Generalized stationary reflection and cardinal arithmetic

The reflection of stationary subsets of P_ω₁(H) P ω 1 ( H ) for all sets H ⊇ ω 1 , which we denote by SR_ω₁ 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 Pκ(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 Pκ(H) P κ ( H ) consisting of internally approachable sets of uncountable cofinalities.