Intermediate Square Principles Chris Lambie-Hanson NY Graduate Student Logic Conference 19 April 2013

(λ) Definition Let λ be a regular, uncountable cardinal. − → C = Cα | α < λ is a coherent sequence if, for all limit ordinals α < β < λ, Cα is club in α If α is a limit point of Cβ (α ∈ Cβ ), then Cα = Cβ ∩ α.

(λ) Definition Let λ be a regular, uncountable cardinal. − → C = Cα | α < λ is a coherent sequence if, for all limit ordinals α < β < λ, Cα is club in α If α is a limit point of Cβ (α ∈ Cβ ), then Cα = Cβ ∩ α. T is a thread through − → C if T is club in λ and, for every α ∈ T , Cα = T ∩ α.

(λ) Definition Let λ be a regular, uncountable cardinal. − → C = Cα | α < λ is a coherent sequence if, for all limit ordinals α < β < λ, Cα is club in α If α is a limit point of Cβ (α ∈ Cβ ), then Cα = Cβ ∩ α. T is a thread through − → C if T is club in λ and, for every α ∈ T , Cα = T ∩ α. − → C is a (λ)-sequence if it is a coherent sequence and there is no thread through − → C .

κ Definition Let κ be an infinite cardinal. − → C = Cα | α < κ+ is a κ-sequence if it is a coherent sequence and, for all limit ordinals α < κ+, otp(Cα) ≤ κ.

κ Definition Let κ be an infinite cardinal. − → C = Cα | α < κ+ is a κ-sequence if it is a coherent sequence and, for all limit ordinals α < κ+, otp(Cα) ≤ κ. A κ sequence is a (κ+)-sequence.

κ Definition Let κ be an infinite cardinal. − → C = Cα | α < κ+ is a κ-sequence if it is a coherent sequence and, for all limit ordinals α < κ+, otp(Cα) ≤ κ. A κ sequence is a (κ+)-sequence. ω is always true.

κ Definition Let κ be an infinite cardinal. − → C = Cα | α < κ+ is a κ-sequence if it is a coherent sequence and, for all limit ordinals α < κ+, otp(Cα) ≤ κ. A κ sequence is a (κ+)-sequence. ω is always true. Square principles provide instances of incompactness.

Consistency Results If V = L, then κ holds for every infinite cardinal κ, and (λ) holds for every infinite regular cardinal λ that is not weakly compact.

Consistency Results If V = L, then κ holds for every infinite cardinal κ, and (λ) holds for every infinite regular cardinal λ that is not weakly compact. PFA implies that (λ) fails for every regular λ > ω1.

Consistency Results If V = L, then κ holds for every infinite cardinal κ, and (λ) holds for every infinite regular cardinal λ that is not weakly compact. PFA implies that (λ) fails for every regular λ > ω1. Failure of κ for a regular, uncountable κ is equiconsistent with a Mahlo cardinal.

Consistency Results If V = L, then κ holds for every infinite cardinal κ, and (λ) holds for every infinite regular cardinal λ that is not weakly compact. PFA implies that (λ) fails for every regular λ > ω1. Failure of κ for a regular, uncountable κ is equiconsistent with a Mahlo cardinal. Failure of (κ+) for a regular, uncountable κ is equiconsistent with a weakly compact cardinal.

Stationary Reflection Definition Let λ be a regular, uncountable cardinal and let S ⊆ λ be stationary. If α < λ and cf(α) > ω, then S reflects at α if S ∩ α is stationary in α.

Stationary Reflection Definition Let λ be a regular, uncountable cardinal and let S ⊆ λ be stationary. If α < λ and cf(α) > ω, then S reflects at α if S ∩ α is stationary in α. Definition Let − → C = Cα | α < λ be a coherent sequence, and let S ⊆ λ. − → C avoids S if, for every limit ordinal α < λ, Cα ∩ S = ∅.

Stationary Reflection Definition Let λ be a regular, uncountable cardinal and let S ⊆ λ be stationary. If α < λ and cf(α) > ω, then S reflects at α if S ∩ α is stationary in α. Definition Let − → C = Cα | α < λ be a coherent sequence, and let S ⊆ λ. − → C avoids S if, for every limit ordinal α < λ, Cα ∩ S = ∅. Note that, if S ⊆ λ is stationary and there is a coherent sequence of length λ that avoids S, then S does not reflect at any α < λ.

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T.

