Dense non-reflection for stationary collections of countable sets

Abstract

We present several forcing posets for adding a non-reflecting stationary subset of P ω 1 ( λ ) , where λ ≥ ω 2 . We prove that PFA is consistent with dense non-reflection in P ω 1 ( λ ) , which means that every stationary subset of P ω 1 ( λ ) contains a stationary subset which does not reflect to any set of size ℵ 1 . If λ is singular with countable cofinality, then dense non-reflection in P ω 1 ( λ ) follows from the existence of squares.