Internal Approachability and Reflection

We prove that the Weak Reflection Principle does not imply that every stationary set reflects to an internally approachable set. We show that several variants of internal approachability, namely internally unbounded, internally stationary, and internally club, are not provably equivalent. Let λ ≥ ω2 be a regular cardinal, and consider an elementary substructure N ≺ H(λ) with size א1. Then N is internally approachable if N is the union of an increasing and continuous chain 〈ai : i < ω1〉 of countable sets such that for all β < ω1, 〈ai : i < β〉 is in N . The Weak Reflection Principle or WRP is the statement that for all regular λ ≥ ω2, for every stationary set S ⊆ Pω1(H(λ)) there is a set N ⊆ H(λ) with size א1 which contains א1 such that S ∩Pω1(N) is stationary in Pω1(N) (that is, S reflects to N). This principle follows from Martin’s Maximum and captures some of its strength; for example, WRP implies Chang’s Conjecture, the presaturation of the non-stationary ideal on ω1, and the Singular Cardinal Hypothesis (see [3] and [6]). An apparent strengthening of WRP is the statement that for all regular λ ≥ ω2, every stationary subset of Pω1(H(λ)) reflects to an internally approachable set with size א1. In practice it tends to be easier to draw strong consequences from this principle than from WRP. Foreman and Todorčević [4] asked whether the two reflection principles are equivalent. We answer this question in the negative. Theorem 0.1. Suppose κ is supercompact. Then there is a forcing poset which forces κ = ω2, WRP holds, and for all regular λ ≥ ω2 there is a stationary subset of Pω1(H(λ)) which does not reflect to any internally approachable set with size א1. Foreman and Todorčević [4] described several variations of the notion of internal approachability, and asked whether these properties are equivalent. Let N ≺ H(λ) be a set with size א1. Then N is internally club if N∩Pω1(N) contains a club subset of Pω1(N). N is internally stationary if N ∩ Pω1(N) is stationary in Pω1(N). N is internally unbounded if N ∩ Pω1(N) is cofinal in Pω1(N). We prove that these properties are not equivalent. Theorem 0.2. MM implies that for all regular λ ≥ ω2, there is a stationary set of N in Pω2(H(λ)) which are internally unbounded but not internally stationary. Theorem 0.3. PFA2 implies that for all regular λ ≥ ω2, there is a stationary set of N in Pω2(H(λ)) which are internally stationary but not internally club. I would like to thank James Cummings, Matt Foreman, and Assaf Sharon for discussing the material in this paper with me. I note that Theorem 0.2 was proved independently by Assaf Sharon with essentially the same argument. This work was partially supported by FWF project number P16790-N04.