Around Jensen’s square principle Young Researchers in Set Theory K¨ onigswinter, Germany 22-March-2011 Assaf Rinot Ben-Gurion University of the Negev 1 / 36
Introduction 2 / 36
Ladder systems. A discussion Definition A ladder for a limit ordinal α is a cofinal subset of α . 3 / 36
Ladder systems. A discussion Definition A ladder for a limit ordinal α is a cofinal subset of α . A ladder for a successor ordinal α + 1 is the singleton { α } . 3 / 36
Ladder systems. A discussion Definition A ladder for a limit ordinal α is a cofinal subset of α . A ladder for a successor ordinal α + 1 is the singleton { α } . Definition A ladder system over a cardinal κ is a sequence, � A α | α < κ � , such that each A α is a ladder for α . 3 / 36
Ladder systems. A discussion Definition A ladder for a limit ordinal α is a cofinal subset of α . A ladder for a successor ordinal α + 1 is the singleton { α } . Definition A ladder system over a cardinal κ is a sequence, � A α | α < κ � , such that each A α is a ladder for α . Remark The existence of ladder systems follows from the axiom of choice. 3 / 36
Ladder systems. Famous applications Partitioning a stationary set The standard proof of the fact that any stationary subset of ω 1 can be partitioned into uncountably many mutually disjoint stationary sets builds on an analysis of ladder systems over ω 1 . Strong colorings, ω 1 �→ [ ω 1 ] 2 ω 1 Todorcevic established the existence of a function f : [ ω 1 ] 2 → ω 1 such that f “[ U ] 2 = ω 1 for every uncountable U ⊆ ω 1 . This function f is determined by a ladder system over ω 1 . 4 / 36
A particular ladder system Definition (Jensen, 1960’s) � λ asserts the existence of a ladder system over λ + , � C α | α < λ + � , such that for all α < λ + : ◮ (Ladders are closed) C α is a club in α ; ◮ (Ladders are of bounded type) otp( C α ) ≤ λ ; ◮ (Coherence) if sup( C α ∩ β ) = β , then C α ∩ β = C β . 5 / 36
A particular ladder system Definition (Jensen, 1960’s) � λ asserts the existence of a ladder system over λ + , � C α | α < λ + � , such that for all α < λ + : ◮ (Ladders are closed) C α is a club in α ; ◮ (Ladders are of bounded type) otp( C α ) ≤ λ ; ◮ (Coherence) if sup( C α ∩ β ) = β , then C α ∩ β = C β . Famous applications The existence of various sorts of λ + -trees; The existence of non-reflecting stationary subsets of λ + ; The existence of other incompact objects. 5 / 36
A particular ladder system Definition (Jensen, 1960’s) � λ asserts the existence of a ladder system over λ + , � C α | α < λ + � , such that for all α < λ + : ◮ (Ladders are closed) C α is a club in α ; ◮ (Ladders are of bounded type) otp( C α ) ≤ λ ; ◮ (Coherence) if sup( C α ∩ β ) = β , then C α ∩ β = C β . Today’s talk would be centered around the above principle, but let us dedicate some time to discuss abstract ladder systems. 5 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. 6 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. Example of such sense: “There exists A ⊆ κ such that A α = A ∩ α for club many α < κ .” 6 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. Example of such sense: “There exists A ⊆ κ such that A α = A ∩ α for club many α < κ .” If κ is a large cardinal, then we may necessarily face means of trivi- ality. Fact (Rowbottom, 1970’s) If κ is measurable, then every ladder system � A α | α < κ � , admits a set A ⊆ κ such that A α = A ∩ α for stationary many α < κ . 6 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. Example of such sense: “There exists A ⊆ κ such that A α = A ∩ α for club many α < κ .” On the other hand, if κ is non-Mahlo, then for every cofinal A ⊆ κ , the following set contains a club: { α < κ | cf( α ) < otp( A ∩ α ) } . This suggests that non-triviality may be insured here, by setting a global bound on otp( A α ), e.g., letting otp( A α ) = cf( α ) for all α . 6 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. It turns out that requiring that otp( A α ) = cf( α ) for all α does not eliminate all means of triviality. For instance, it may be the case that any sequence of functions defined on the ladders is necessarily induced from a single κ -sized object. 7 / 36