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T. Proof. Let − → C = Cα | α < κ+ be a κ-sequence. By Fodor’s lemma, we can find a stationary T ⊆ S and η ≤ κ such that otp(Cα) = η for all α ∈ T.

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T. Proof. Let − → C = Cα | α < κ+ be a κ-sequence. By Fodor’s lemma, we can find a stationary T ⊆ S and η ≤ κ such that otp(Cα) = η for all α ∈ T. Now adjust − → C to obtain − → D = Dα | α < κ+ as follows:

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T. Proof. Let − → C = Cα | α < κ+ be a κ-sequence. By Fodor’s lemma, we can find a stationary T ⊆ S and η ≤ κ such that otp(Cα) = η for all α ∈ T. Now adjust − → C to obtain − → D = Dα | α < κ+ as follows: If otp(Cα) ≤ η, let Dα = Cα.

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T. Proof. Let − → C = Cα | α < κ+ be a κ-sequence. By Fodor’s lemma, we can find a stationary T ⊆ S and η ≤ κ such that otp(Cα) = η for all α ∈ T. Now adjust − → C to obtain − → D = Dα | α < κ+ as follows: If otp(Cα) ≤ η, let Dα = Cα. If otp(Cα) > η, let γ ∈ Cα be such that otp(Cα ∩ γ) = η, and let Dα = Cα \ (γ + 1).

Stationary Reflection Proposition Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S and a κ sequence that avoids T. Proof. Let − → C = Cα | α < κ+ be a κ-sequence. By Fodor’s lemma, we can find a stationary T ⊆ S and η ≤ κ such that otp(Cα) = η for all α ∈ T. Now adjust − → C to obtain − → D = Dα | α < κ+ as follows: If otp(Cα) ≤ η, let Dα = Cα. If otp(Cα) > η, let γ ∈ Cα be such that otp(Cα ∩ γ) = η, and let Dα = Cα \ (γ + 1). − → D is a κ-sequence and avoids T.

Stationary Reflection Corollary Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S that does not reflect.

Stationary Reflection Corollary Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S that does not reflect. Proposition Suppose λ is an uncountable regular cardinal and − → C = Cα | α < λ is a coherent sequence. TFAE:

Stationary Reflection Corollary Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S that does not reflect. Proposition Suppose λ is an uncountable regular cardinal and − → C = Cα | α < λ is a coherent sequence. TFAE: 1. − → C is a (λ)-sequence.

Stationary Reflection Corollary Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S that does not reflect. Proposition Suppose λ is an uncountable regular cardinal and − → C = Cα | α < λ is a coherent sequence. TFAE: 1. − → C is a (λ)-sequence. 2. For every stationary S ⊆ λ, there are stationary T0, T1 ⊆ S such that, for every limit ordinal α < λ, Cα ∩ T0 = ∅ or Cα ∩ T1 = ∅.

Stationary Reflection Corollary Let κ be an uncountable cardinal and suppose κ holds. If S ⊆ κ+ is stationary, then there is a stationary T ⊆ S that does not reflect. Proposition Suppose λ is an uncountable regular cardinal and − → C = Cα | α < λ is a coherent sequence. TFAE: 1. − → C is a (λ)-sequence. 2. For every stationary S ⊆ λ, there are stationary T0, T1 ⊆ S such that, for every limit ordinal α < λ, Cα ∩ T0 = ∅ or Cα ∩ T1 = ∅. Corollary Suppose λ is an uncountable regular cardinal and (λ) holds. If S ⊆ λ is stationary, then there are stationary T0, T1 ⊆ S such that T0 and T1 do not reflect simultaneously.

Intermediate Square Principles

Intermediate Square Principles Definition Let µ < λ be infinite, regular cardinals. µ(λ) holds if there is − → C such that − → C is a (λ)-sequence and {α < λ | otp(Cα) = µ} is stationary.

Intermediate Square Principles Definition Let µ < λ be infinite, regular cardinals. µ(λ) holds if there is − → C such that − → C is a (λ)-sequence and {α < λ | otp(Cα) = µ} is stationary. Definition Let λ be a regular, uncountable cardinal, and let S ⊆ λ. (λ, S) holds if there is − → C such that − → C is a (λ)-sequence that avoids S.

Intermediate Square Principles Definition Let µ < λ be infinite, regular cardinals. µ(λ) holds if there is − → C such that − → C is a (λ)-sequence and {α < λ | otp(Cα) = µ} is stationary. Definition Let λ be a regular, uncountable cardinal, and let S ⊆ λ. (λ, S) holds if there is − → C such that − → C is a (λ)-sequence that avoids S. We are interested in knowing what implications and non-implications hold between these various intermediate square principles.

