Ordinary infinitary languages L_{ lambda, kappa} satisfy the Interpolation Theorem only in the case lambda <= { aleph_1}, kappa = { aleph_0}, this include first order logic of course. There are also some pairs of such logics satifying interpolation, e.g. (L_{ lambda^+,{ aleph_0}}, L_{ (2^lambda)^+, lambda^+}) . Does this come from an intermidiate logic satisfying it? Is it nice? unique? We define for kappa = beth_kappa a new logic L^1_kappa such that L_{ kappa omega}< L^1_kappa LL_{ kappa kappa} and L^1_kappa is very nice; in particular satisfies the Interpolation Theorem. Moreover, L^1_kappa has a model--theoretic characterization in the style of Lindstrom's Theorem in terms of a form of undefinability of well--order. We also define for strong limit kappa of cofinality aleph_0 a logic L^2_{ kappa^+} such that L_{ kappa^+, { aleph_0}}<L^2_{ kappa^+}<L_{ kappa^+, kappa} and L^2_{ kappa^+} satisfies the Interpolation Theorem.
Categories: math.LO