree if there is a function F : [Tλ]2 → λ+ such that, for all (a, b), (a, c) ∈ [Tλ]2, if F(a, b) = F(a, c), then there is i < λ such that b(i) and c(i) are ≤T -comparable.
32. ### Souslin trees and morasses Lemma (Shelah) If (T, <T )

is a λ++-super-Souslin tree, then it has a λ++-Souslin subtree
33. ### Souslin trees and morasses Lemma (Shelah) If (T, <T )

is a λ++-super-Souslin tree, then it has a λ++-Souslin subtree and continues to do so in any outer model with the same P(λ) and λ++.
34. ### Souslin trees and morasses Lemma (Shelah) If (T, <T )

is a λ++-super-Souslin tree, then it has a λ++-Souslin subtree and continues to do so in any outer model with the same P(λ) and λ++. Theorem (Shelah-Stanley) If 2λ = λ+ and there is a (λ+, 1)-morass, then there is a λ++-super-Souslin tree.
35. ### Souslin trees and morasses Lemma (Shelah) If (T, <T )

is a λ++-super-Souslin tree, then it has a λ++-Souslin subtree and continues to do so in any outer model with the same P(λ) and λ++. Theorem (Shelah-Stanley) If 2λ = λ+ and there is a (λ+, 1)-morass, then there is a λ++-super-Souslin tree. Recall that SHµ is the statement that there are no µ-Souslin trees. Since the non-existence of a (κ, 1)-morass implies that κ+ is inaccessible in L, we obtain the following corollary.
36. ### Souslin trees and morasses Lemma (Shelah) If (T, <T )

is a λ++-super-Souslin tree, then it has a λ++-Souslin subtree and continues to do so in any outer model with the same P(λ) and λ++. Theorem (Shelah-Stanley) If 2λ = λ+ and there is a (λ+, 1)-morass, then there is a λ++-super-Souslin tree. Recall that SHµ is the statement that there are no µ-Souslin trees. Since the non-existence of a (κ, 1)-morass implies that κ+ is inaccessible in L, we obtain the following corollary. Corollary (Shelah-Stanley) If λ is an inﬁnite cardinal and 2λ = λ+ and SHλ++ both hold, then λ++ is inaccessible in L.
37. ### 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:
38. ### 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.
39. ### A lower bound Using the forcing axiom from Section I,

we obtain the following theorem.
40. ### 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.
41. ### 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.
42. ### A lower bound The failure of B κ implies that

κ+ is Mahlo in L.
43. ### 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+.
44. ### 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.
45. ### 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.