Kripke Models Built from Models of Arithmetic

Henk, Paula

doi:10.1007/978-3-662-46906-4_10

Cited by 1 publication

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“…To the best of our knowledge, Theorem 6 (or its analogue (2)) has never been formulated explicitly before, although similar constructions related to frames of arithmetic theories were considered earlier (V. Yu. Shavrukov, an unpublished note, 2013; [19,Remark 3]).…”

Section: Definitionmentioning

confidence: 99%

“…During the last years, modal logics of various relations between models of set theory have been studied, see, e.g., [15,18,7,17,21]. A well established area in provability logic deals with modal axiomatizations of relations between models of arithmetic (and between arithmetic theories), see, e.g., [27,5,20,34,35,19,16]. In another extensively studied area modalities are interpreted by relations between Kripke and temporal models, see, e.g., [33,11,4] or the monograph [32].…”

Section: Introductionmentioning

confidence: 99%

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On Modal Logics of Model-Theoretic Relations

Saveliev

Shapirovsky

2019

Stud Logica

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Given a class C of models, a binary relation R between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of C in L where the modal operator is interpreted via R. We discuss how modal theories of C and R depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside L. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim-Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.Keywords: modal logic, modal algebra, robust modal theory, logic of submodels, logic of quotients, logic of forcing, provability logic, modeltheoretic logic * The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences.1 Modal algebras of theories are broadly used in provability logic, where they are called Magari (or diagonizable) algebras; they are known to keep a lot of information about theories containing arithmetic (see, e.g., [28]).2 A significant part of Sections 4 and 5 was motivated by reviewers' questions on an earlier version of this paper.

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Section: Definitionmentioning

confidence: 99%