Weak square and stationary reflection

Joint work with Gunter Fuchs.

Abstract. It is well-known that the square principle ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$ entails the existence of a non-reflecting stationary subset of ${\lambda }^{+}$, whereas the weak square principle ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ does not.
Here we show that if ${\mu }^{cf(\lambda )}<\lambda$ for all $\mu <\lambda$, then ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ entails the existence of a non-reflecting stationary subset of $E_{cf(\lambda )}^{{\lambda }^{+}}$ 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 ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ for every singular cardinal $\lambda$ of countable cofinality.

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Citation information:

G. Fuchs and A. Rinot, Weak square and stationary reflection, Acta. Math. Hungar., 155(2): 393-405, 2018.