A well-known polymodal provability logic GLP is complete w.r.t.\ud the arithmetical semantics where modalities correspond to reflection\ud principles of restricted logical complexity in arithmetic [9, 5, 8]. This\ud system plays an important role in some recent applications of provability algebras in proof theory [2, 3]. However, an obstacle in the study\ud of GLP is that it is incomplete w.r.t. any class of Kripke frames. In\ud this paper we provide a complete Kripke semantics for GLP. First, we\ud isolate a certain subsystem J of GLP that is sound and complete w.r.t.\ud a nice class of finite frames. Second, appropriate models for GLP are\ud defined as the limits of chains of finite expansions of models for J. The\ud techniques involves unions of n-elementary chains and inverse limits of\ud Kripke models. All the results are obtained by purely modal-logical\ud methods formalizable in elementary arithmetic
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