Nice infinitary logics
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Publication:2879888
DOI10.1090/S0894-0347-2011-00712-1zbMATH Open1251.03044arXiv1005.2806OpenAlexW1979392595MaRDI QIDQ2879888
Author name not available (Why is that?)
Publication date: 5 April 2012
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1005.2806
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