Paper detail

The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $Π^1_1$ formulas, which we call the $Π^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $Π^1_1 \! \! \downarrow$-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.