@MISC{Mints98modularizationand, author = {G. Mints}, title = {Modularization and Interpolation}, year = {1998} }

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Abstract

Interpolation does not hold for first order arithmetic, but this does not affect modularization theorem due to Maibaum and Turski (dealing with pushouts of conservative extensions) for theories containing arithmetic, since this theorem does not in fact use Interpolation Theorem. We present a short proof of modularization theorem for the case of refinements which can substitute predicates by arbitrary formulas. This proof uses interpolation in an essential way. 1 Interpolation Theorem and Arithmetic This section reproduces the proof of a folklore result of proof theory. Probably there is a hint in [1]. Theorem 1 Interpolation Theorem does not hold in first order arithmetic with additional predicate symbols. Proof . Let T (P ) be a formula of first-order arithmetic with an additional unary predicate P stating that P satisfies standard inductive clauses of the truth definition for prenex sentences of arithmetic: T (x) j 8x(Sentence \Gamma code(x) ! [(Quantifier \Gamma free(x) ! (P (...