Strong downward Löwenheim–Skolem theorems for stationary logics, I

This note concerns the model theoretic properties of logics extending the first-order logic with monadic (weak) second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem (SDLS) down to <ℵ _2 for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into four principles. The strongest of these four is shown to be equivalent to the conjunction of CH and the Diagonal Reflection Principle for internally clubness of S. Cox. We show that a further strengthening of this SDLS and its variations follow from the Game Reflection Principle of B. König and its generalizations.