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 μ^cf(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.