Non-good Base Sets And Modal Logic

In the Zermelo-Fraenkel system ZF of set theory, the foundation axiom restricts the domain of set theory to well-founded sets. Thus there are no infinitely descending∈-chains, and no set belonging to itself. In 1989, Aczel established non-well-foundedsystem ZFA of set theory by replacing the foundation axiom in classical system ZF of set theory by the anti-foundation axiom AFA. Since non-well-founded sets can be used to model circular phenomena that cannot be described in classical set theory, they have vast applications in philosophy, economics, modal logic, linguistics and theoretical computer sciences. It has very important theoretical implications and value for application.This dissertation studies the relationship between non-well-founded sets and modal logic. It investigates some important semantic notions, and then investigates the model-theoretic problems of set (model) construction and definability. In general, the work in my dissertation contains the following three points.First, it sketches the theoretical background, history, and recent research of non-well-founded set theory. And it also discusses vast applications of non-well-founded sets.Second, it explains the basic theory of non-well-founded sets according to Aczel and Barwise etc. And it introduces anti-foundation axiom from two perspectives: the relation between set and graph, and the solution of an equation system. Thus non-well-founded sets exist. Then two important notions are introduced. One is the notion of urelement. Urelements are neither sets not classes, but can be elements of sets. By adding urelements into graph we can obtain labeled graphs. The other notion is the bisimulation relation on sets which is used to decide whether two non-well-founded sets are the same or not.Third, the dissertation investigates modal logic under set-theoretic semantics. It explores the relation between frame, model and set. Then it studies set constructions and the problem of modal definability and it also translates basic modal language into classical (first-order or second-order) set theoretic languages. It also introduces the notion of bisimulation on sets and on equation systems. And then it investigates the relation between bisimulation and modal equivalence between labeled graphs. Finally, it explores some meta-logical properties of modal logic.The main creative points of this dissertation are the following.First, it defines under the set-theoretic semantics of modal logic some non-standard operations between sets, including disjoint union, generated subset, p-morphism, tree unraveling etc., and proves preservation or invariance results under those operations.Second, it translates basic modal language into classical set-theoretic languages, and proves some semantically corresponding results. It also touches van Benthem type characterization theorems.Third, it investigates definability theory under set-theoretic semantics for modal logic. It mainly investigates how to use modal formulas to characterize sets or classes of sets. It extends definability results to other characteristic formulas of modal logics from the characteristic formula of modal logic T, including modal logics KD, KB, K4, K5, GL and so on.Forth, some meta-logical properties of modal logic, including completeness, finite labeled graph property and decidability, are investigated under set-theoretic semantics. It also tries to study those properties by using pure sets and sets with urelements.In recent years, almost no logician in China has studied non-well-founded sets and their applications. It stays in the initial stage. In this sense, the meaning of the dissertation is to introduce results obtained from combination non-well-founded sets and modal logic into China by studying abroad investigations. What more important is the following. It gives a new research for modal logic under non-well-founded set-theoretic semantics, and solves some model-theoretic problems. This is helpful for the development of various logical directions and making the role of logic as tool more important. I think that the combination of non-well-founded sets and modal logic is a research direction with profound theoretical meanings and value of applications.