Approachability at the second successor of a singular cardinal

Abstract

We prove that if μ is a regular cardinal and ℛ is a μ-centered forcing poset, then ℛ forces that (I[μ⁺⁺])^V generates I[μ⁺⁺] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.