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The Proper Forcing Axiom and the Singular Cardinal Hypothesis

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The Proper Forcing Axiom and the Singular Cardinal Hypothesis The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria The Proper Forcing Axiom and the Singular Cardinal Hypothesis Matteo Viale Vienna, Preprint ESI 1697 (2005) September 14, 2005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at The Proper Forcing Axiom and the Singular Cardinal Hypothesis Matteo Viale ∗ Abstract We show that the Proper Forcing Axiom implies the Singular Cardinal Hy- pothesis. The proof uses ideas of Moore from [11] and the notion of a relativized trace function on pairs of ordinals. Introduction In one of the first applications of the forcing techniques, Easton [3] showed that the exponential function κ 7→ 2κ on regular cardinals can be arbitrary modulo some mild restrictions. The situation for singular cardinals is much more subtle. For instance, Silver [14] showed that the Singular Cardinal Hypothesis SCH cannot first fail at a singular cardinal of uncountable cofinality. Recall that SCH states that 2κ = κ+, for all singular strong limit cardinals κ. However, it is known that SCH can first fail even at ℵω. The role of large cardinals in this context is twofold. On the one hand they are necessary for the construction of models of the negation of SCH (see [5] for a survey of Prikry type forcings and applications to SCH). On the other hand it is a theorem of Solovay [15] that SCH holds above a strongly compact cardinal. Forcing axioms imply reflection principles similar to the one used in Solovay’s proof, thus it was reasonable to expect that they would also settle SCH. Indeed, in [4], Foreman, Magidor and Shelah showed that the strongest forcing axiom, Martin’s Maximum MM, implies SCH. This was later improved by Veliˇckovi´c [17] who also showed that SCH follows from PFA+. In fact, what is shown in [17] ω is that if θ > ℵ1 is regular and stationary subsets of [θ] reflect to an internally closed and unbounded set, then θω = θ. This, combined with Silver’s theorem, implies SCH. At this point, it was left open whether SCH is a consequence of PFA. Very little progress was made on this problem for over fifteen years. Then, in 2003, Moore [10] introduced a new reflection principle, the Mapping Reflection Principle MRP and deduced it from PFA. He showed that MRP implies the continuum is equal to ℵ2 and the failure of 2(κ), for all κ > ℵ1. MRP has ∗ Dipartimento di Matematica, Universit´adi Torino and Equipe de Logique Math´ematique, Uni- versit´eParis 7, email: [email protected]. 2000 Mathematical Subject Classification: 03E05, 03E10, 03E65, 03E75. Key words: MRP, PFA, SCH, minimal walks. 1 many features in common with the reflection principles which follow from MM, so it should be expected that MRP could affect the behaviour of the exponential function also on higher cardinals. In fact, Moore showed in [11] that if MRP holds and κ > ω1 is a regular cardinal with a nonreflecting stationary set consisting of points of countable cofinality, then κω1 = κ. This, combined with the above result of Veliˇckovi´c, strongly suggests that PFA implies SCH. In this paper we confirm this conjecture. The new technical tool we introduce is the notion of a relativized trace function defined on pairs of ordinals. This allows us to get rid of the nonreflecting stationary set in Moore’s argument. The paper is organized as follows. In section §1 we introduce the relativized trace, which is a generalization of the functions considered in [16], and we outline the properties of this function which are relevant for our purposes. In section ω §2 we prove that if MRP holds then κ = κ for all regular cardinals κ > ω1. This, together with Silver’s theorem, implies that SCH follows from PFA. In the final section we show that the techniques introduced in this paper can be applied to study another interesting problem in the area of forcing axioms, i.e. to investigate what kind of forcing notions can preserve this type of axioms. Our notation is standard and follows [7] and [11]. For a regular cardinal θ, we use H(θ) to denote the structure (H(θ), ∈, <) whose domain is the collection of sets whose transitive closure is of size less than θ and where < is a predicate for a fixed well ordering of H(θ). If X is an uncountable set, C ⊆ [X]ω is closed and unbounded (club) if there is f : [X]<ω → X such that C is the set of all Y ∈ [X]ω such that f[Y ]<ω ⊆ Y . S ⊆ [X]ω is stationary if it intersects all club subsets of [X]ω. The f-closure of X is the smallest Y containing X such that f[Y ]<ω ⊆ Y . If X is a set of ordinals then X denotes the topological closure of X in the order topology. 