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

Abstract

proved that the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets (DRPIC) introduced in [4]. We prove that the restriction of the LST property to (downward) reflection of Π 1 1 formulas, which we call the Π 1 1 ↓-LST property, is equivalent to the internal version of DRP from [2]. Combined with results from [2], this shows that the Π 1 1 ↓-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.