Nice infinitary logics

Abstract

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.