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 ${\lambda }^{+}$, whereas the weak square principle ${\square }_{\lambda }^{\ast }$ does not. Here we show that if ${\mu }^{\mathrm{cf}(\lambda )}<\lambda$ for all $\mu <\lambda$, then ${\square }_{\lambda }^{\ast }$ entails the existence of a non-reflecting stationary subset of $E_{\mathrm{cf}(\lambda )}^{{\lambda }^{+}}$ in the forcing extension for adding a single Cohen subset of ${\lambda }^{+}$. 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 ${\square }_{\lambda }^{\ast }$ for every singular cardinal $\lambda$ of countable cofinality.

11 pages