The modal logic of inner models (Q2805035)

In [Trans. Am. Math. Soc. 360, No. 4, 1793--1817 (2008; Zbl 1139.03039)], \textit{J. D. Hamkins} and \textit{B. Löwe} identified the modal logic of forcing extensions; it turned out to be the well-known modal logic \(\mathsf {S4.2}\). Later, in [Isr. J. Math. 207, Part 2, 617--651 (2015; Zbl 1367.03095)] \textit{J. D. Hamkins} et al. studied the structural connections between a notion of forcing \(\Gamma\) and its modal logic.NEWLINENEWLINEThe authors of the paper under review determine the model logic of inner models. In particular they prove that if \(M\) is a transitive model of \(\mathsf{ZFC}\), then the modal logic of the inner models of \(M\) is \(\mathsf {S4.2Top}\).NEWLINENEWLINEThe paper begins with the definition of the modal propositional logic \(\mathsf {S4.2Top}\), which is the normal modal logic \(\mathsf {S4.2}\) plus the axiom NEWLINE\[NEWLINE\mathsf{Top}: \lozenge((\square\varphi\leftrightarrow\varphi)\wedge(\square\neg\varphi\leftrightarrow\neg\varphi)).NEWLINE\]NEWLINE The authors prove that a certain class of preorders that they call \textit{inverted lollipops} characterizes \(\mathsf {S4.2Top}\). They employ this characterization and a version of the machinery of ``buttons'' and ``switches'', which played a pivotal role in the paper of Hamkins et al. [loc. cit.], to prove that the modal logic of the inner models of \(M\) is contained in \(\mathsf{S4.2Top}\). Using some methods from the Ph.D. Thesis of \textit{J. Reitz} [The ground axiom. New York, NY: The Graduate Center of the City University of New York (Ph.D. Thesis) (2006)], the authors complete the proof of their main theorem.