Abstract. Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at ℵω+1, and at ℵn for all 2 ≤ n < ω. A model with the former was obtainedby Magidor–Shelahfrom a huge cardinaland ω supercompactcardinals above it, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals. MSC-2010: 03E35, 03E05, 03E55
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