Implications Proposition Let λ be an uncountable regular cardinal.

Implications Proposition Let λ be an uncountable regular cardinal. 1. If µ < ν < λ are infinite, regular cardinals and ν(λ) holds, then µ(λ) holds.

Implications Proposition Let λ be an uncountable regular cardinal. 1. If µ < ν < λ are infinite, regular cardinals and ν(λ) holds, then µ(λ) holds. 2. If µ < λ is regular and µ(λ) holds, then there is a stationary S ⊆ λ ∩ cof(µ) such that (λ, S) holds.

Implications Proposition Let λ be an uncountable regular cardinal. 1. If µ < ν < λ are infinite, regular cardinals and ν(λ) holds, then µ(λ) holds. 2. If µ < λ is regular and µ(λ) holds, then there is a stationary S ⊆ λ ∩ cof(µ) such that (λ, S) holds. 3. If there is a stationary S ⊆ λ ∩ cof(ω) such that (λ, S) holds, then ω(λ) holds.

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where 1. βq < λ. 2. For all α ≤ βq, Cq α is club in α. 3. For all α < α ≤ βq, if α ∈ C α , then Cα = Cα ∩ α.

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where 1. βq < λ. 2. For all α ≤ βq, Cq α is club in α. 3. For all α < α ≤ βq, if α ∈ C α , then Cα = Cα ∩ α. p ≤ q if and only if p end-extends q.

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where 1. βq < λ. 2. For all α ≤ βq, Cq α is club in α. 3. For all α < α ≤ βq, if α ∈ C α , then Cα = Cα ∩ α. p ≤ q if and only if p end-extends q. Fact Forcing with Q(λ) does not add sequences of ordinals of length < λ.

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where 1. βq < λ. 2. For all α ≤ βq, Cq α is club in α. 3. For all α < α ≤ βq, if α ∈ C α , then Cα = Cα ∩ α. p ≤ q if and only if p end-extends q. Fact Forcing with Q(λ) does not add sequences of ordinals of length < λ. If λ<λ = λ, then |Q(λ)| = λ.

Forcing Square Sequences Let λ be an uncountable regular cardinal. Let Q(λ) be the partial order whose elements are of the form q = Cq α | α ≤ βq , where 1. βq < λ. 2. For all α ≤ βq, Cq α is club in α. 3. For all α < α ≤ βq, if α ∈ C α , then Cα = Cα ∩ α. p ≤ q if and only if p end-extends q. Fact Forcing with Q(λ) does not add sequences of ordinals of length < λ. If λ<λ = λ, then |Q(λ)| = λ. Forcing with Q(λ) adds a (λ)-sequence which is moreover a µ(λ)-sequence for every regular µ < λ.

Threading Square Sequences Let λ be an uncountable regular cardinal, and let − → C be a (λ)-sequence. Let T( − → C ) be the partial order consisting of elements t such that

Threading Square Sequences Let λ be an uncountable regular cardinal, and let − → C be a (λ)-sequence. Let T( − → C ) be the partial order consisting of elements t such that 1. t is a closed, bounded subset of λ. 2. For every α ∈ t , t ∩ α = Cα.

Threading Square Sequences Let λ be an uncountable regular cardinal, and let − → C be a (λ)-sequence. Let T( − → C ) be the partial order consisting of elements t such that 1. t is a closed, bounded subset of λ. 2. For every α ∈ t , t ∩ α = Cα. s ≤ t if and only if s end-extends t.

Threading Square Sequences Let λ be an uncountable regular cardinal, and let − → C be a (λ)-sequence. Let T( − → C ) be the partial order consisting of elements t such that 1. t is a closed, bounded subset of λ. 2. For every α ∈ t , t ∩ α = Cα. s ≤ t if and only if s end-extends t. Fact If ˙ − → C is a name for the (λ)-sequence added by Q(λ), then Q(λ) ∗ T( ˙ − → C ) has a λ-closed dense subset and T( − → C ) adds a thread through − → C .

Non-implications

Non-implications Suppose µ < κ are regular cardinals and λ > κ is a measurable cardinal. Let P = Coll(κ, < λ) and, in V P, let Q = Q(λ).

Non-implications Suppose µ < κ are regular cardinals and λ > κ is a measurable cardinal. Let P = Coll(κ, < λ) and, in V P, let Q = Q(λ). Theorem (L-H) In V P∗Q, λ = κ+, µ(λ) holds, and κ fails. (In fact, κ,<κ fails, though ∗ κ holds.)

