Generalized stationary reflection and cardinal arithmetic

Abstract The reflection of stationary subsets of P_₁ (H) P ω 1 (H) for all sets H ⊇ ω 1, which we denote by SR_₁ SR ω 1, is known to imply that λ ω = λ for all regular cardinals λ ≥ ω 2. In particular, it implies 2 ω ≤ ω 2 and the Singular Cardinal Hypothesis. For a regular cardinal κ ≥ ω 2, the reflection of stationary subsets of P_ (H) P κ (H) for all H ⊇ κ is inconsistent with ZFC. But its restriction to stationary sets consisting of internally approachable sets, which we denote by SR κ ↾ IA, is consistent with ZFC. In this paper, we study consequences of SR κ ↾ IA on cardinal arithmetic. We prove that SR κ ↾ IA does not give any bound on 2 μ for any regular uncountable cardinal μ, while it implies λ ω = λ for all regular cardinals λ ≥ κ +. We also prove that SR κ ↾ IA > ω does not give any bound on 2 ω and does not imply the Singular Cardinal Hypothesis, where SR κ ↾ IA > ω denotes the reflection of stationary subsets of P_ (H) P κ (H) consisting of internally approachable sets of uncountable cofinalities.