Generalized Interpolation Theorems

(Note that the Craig interpolation theorem is the special case of Chang's theorem in which Q19,.., Qn are all universal quantifiers.) Chang also raised the question [2, Remark (k)] as to whether the Lopez-Escobar interpolation theorem [6] for the infinitary language L.1. possesses a similar generalization. In this paper, we show that the answer to Chang's question is affirmative and, moreover, that several interpolation theorems for applied second-order languages for number theory also possess such generalizations. Maehara and Takeuti [7] have established independently proof-theoretic interpolation theorems for first-order logic and L,, which have as corollaries both Chang's theorem and its analog for L.1,. Our proofs are quite different from theirs and rely on model-theoretic techniques stemming from the analogy between the theory of definability in L.1. and the theory of Borel and analytic sets of real numbers, rather than the technique of cut-elimination. In addition, we are able to show that slight variants of the seemingly more general results of Takeuti and Maehara can in fact be derived from Chang's theorem and its L1,. analog without the use of proof theory.