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.
Moti Gitik. John Krueger. "Approachability at the second successor of a singular cardinal." J. Symbolic Logic 74 (4) 1211 - 1224, December 2009. https://doi.org/10.2178/jsl/1254748688