Triviality of ladder systems Means of triviality A ladder system � A α | α < κ � is considered to be trivial, if, in some sense, it is determined by a single κ -sized object. It turns out that requiring that otp( A α ) = cf( α ) for all α does not eliminate all means of triviality. For instance, it may be the case that any sequence of functions defined on the ladders is necessarily induced from a single κ -sized object. Fact (Devlin-Shelah, 1978) MA ω 1 implies that any ladder system � A α | α < ω 1 � satisfying otp( A α ) = cf( α ) for every α , is trivial in the following sense. For every sequence of local functions � f α : A α → 2 | α < ω 1 � there exists a global function f : ω 1 → 2 such that for each α : f α = f ↾ A α (mod finite). 7 / 36
Nontrivial ladder systems over ω 1 In contrast, the following concept yields a ladder system which is resistant to Devlin and Shelah’s notion of triviality. Definition (Ostaszweski’s ♣ ) ♣ asserts the existence of a ladder system � A α | α < ω 1 � such that for every cofinal A ⊆ ω 1 , there exists a limit α < ω 1 with A α ⊆ A . 8 / 36
Nontrivial ladder systems over ω 1 In contrast, the following concept yields a ladder system which is resistant to Devlin and Shelah’s notion of triviality. Definition (Ostaszweski’s ♣ ) ♣ asserts the existence of a ladder system � A α | α < ω 1 � such that for every cofinal A ⊆ ω 1 , there exists a limit α < ω 1 with A α ⊆ A . Indeed, if � A α | α < ω 1 � is a ♣ -sequence, then for every global f : ω 1 → 2, there exists a limit α < ω 1 for which f ↾ A α is constant. Thus, if f α : A α → 2 partitions A α into two cofinal subsets for all limit α , then no global f trivializes the sequence � f α | α < ω 1 � . 8 / 36
Improve your square! Suppose that κ = λ + is a successor cardinal. Thus, we are interested in a ladder system � A α | α < κ � with ALL of the following features: 1. the set { otp( A α ) | α < κ } is bounded below κ ; 2. the ladders are closed; 3. the ladders cohere; 4. yields a canonical partition of κ into mutually disjoint stationary sets; 5. induces strong colorings; 6. a non-triviality condition ` a la Devlin-Shelah. 9 / 36
The Ostaszewski square 10 / 36
λ -sequences We propose a principle which combines � λ together with ♣ λ + . 11 / 36
λ -sequences We propose a principle which combines � λ together with ♣ λ + . For clarity, let us adopt the next ad-hoc terminology: Definition A sequence � A i | i < λ � is a λ -sequence if the following two holds: 1. each A i is a cofinal subset of λ + ; 2. if i < λ is a limit ordinal, then A i is moreover closed. Remark. Clause (2) may be viewed as a continuity condition. 11 / 36
The Ostaszewski square Definition ♣ λ asserts the existence of a ladder system − → C = � C α | α < λ + � such that: ◮ otp( C α ) ≤ λ for all α < λ + ; ◮ C α is a club in α for all limit α < λ + ; ◮ if sup( C α ∩ β ) = β , then C α ∩ β = C β ; 12 / 36
The Ostaszewski square Definition ♣ λ asserts the existence of a ladder system − → C = � C α | α < λ + � such that: ◮ − → C is a � λ -sequence. Let C α ( i ) denote the i th element of C α . 12 / 36
The Ostaszewski square Definition ♣ λ asserts the existence of a ladder system − → C = � C α | α < λ + � such that: ◮ − → C is a � λ -sequence. Let C α ( i ) denote the i th element of C α . ◮ Suppose that � A i | i < λ � is a λ -sequence. Then for every cofinal B ⊆ λ + , and every limit θ < λ , there exists some α < λ + such that: 1. otp( C α ) = θ ; 2. for all i < θ , C α ( i ) ∈ A i ; 3. for all i < θ , there exists β i ∈ B with C α ( i ) < β i < C α ( i + 1). 12 / 36
The Ostaszewski square (cont.) ♣ λ asserts the existence of a � λ -sequence � C α | α < λ + � such that for every λ -sequence � A i | i < λ � , every cofinal B ⊆ λ + , and every limit θ < λ , there exists some α < λ + such that: 1. the inverse collapse of C α is an element of � i <θ A i ; 13 / 36
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