Approachability at the second successor of a singular cardinal

We prove that if μ is a regular cardinal and P is a μ-centered forcing poset, then P 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. In this paper we construct models in which the approachability property APμ+ fails, where μ is a singular cardinal. We also obtain models where the properties of being internally club and internally approachable are distinct for models of size the successor of a singular cardinal. These results are related to the approachability ideal I[μ], where μ is a singular cardinal. Theorem 1. Suppose μ is a regular cardinal and P is a μ-centered forcing poset. Let λ = μ. Then P forces that (I[λ]) generates I[λ] modulo clubs. Theorem 2. Suppose μ is supercompact and λ > μ is Mahlo. (1) For any regular cardinal ν < μ, there is a forcing poset which forces that μ is a singular strong limit cardinal with cofinality ν and APμ+ fails. (2) For any limit ordinal α < μ, there is a forcing poset which forces that μ = אα and APμ+ fails. Theorem 3. Suppose μ < λ are supercompact cardinals. (1) For any regular ν < μ, there is a forcing poset which forces that μ is a singular strong limit cardinal with cofinality ν, and for all regular θ ≥ μ, there are stationarily many sets N in [H(θ)] + which are internally club but not internally approachable. (2) For any limit ordinal α < μ, there is a forcing poset which forces that μ = אα, and for all regular θ ≥ μ, there are stationarily many sets N in [H(θ)]μ+ which are internally club but not internally approachable. We assume that the reader has some basic familiarity with iterated forcing, generalized stationarity, generic elementary embeddings, and the interaction of forcing with elementary substructures. For a cardinal μ and a set X containing μ, a set S ⊆ [X]μ+ is stationary if for any function F : [X] → X there is N in S such that μ ⊆ N and N is closed under F . This is equivalent to the property of having non-empty intersection with every closed and unbounded subset of [X] + . For a regular cardinal μ we let Add(μ) denote the Cohen forcing for adding a Cohen