# Robust reflection principles

ASL Winter Meeting, January 2016

January 09, 2016

## Transcript

1. ### Robust reﬂection principles Chris Lambie-Hanson Einstein Institute of Mathematics Hebrew

University of Jerusalem 2016 ASL Winter Meeting Seattle, WA 9 January 2016
2. ### Reﬂection/compactness principles The study of reﬂection and compactness principles has

been a central theme in modern set theory.
3. ### Reﬂection/compactness principles The study of reﬂection and compactness principles has

been a central theme in modern set theory. Very roughly speaking, a reﬂection principle at a cardinal κ takes the following form: If (something) holds for κ, then it holds for some (many) α < κ.
4. ### Reﬂection/compactness principles The study of reﬂection and compactness principles has

been a central theme in modern set theory. Very roughly speaking, a reﬂection principle at a cardinal κ takes the following form: If (something) holds for κ, then it holds for some (many) α < κ. Compactness is the dual notion: If (something) holds for all (most) α < κ, then it holds for κ.
5. ### Reﬂection/compactness principles The study of reﬂection and compactness principles has

been a central theme in modern set theory. Very roughly speaking, a reﬂection principle at a cardinal κ takes the following form: If (something) holds for κ, then it holds for some (many) α < κ. Compactness is the dual notion: If (something) holds for all (most) α < κ, then it holds for κ. Canonical inner models, such as L, typically exhibit large degrees of incompactness, while large cardinals tend to imply compactness and reﬂection principles.
6. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅.
7. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β.
8. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β. 3 Suppose κ is a regular, uncountable cardinal. Reﬂ(κ) is the assertion that every stationary subset of κ reﬂects.
9. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β. 3 Suppose κ is a regular, uncountable cardinal. Reﬂ(κ) is the assertion that every stationary subset of κ reﬂects. • If κ is weakly compact, then Reﬂ(κ) holds.
10. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β. 3 Suppose κ is a regular, uncountable cardinal. Reﬂ(κ) is the assertion that every stationary subset of κ reﬂects. • If κ is weakly compact, then Reﬂ(κ) holds. • If κ is the successor of a singular limit of strongly compact cardinals, then Reﬂ(κ) holds.
11. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β. 3 Suppose κ is a regular, uncountable cardinal. Reﬂ(κ) is the assertion that every stationary subset of κ reﬂects. • If κ is weakly compact, then Reﬂ(κ) holds. • If κ is the successor of a singular limit of strongly compact cardinals, then Reﬂ(κ) holds. • (Jensen) If V = L, then Reﬂ(κ) holds iﬀ κ is weakly compact.
12. ### Stationary reﬂection Deﬁnition Let α be an ordinal with cf(α)

> ω. 1 S ⊆ α is stationary in α if, for every closed, unbounded C ⊆ α, C ∩ S = ∅. 2 Suppose S ⊆ α is stationary, β < α, and cf(β) > ω. Then S reﬂects at β if S ∩ β is stationary in β. 3 Suppose κ is a regular, uncountable cardinal. Reﬂ(κ) is the assertion that every stationary subset of κ reﬂects. • If κ is weakly compact, then Reﬂ(κ) holds. • If κ is the successor of a singular limit of strongly compact cardinals, then Reﬂ(κ) holds. • (Jensen) If V = L, then Reﬂ(κ) holds iﬀ κ is weakly compact. • (Magidor) Con(ZFC+ there are inﬁnitely many supercompact cardinals) ⇒ Con(ZFC + Reﬂ(ℵω+1)).
13. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T .
14. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T . If (T, <T ) is a tree and x ∈ T, then htT (x) is the order-type of predT (x).
15. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T . If (T, <T ) is a tree and x ∈ T, then htT (x) is the order-type of predT (x). If α is an ordinal, then the αth level of T, Tα, is {x ∈ T | htT (x) = α}.
16. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T . If (T, <T ) is a tree and x ∈ T, then htT (x) is the order-type of predT (x). If α is an ordinal, then the αth level of T, Tα, is {x ∈ T | htT (x) = α}. The height of T, ht(T), is the least α such that Tα = ∅.
17. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T . If (T, <T ) is a tree and x ∈ T, then htT (x) is the order-type of predT (x). If α is an ordinal, then the αth level of T, Tα, is {x ∈ T | htT (x) = α}. The height of T, ht(T), is the least α such that Tα = ∅. Deﬁnition If (T, <T ) is a tree and b ⊆ T, then b is a branch through T if b is linearly ordered by <T . It is a coﬁnal branch if, for all α < ht(T), b ∩ Tα = ∅.
18. ### The tree property Deﬁnition A partial order (T, <T )

