Weak square and stationary reflection

TL;DR

This paper explores the relationship between weak square principles and stationary reflection, showing that certain forcing extensions and axioms imply the failure of weak square at singular cardinals.

Contribution

It establishes that indestructible stationary reflection implies the failure of weak square principles at singular cardinals, settling a question related to the subcomplete forcing axiom.

Findings

Weak square implies non-reflecting stationary subsets in certain extensions.

Indestructible stationary reflection entails the failure of weak square.

SCFA implies the failure of _ for all singular  of countable cofinality.

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 $\mu^{\mathrm{cf}(\lambda)} < \lambda$ for all $\mu < \lambda$, then $\square^*_\lambda$ entails the existence of a non-reflecting stationary subset of $E^{\lambda^+}_{\mathrm{cf}(\lambda)}$ 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.