A many-sorted interpolation theorem for 𝐿(𝑄)

Abstract

Let L be a many-sorted relational language with ∈ \in and consider the logic L ω 1 ω ( Q ) {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 {L_A} of L ω 1 ω ( Q ) {L_{{\omega _1}\omega }}(Q) , any sentence which persists for end extensions is equivalent to a Σ \Sigma sentence.