Internally Club and Approachable for Larger Structures

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Suppose μ < κ, μ<μ = μ, and κ is supercompact. Then there is a generic extension in which κ = μ++, and for all regular λ ≥ μ++, there are stationarily many N in [H(λ)]μ which are internally club but not internally approachable. Suppose μ is an infinite cardinal. A set N is internally approachable with length μ if N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that for all α < μ, 〈Ni : i < α〉 is in N . A related idea is that of an internally club set. A set N with size μ is internally club if N ∩ [N ] contains a club subset of [N ]. In other words, N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that each Ni is in N . Foreman and Todorčević [3] asked whether the properties of being internally approachable and internally club are equivalent. In [5] we proved that under PFA, for all regular λ ≥ ω2 there are stationarily many structures N ≺ H(λ) with size א1 such that N is internally club but not internally approachable. In this paper we generalize this result to larger structures. Theorem 1. Suppose μ < κ, μ = μ, and κ is supercompact. Then there is a μ-closed, μ-proper forcing poset which collapses κ to become μ, and forces that for all regular λ ≥ μ, there are stationarily many N in [H(λ)]μ+ which are internally club but not internally approachable. In the model we construct to prove Theorem 1, we have that 2 = μ. In fact, if 2 = μ, then any elementary substructure N ≺ H(λ) with size μ and which contains μ is internally club iff it is internally approachable; this is shown at the end of the paper. In Section 1 we review notation and some background material. Section 2 generalizes the idea of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Section 3 presents the basic forcing poset we use in our consistency result, and in Section 4 we describe how to iterate this poset with a mixed support forcing iteration. In Section 5 we prove Theorem 1.