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Algebraic and topological models of Solovay's modal system
by
Leo Esakia, David Gabelaia
Razmadze Mathematical Institute
Robert Solovay, in his classic work on provability, among other things, presented a set-theoretical interpretation of modal formulas by putting the modal [¯] to mean `true in each transitive model of Zermelo-Fraenkel Set Theory ZF'. More precisely, an interpretation is understood as a function *, which sends modal formulas to sentences of ZF, commutes with the Boolean connectives and takes ([¯] p)* to be equal to the sentence of the language of set theory that expresses the statement: `p* is true in each transitive model of ZF'.
Solovay formulated a modal system, which we call here GL.S, and announced its ZF-completeness. The system GL.S results from the Gödel-Löb system GL by adding the formula [¯]([¯] p→ [¯] q)→([¯] q→ p) as a new axiom.
We will present a study of models of the system GL.S in various semantics. In particular, we investigate algebraic, Kripke and topological models of GL.S. We employ a particular view of scattered spaces to show that strongly connected topological models of GL.S are Alexandroff. Connections with the logic of preference relation and ordinals will be discussed.
Date received: May 9, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caug-37.