Nice infinitary logics

الأصل · EN

Ordinary infinitary languages Llambda, kappa satisfy the Interpolation Theorem only in the case lambda <= aleph₁, kappa = aleph₀, this include first order logic of course. There are also some pairs of such logics satifying interpolation, e.g. (Llambda+,aleph₀, L(2ˡambda)+, lambda+). Does this come from an intermidiate logic satisfying it? Is it nice? unique? We define for kappa = bethₖappa a new logic L¹ₖappa such that Lkappa omega< L¹ₖappa LLkappa kappa and L¹ₖappa is very nice; in particular satisfies the Interpolation Theorem. Moreover, L¹ₖappa 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 aleph₀ a logic L²kappa+ such that Lkappa+, aleph₀<L²kappa+<Lkappa+, kappa and L²kappa+ satisfies the Interpolation Theorem.