The reflection of stationary subsets of \({\cal{P}}_{\omega_{1}}(H)\) for all sets H ⊇ ω1, which we denote by \(\text{SR}_{\omega_{1}}\) , is known to imply that λω = λ for all regular cardinals λ ≥ ω2. In particular, it implies 2ω ≤ ω2 and the Singular Cardinal Hypothesis. For a regular cardinal κ ≥ ω2, the reflection of stationary subsets of \({\cal{P}}_{\kappa}(H)\) for all H ⊇ κ is inconsistent with ZFC. But its restriction to stationary sets consisting of internally approachable sets, which we denote by SRκ ↾ IA, is consistent with ZFC. In this paper, we study consequences of SRκ ↾ IA on cardinal arithmetic.
We prove that SRκ ↾ IA does not give any bound on 2μ for any regular uncountable cardinal μ, while it implies λω = λ for all regular cardinals λ ≥ κ+. We also prove that SRκ ↾ IA>ω does not give any bound on 2ω and does not imply the Singular Cardinal Hypothesis, where SRκ ↾ IA>ω denotes the reflection of stationary subsets of \({\cal{P}}_{\kappa}(H)\) consisting of internally approachable sets of uncountable cofinalities.