Non-implications Suppose µ < κ are regular cardinals and λ > κ is a measurable cardinal. Let P = Coll(κ, < λ) and, in V P, let Q = Q(λ). Theorem (L-H) In V P∗Q, λ = κ+, µ(λ) holds, and κ fails. (In fact, κ,<κ fails, though ∗ κ holds.) In V P∗Q, let T be the forcing to thread the (λ)-sequence added by Q. Let ν ≤ κ be a regular cardinal. We can define a forcing iteration Sν of length λ+ such that, in V P∗Q∗Sν , if S ⊆ λ ∩ cof(ν) is stationary, then T “S is stationary”.

Non-implications Suppose µ < κ are regular cardinals and λ > κ is a measurable cardinal. Let P = Coll(κ, < λ) and, in V P, let Q = Q(λ). Theorem (L-H) In V P∗Q, λ = κ+, µ(λ) holds, and κ fails. (In fact, κ,<κ fails, though ∗ κ holds.) In V P∗Q, let T be the forcing to thread the (λ)-sequence added by Q. Let ν ≤ κ be a regular cardinal. We can define a forcing iteration Sν of length λ+ such that, in V P∗Q∗Sν , if S ⊆ λ ∩ cof(ν) is stationary, then T “S is stationary”. Theorem (L-H) Let µ < ν ≤ κ be regular cardinals. Then, in V P∗Q∗Sν , µ(λ) holds but, for every stationary S ⊆ λ ∩ cof(ν), (λ, S) fails.

Non-implications Theorem (Jensen) Suppose V = L. Let λ be an inaccessible cardinal which is not weakly compact, and let S ⊆ λ be stationary. Then there is a stationary T ⊆ S such that (λ, T) holds.

Non-implications Theorem (Jensen) Suppose V = L. Let λ be an inaccessible cardinal which is not weakly compact, and let S ⊆ λ be stationary. Then there is a stationary T ⊆ S such that (λ, T) holds. Theorem (Harrington-Shelah) Let κ be an uncountable regular cardinal, and let λ > κ be Mahlo. In V Coll(κ,<λ), there is a cardinal-preserving forcing iteration S of length λ+ such that, in V Coll(κ,<λ)∗S, every stationary subset of λ ∩ cof(< κ) reflects.

Non-implications Theorem (Jensen) Suppose V = L. Let λ be an inaccessible cardinal which is not weakly compact, and let S ⊆ λ be stationary. Then there is a stationary T ⊆ S such that (λ, T) holds. Theorem (Harrington-Shelah) Let κ be an uncountable regular cardinal, and let λ > κ be Mahlo. In V Coll(κ,<λ), there is a cardinal-preserving forcing iteration S of length λ+ such that, in V Coll(κ,<λ)∗S, every stationary subset of λ ∩ cof(< κ) reflects. Theorem (L-H) Suppose V = L. Let κ be an uncountable regular cardinal, and let λ be the least Mahlo cardinal greater than κ. If S is as in H-S, then, in V Coll(κ,<λ)∗S, there is a stationary S ⊆ λ ∩ cof(κ) such that (λ, S) holds.

Non-implications Theorem (Jensen) Suppose V = L. Let λ be an inaccessible cardinal which is not weakly compact, and let S ⊆ λ be stationary. Then there is a stationary T ⊆ S such that (λ, T) holds. Theorem (Harrington-Shelah) Let κ be an uncountable regular cardinal, and let λ > κ be Mahlo. In V Coll(κ,<λ), there is a cardinal-preserving forcing iteration S of length λ+ such that, in V Coll(κ,<λ)∗S, every stationary subset of λ ∩ cof(< κ) reflects. Theorem (L-H) Suppose V = L. Let κ be an uncountable regular cardinal, and let λ be the least Mahlo cardinal greater than κ. If S is as in H-S, then, in V Coll(κ,<λ)∗S, there is a stationary S ⊆ λ ∩ cof(κ) such that (λ, S) holds. (But (λ, T) fails for every stationary T ⊆ λ ∩ cof(< κ) because of stationary reflection.)

Summary at λ = ω2 ω1 ω(ω2) ω1 (ω2) ∃ stationary S ⊆ Sℵ2 ℵ0 ( (ω2, S)) ∃ stationary T ⊆ Sℵ2 ℵ1 ( (ω2, T)) (ω2)

Thank you!