A many-sorted interpolation theorem for $L(Q)$

Abstract:

Let L be a many-sorted relational language with $\in$ and consider the logic ${L_{{\omega _1}\omega }}(Q)$, infinitary logic with a monotone quantifier. We prove a version of Feferman’s Interpolation Theorem for this logic. We then use the theorem to show that for a one-sorted language L and a countable admissible fragment ${L_A}$ of ${L_{{\omega _1}\omega }}(Q)$, any sentence which persists for end extensions is equivalent to a $\Sigma$ sentence.