The weak □* is really weaker than the full □

Abstract

Abstract We show that relative to the consistency of a supercompact cardinal does not imply . The model-theoretic transfer property ⟨ℵ 1 , ℵ 0 ⟩ → ⟨ℵ ω + 1 , ℵ ω ⟩ does not imply , and it is consistent to have an ultrafilter on ℵ ω + 1 which is λ -indecomposible for all ω < λ < ℵ ω .