is a tree if, for every x ∈ T, predT (x) := {y ∈ T | y <T x} is well-ordered by <T . If (T, <T ) is a tree and x ∈ T, then htT (x) is the order-type of predT (x). If α is an ordinal, then the αth level of T, Tα, is {x ∈ T | htT (x) = α}. The height of T, ht(T), is the least α such that Tα = ∅. Deﬁnition If (T, <T ) is a tree and b ⊆ T, then b is a branch through T if b is linearly ordered by <T . It is a coﬁnal branch if, for all α < ht(T), b ∩ Tα = ∅. Deﬁnition Let κ be a regular, uncountable cardinal. A tree (T, <T ) is a κ-tree if ht(T) = κ and, for all α < κ, |Tα| < κ.

cardinal.
20. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch.
21. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees.
22. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property.
23. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property. • (Aronszajn) ℵ1 does not have the tree property.
24. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property. • (Aronszajn) ℵ1 does not have the tree property. • (Erd˝ os-Tarski) κ is weakly compact ⇔ κ is inaccessible and has the tree property.
25. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property. • (Aronszajn) ℵ1 does not have the tree property. • (Erd˝ os-Tarski) κ is weakly compact ⇔ κ is inaccessible and has the tree property. • (Magidor-Shelah) If κ is the successor of a singular limit of strongly compact cardinals, then κ has the tree property.
26. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property. • (Aronszajn) ℵ1 does not have the tree property. • (Erd˝ os-Tarski) κ is weakly compact ⇔ κ is inaccessible and has the tree property. • (Magidor-Shelah) If κ is the successor of a singular limit of strongly compact cardinals, then κ has the tree property. • (Mitchell, Silver) Con(ZFC+ there is a weakly compact cardinal) ⇔ Con(ZFC + TP(ℵ2)).
27. ### The tree property Deﬁnition Let κ be a regular, uncountable

cardinal. 1 A tree (T, <T ) is a κ-Aronszajn tree if T is a κ-tree and T has no coﬁnal branch. 2 κ has the tree property if there are no κ-Aronszajn trees. • (K¨ onig) ℵ0 has the tree property. • (Aronszajn) ℵ1 does not have the tree property. • (Erd˝ os-Tarski) κ is weakly compact ⇔ κ is inaccessible and has the tree property. • (Magidor-Shelah) If κ is the successor of a singular limit of strongly compact cardinals, then κ has the tree property. • (Mitchell, Silver) Con(ZFC+ there is a weakly compact cardinal) ⇔ Con(ZFC + TP(ℵ2)). • (Sinapova) Con(ZFC+ there are inﬁnitely many supercompact cardinals) ⇒ Con(ZFC + TP(ℵω+1)).
28. ### Robustness Deﬁnition Suppose κ is a cardinal and P is

a property that a cardinal can satisfy (e.g. being weakly compact, having the tree property, etc.). κ satisﬁes P robustly (or κ has the robust property P) if, whenever Q is a forcing poset and |Q|+ < κ, κ satisﬁes P in V Q.
29. ### Robustness Deﬁnition Suppose κ is a cardinal and P is

