Robust reflection principles Chris Lambie-Hanson Einstein Institute of Mathematics Hebrew University of Jerusalem 2016 ASL Winter Meeting Seattle, WA 9 January 2016

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

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.

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.

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.

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 = ∅.

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.

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.

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.

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.

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.

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)).

The tree property Definition A partial order (T,

The tree property Definition A partial order (T,

The tree property Definition A partial order (T,

The tree property Definition A partial order (T,

The tree property Definition A partial order (T,

The tree property Definition A partial order (T,

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

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

The tree property Definition Let κ be a regular, uncountable cardinal. 1 A tree (T,

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.

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.

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.

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?

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.

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?

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

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).

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

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.

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.

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.

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 = ∅.

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.

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.

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:

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.

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.

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.

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

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.

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 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.

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.

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

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

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?

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