We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω1 + 1)-game-closed forcings. PFA can be destroyed by a strongly (ω1 + 1)-game-closed forcing but not by an ω2-closed.
We present several forcing posets for adding a non-reflecting stationary subset of Pω 1 (λ), where λ ≥ ω 2. We prove that PFA is consistent with dense non-reflection in Pω 1 (λ), which means that every stationary subset of Pω 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… (More)