a property that a cardinal can satisfy (e.g. being weakly compact, having the tree property, etc.). κ satisﬁes P robustly (or κ has the robust property P) if, whenever Q is a forcing poset and |Q|+ < κ, κ satisﬁes P in V Q. Theorem (Levy-Solovay) Most large cardinal properties (e.g. being inaccessible, Mahlo, weakly compact, measurable, strongly compact, supercompact, etc.) are robust.
30. ### Robustness Deﬁnition Suppose κ is a cardinal and P is

a property that a cardinal can satisfy (e.g. being weakly compact, having the tree property, etc.). κ satisﬁes P robustly (or κ has the robust property P) if, whenever Q is a forcing poset and |Q|+ < κ, κ satisﬁes P in V Q. Theorem (Levy-Solovay) Most large cardinal properties (e.g. being inaccessible, Mahlo, weakly compact, measurable, strongly compact, supercompact, etc.) are robust. Thus, reﬂection principles, when they hold due to the existence of large cardinals, are themselves robust.
31. ### Robustness Deﬁnition Suppose κ is a cardinal and P is

a property that a cardinal can satisfy (e.g. being weakly compact, having the tree property, etc.). κ satisﬁes P robustly (or κ has the robust property P) if, whenever Q is a forcing poset and |Q|+ < κ, κ satisﬁes P in V Q. Theorem (Levy-Solovay) Most large cardinal properties (e.g. being inaccessible, Mahlo, weakly compact, measurable, strongly compact, supercompact, etc.) are robust. Thus, reﬂection principles, when they hold due to the existence of large cardinals, are themselves robust. This raises the question: to what extent can reﬂection principles hold robustly or hold non-robustly at small cardinals?
32. ### Robust stationary reﬂection Deﬁnition Let κ be a regular, uncountable

cardinal, and let S ⊆ κ be stationary. S reﬂects at arbitrarily large coﬁnalities if, for every regular cardinal λ < κ, there is α < κ such that cf(α) ≥ λ and S reﬂects at α.
33. ### Robust stationary reﬂection Deﬁnition Let κ be a regular, uncountable

cardinal, and let S ⊆ κ be stationary. S reﬂects at arbitrarily large coﬁnalities if, for every regular cardinal λ < κ, there is α < κ such that cf(α) ≥ λ and S reﬂects at α. Question (Eisworth) Suppose µ is a singular cardinal, κ = µ+, and Reﬂ(κ) holds. Must it be the case that every stationary subset of κ reﬂects at arbitrarily high coﬁnalities?
34. ### Robust stationary reﬂection Proposition If Reﬂ(ℵω+1) holds, then every stationary

subset of ℵω+1 reﬂects at arbitrarily high coﬁnalities.
35. ### Robust stationary reﬂection Proposition If Reﬂ(ℵω+1) holds, then every stationary

subset of ℵω+1 reﬂects at arbitrarily high coﬁnalities. Theorem (Cummings-LH) Con(ZFC+ there is an ω · 2-sequence of supercompact cardinals) ⇒ Con(ZFC + Reﬂ(ℵω·2+1) + there is a stationary subset of ℵω·2+1 that does not reﬂect at any ordinal of coﬁnality > ℵω).
36. ### Robust stationary reﬂection Theorem (LH) Suppose κ is a regular,

uncountable cardinal. Then the following are equivalent.
37. ### Robust stationary reﬂection Theorem (LH) Suppose κ is a regular,

uncountable cardinal. Then the following are equivalent. 1 Every stationary subset of κ reﬂects at arbitrarily high coﬁnalities.
38. ### Robust stationary reﬂection Theorem (LH) Suppose κ is a regular,

uncountable cardinal. Then the following are equivalent. 1 Every stationary subset of κ reﬂects at arbitrarily high coﬁnalities. 2 Reﬂ(κ) holds robustly.
39. ### Robust stationary reﬂection Theorem (LH) Suppose κ is a regular,

uncountable cardinal. Then the following are equivalent. 1 Every stationary subset of κ reﬂects at arbitrarily high coﬁnalities. 2 Reﬂ(κ) holds robustly. Theorem (LH) It is consistent, relative to large cardinals, that Reﬂ(κ) holds non-robustly, where κ is the least inaccessible cardinal.
40. ### Strong systems Deﬁnition Let κ be a regular, uncountable cardinal.

