2010/05/17 by Shelah, Saharon · 1 citation
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1005.2806
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. (L_lambda+,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 L_kappa+, aleph0