1 The relativized trace function In this section we introduce a relativized version of the trace function considered by Todorˇcevi´c(see [16]) and outline some of its relevant properties. This function describes a descending walk from a larger ordinal β to a smaller ordinal α similar to the standard trace function. The rules of the walk are determined by a parameter X. This relativized trace function is designed in order to control very well its behavior on the points of the walk belonging to X. Given κ regular cardinal, fix a sequence (Cα : α ∈ κ & cof α = ω) such that Cα is a cofinal subset of α of order type ω for all α < κ of cofinality ω. ≤ω Definition 1 Let X ∈ [κ] , then for all α, β ∈ κ, trX (α, β) is defined as follows: (i) if β < α then trX (α, β) = ∅ (ii) trX (α, α) = {α} (iii) trX (α, β +1) = trX (α, β) ∪{β +1} (iv) if β > α is a limit ordinal then: if cof β > ω then trX (α, β) = trX (α, sup(β ∩ X)) ∪{β}, 2 if cof β = ω then set trX(α, β) = trX (α, min(Cβ \ α)) ∪{β}. So, walking from β to α we treat points which are either successors or of countable cofinality as in the usual walk. However, points of uncountable cofinality are treated as successors in the sense that we go from β to sup(β ∩ X). We now state the key properties of the function trX which we will use. Fact 1 For α, β < κ we have the following. (i) If α ∈ X and β ≥ α, then α ∈ trX (α, β) (ii) For all α ≤ β there is a γ < α such that trX (α, β) ⊆ trX (η,β), for all η ∈ (γ, α]. Proof: (i) If α ∈ X then sup(β ∩ X) ≥ α for any β > α. We proceed by induction. The base case is trivial by (ii) in the definition of trX. If the claim holds for all γ such that α ≤ γ<β, we have three cases, β successor, β of countable cofinality, β of uncountable cofinality. (a) If β = γ+1, the induction hypothesis applies to trX(α, γ). Since trX(α, β) = trX (α, γ) ∪{β} we are done. (b) If cof(β) = ω, then the induction hypothesis applies to trX (α, min(Cβ \ α)), since α ≤ min(Cβ \ α) < β. Now trX (α, β) = trX(α, sup(Cβ \ α)) ∪{β}, so we are done. (c) If cof(β) > ω then α ≤ sup(β ∩ X) <β and trX (α, β) = trX (α, sup(β ∩ X)) ∪{β} now the induction hypothesis applies to trX (α, sup(β ∩ X)). This concludes the proof of (i). (ii) We may assume that α is a limit ordinal. Then there are two cases: (a) sup(α ∩ X) = α. Then let γβ = max{max(Cη ∩ α) : η ∈ trX (α, β) ∩ cof(ω),η 6= α}. It is easy to check by induction on β ≥ α that if ρ ∈ (γβ, α] then trX(α, β) ⊆ trX(ρ,β), using the fact that γβ ≥ γη and that sup(η ∩ X) ≥ α, for all η ∈ trX(α, β). (b) sup(α ∩ X) < α. Then proceed as above taking γβ = max({sup(α ∩ X)}∪{max(Cη ∩ α) : η ∈ trX (α, β) ∩ cof(ω),η 6= α}). 3 2 The main result The purpose of this section is to show that MRP implies that λω = λ, for every cardinal λ > ω1 with cof(λ) > ω. We start by recalling the relevant definitions from [10]. Definition 2 Let θ be a regular cardinal, let X be uncountable, and let M ≺ H(θ) be countable such that [X]ω ∈ M. A subset Σ of [X]ω is M-stationary if for all E ∈ M such that E ⊆ [X]ω is club, Σ ∩ E ∩ M 6= ∅. Recall that the Ellentuck topology on [X]ω is obtained by declaring a set open if it is the union of sets of the form [x,N] = {Y ∈ [X]ω : x ⊆ Y ⊆ N} where N ∈ [X]ω and x ⊆ N is finite. When we say ‘open’ in this paper we refer to this topology. Definition 3 Σ is an open stationary set mapping if there is an uncountable set X and a regular cardinal θ such that [X]ω ∈ H(θ), the domain of Σ is a club in [H(θ)]ω of countable elementary submodels M such that X ∈ M and for all M, Σ(M) ⊆ [X]ω is open and M-stationary. The Mapping Reflection Principle (MRP) asserts that: If Σ is an open stationary set mapping, there is a continuous ∈-chain N~ = (Nξ : ξ < ω1) of elements in the domain of Σ such that for all limit ordinals ξ < ω1 there is ν<ξ such that Nη ∩ X ∈ Σ(Nξ) for all η such that ν<η<ξ.
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