Weak square and stationary reflection

Joint work with Gunter Fuchs.

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 $\square^*_\lambda$ does not.
Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, then $\square^*_\lambda$ entails the existence of a non-reflecting stationary subset of $E^{\lambda^+}_{cf(\lambda)}$ 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 $\square^*_\lambda$ for every singular cardinal $\lambda$ of countable cofinality.

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