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.
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.
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.
> 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.
> 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.
> 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.
> 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.
> 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.
> 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)).
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).
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}.
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 = ∅.
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 = ∅.
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. 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.
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.
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.
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.
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.
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)).
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)).
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.
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.
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.
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?
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.
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?
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).
uncountable cardinal. Then the following are equivalent. 1 Every stationary subset of κ reﬂects at arbitrarily high coﬁnalities. 2 Reﬂ(κ) holds robustly.
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.
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 = ∅.
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.
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:
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.
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.
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 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 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 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 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.
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?