A model of Cummings and Foreman revisited

Abstract This paper concerns the model of Cummings and Foreman where from ω supercompact cardinals they obtain the tree property at each ℵ n for 2 ≤ n ω . We prove some structural facts about this model. We show that the combinatorics at ℵ ω + 1 in this model depend strongly on the properties of ω 1 in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either ℵ ω + 1 ∈ I [ ℵ ω + 1 ] and every stationary subset of ℵ ω + 1 reflects or there are a bad scale at ℵ ω and a non-reflecting stationary subset of ℵ ω + 1 ∩ cof ( ω 1 ) . We also prove that regardless of the ground model a strong generalization of the tree property holds at each ℵ n for n ≥ 2 .