Weak square and stationary reflection

Abstract

It is well-known that the square principle $${\square_\lambda}$$ entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle $${\square^{*} _\lambda}$$ does not. Here we show that if μcf(λ) < λ for all μ < λ, then $${\square^{*} _\lambda}$$ entails the existence of a non-reflecting stationary subset of $${E^{\lambda^+}_{{\rm cf}(\lambda)}}$$ in the forcing extension for adding a single Cohen subset of λ+. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of $${\square^{*} _\lambda}$$ for every singular cardinal λ of countable cofinality.