S = Sα | α < κ , R is a strong κ-system if: 1 For all α < κ, 0 < |Sα| < κ and, if α = β < κ, then Sα ∩ Sβ = ∅. 2 R (sometimes denoted RS) is a set of binary, transitive, tree-like relations on α<κ Sα. 3 For all R ∈ R, α, β < κ, u ∈ Sα, and v ∈ Sβ, if uRv, then α < β. 4 For all α < β < κ and v ∈ Sβ, there is R ∈ R and u ∈ Sα such that uRv. A branch through S is a set b ⊆ {{α} × λα | α < κ} such that, for some R ∈ R, b is linearly ordered by R. b is a coﬁnal branch if, for coﬁnally many α < κ, b ∩ {α} × λα = ∅.
41. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch.
42. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch. A strong κ-system with 1 relation is precisely a κ-tree, so the strong system property implies the tree property.
43. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch. A strong κ-system with 1 relation is precisely a κ-tree, so the strong system property implies the tree property. Theorem (LH) Suppose κ is a regular, uncountable cardinal. The following are equivalent:
44. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch. A strong κ-system with 1 relation is precisely a κ-tree, so the strong system property implies the tree property. Theorem (LH) Suppose κ is a regular, uncountable cardinal. The following are equivalent: 1 κ has the strong system property.
45. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch. A strong κ-system with 1 relation is precisely a κ-tree, so the strong system property implies the tree property. Theorem (LH) Suppose κ is a regular, uncountable cardinal. The following are equivalent: 1 κ has the strong system property. 2 κ has the robust strong system property.
46. ### The strong system property Deﬁnition Let κ be a regular,

uncountable cardinal. κ has the strong system property if every strong κ-system S with |RS|+ < κ has a coﬁnal branch. A strong κ-system with 1 relation is precisely a κ-tree, so the strong system property implies the tree property. Theorem (LH) Suppose κ is a regular, uncountable cardinal. The following are equivalent: 1 κ has the strong system property. 2 κ has the robust strong system property. 3 κ has the robust strong system property for systems with countably many relations.
47. ### The strong system property Thus, the strong system property implies

the robust tree property.
48. ### The strong system property Thus, the strong system property implies

the robust tree property. Also, if κ is inaccessible, then the strong system property is equivalent to weak compactness and hence to the tree property.
49. ### The strong system property Thus, the strong system property implies

the robust tree property. Also, if κ is inaccessible, then the strong system property is equivalent to weak compactness and hence to the tree property. Theorem (LH) It is consistent, relative to large cardinals, that the strong system property holds at ℵω2+1 .
50. ### The strong system property Thus, the strong system property implies

the robust tree property. Also, if κ is inaccessible, then the strong system property is equivalent to weak compactness and hence to the tree property. Theorem (LH) It is consistent, relative to large cardinals, that the strong system property holds at ℵω2+1 . The proof of this theorem uses a forcing notion due to Magidor and Shelah and is heavily indebted to work of Fontanella and Magidor.
51. ### The strong system property Thus, the strong system property implies

the robust tree property. Also, if κ is inaccessible, then the strong system property is equivalent to weak compactness and hence to the tree property. Theorem (LH) It is consistent, relative to large cardinals, that the strong system property holds at ℵω2+1 . The proof of this theorem uses a forcing notion due to Magidor and Shelah and is heavily indebted to work of Fontanella and Magidor. Theorem (LH) Fix α < ω2. It is consistent, relative to large cardinals, that ℵω2+1 has the strong system property for all strong systems S with |RS| ≤ ℵα but ℵω2+1 fails to have the full strong system property.

53. ### Questions Question Can ℵω+1 have the strong system property? Question

Is the strong system property equivalent to the robust tree property?
54. ### Questions Question Can ℵω+1 have the strong system property? Question

Is the strong system property equivalent to the robust tree property? Question Is the tree property equivalent to the strong system property for systems with countably many relations?