NAMBA FORCING AND NO GOOD SCALE

Abstract. We develop a version of Namba forcing which is useful for con-structing models with no good scale on ℵω. A model is produced in which ℵn holds for all finite n ≥ 1, but there is no good scale on ℵω; this strengthens a theorem of Cummings, Foreman, and Magidor [3] on the non-compactness of square. Cummings, Foreman, and Magidor [4] proved that if the square principle ℵn holds for all finite n ≥ 1, then there exists a “square-like ” sequence on ℵω+1 which exhibits uniformity at ordinals of uncountable cofinality. But in [3] they showed that the existence of squares on cardinals less than ℵω fails to imply the existence of a square sequence on ℵω+1. Specifically, they constructed a model in which ℵn holds for all n ≥ 1, but every stationary subset of ℵω+1 ∩ cof(ℵ0) reflects to an ordinal in ℵω+1 of cofinality ℵ1. Since ℵω implies that every stationary subset of ℵω+1 contains a non-reflecting stationary subset, ℵω fails in this model. Stationary set reflection is consistent with weaker forms of square. For example, in [2] it is proven that the weak square principle ∗ℵω is consistent with the state