Modal Logics and Multiverses

This paper is a survey on modal logic of multiverses. It summarizes known results by Hamkins, Inamdar, Leibman, and the second author about the modal logics of forcing, grounds, and inner models in a general abstract setting. Most results in this survey come from a series of papers co-authored by the second author [19, 20, 18, 24]. Exceptions are the discussion of spiked Boolean algebras and the modal logic of c.c.c. forcing in § 7.1 (these results were obtained by Inamdar in his Master’s thesis [23] supervised by the second author) and the determination of the modal logic of symmetric extensions in § 7.4 due to the first author. The study of modal logics of set theoretic constructions started with [15] and [19]. Various other aspects of the modal logic of forcing are considered in [29, 21, 9, 10, 30, 32, 8, 7, 20, 18]. In § 2, we explain the natural (and very general) setting for the study of modal logics of forcing: set theoretic multiverses. After providing some general background in modal logic in § 3 and the abstract definition of modal logics for multiverses in § 4, we then discuss the main result of [19] in § 5 and provide the general proof strategy adapted to our general multiverse setting in § 6. Finally, in § 7, we survey various generalisations in the multiverse setting.