Forcing axioms and cardinal arithmetic II Boban Velickovic EquipedeLogique Universite´ deParis7 Logic Colloquium 2006, Nijmegen, July 27- August 2 2006 TheMappingReflectionPrinciple TheP-idealdichotomy Outline 1 The Mapping Reflection Principle The statement of MRP MRP and the continuum MRP and (cid:3)(κ) 2 The P-ideal dichotomy P-ideals The Singular Cardinal Hypothesis TheMappingReflectionPrinciple TheP-idealdichotomy Outline 1 The Mapping Reflection Principle The statement of MRP MRP and the continuum MRP and (cid:3)(κ) 2 The P-ideal dichotomy P-ideals The Singular Cardinal Hypothesis TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP The statement of MRP Definition Let θ be a regular cardinal, let X be uncountable, and let M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is θ M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club, Σ∩E ∩M 6= ∅. If Σ in M then this is just the usual notion, but we will be interested precisely in the case when Σ is not in M. Example LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype ℵ2 1 < δ. Then δ\A is M-stationary. TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP The statement of MRP Definition Let θ be a regular cardinal, let X be uncountable, and let M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is θ M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club, Σ∩E ∩M 6= ∅. If Σ in M then this is just the usual notion, but we will be interested precisely in the case when Σ is not in M. Example LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype ℵ2 1 < δ. Then δ\A is M-stationary. TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP The statement of MRP Definition Let θ be a regular cardinal, let X be uncountable, and let M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is θ M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club, Σ∩E ∩M 6= ∅. If Σ in M then this is just the usual notion, but we will be interested precisely in the case when Σ is not in M. Example LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype ℵ2 1 < δ. Then δ\A is M-stationary. TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP Recall the Ellentuck topology on [X]ω. Basic open sets are of the form [x,N] = {Y ∈ [X]ω:x ⊆ Y ⊆ N} where N ∈ [X]ω and x ⊆ N is finite. TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP Definition A set mapping Σ is open stationary iff there is an uncountable set X = X and a regular cardinal θ = θ such that [X]ω ∈ H , Σ Σ θ dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and θ M-stationary, for every M ∈ dom(Σ). The Mapping Reflection Principle [Moore] If Σ is an open stationary set mapping whose domain is a club, ~ there is a continuous ∈-chain N = (N :ξ < ω ) of elements ξ 1 dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such 1 that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ. η Σ ξ If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is ξ 1 said to be a reflecting sequence for Σ TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP Definition A set mapping Σ is open stationary iff there is an uncountable set X = X and a regular cardinal θ = θ such that [X]ω ∈ H , Σ Σ θ dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and θ M-stationary, for every M ∈ dom(Σ). The Mapping Reflection Principle [Moore] If Σ is an open stationary set mapping whose domain is a club, ~ there is a continuous ∈-chain N = (N :ξ < ω ) of elements ξ 1 dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such 1 that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ. η Σ ξ If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is ξ 1 said to be a reflecting sequence for Σ TheMappingReflectionPrinciple TheP-idealdichotomy ThestatementofMRP Definition A set mapping Σ is open stationary iff there is an uncountable set X = X and a regular cardinal θ = θ such that [X]ω ∈ H , Σ Σ θ dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and θ M-stationary, for every M ∈ dom(Σ). The Mapping Reflection Principle [Moore] If Σ is an open stationary set mapping whose domain is a club, ~ there is a continuous ∈-chain N = (N :ξ < ω ) of elements ξ 1 dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such 1 that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ. η Σ ξ If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is ξ 1 said to be a reflecting sequence for Σ