Propositional Logic of Continuous Transformation in Cantor Space
Grigori Mints,
Ting Zhang
Abstract
A well-known axiomatization of the basic notions of general topology in
the form of propositional logic S4 was given by McKinsey and (for mathematically more interesting spaces like real line, real plane, etc.) by McKinsey and Tarski. If only open sets are considered
instead of arbitrary sets, one gets intuitionistic logic. L.E.J. Brower (of fixed point theorem) who created the background for this logic, stressed the connection of his principles with continuity considerations. This line of investigation leads to the theory of toposes and other connections with mainstream mathematics. A topological space X often acquires more interesting structures when it is the domain of a
dynamical topological system, that is, a pair &lang X,T &rang where X is the topological space and T is a continuous transformation on X. Dynamical topological logic studies dynamical topological systems by logical means.
We consider here propositional
systems, since predicate extensions tend to be intractable, in particular non-axiomatizable. Propositional formulas are constructed from variables (atomic formulas) by Boolean connectives, necessity [ ] and a monadic operation ο. In a standard interpretation, variables represents subsets of X, Boolean connectives act in a natural way, [ ] is the interior and ο is the pre-image under the operation T. Under this interpretation the axiom schema (C)
ο [ ] A → [ ] ο A
expresses continuity of T.
The propositional system S4C includes S4, (C) and standard axioms relating ο to Boolean connectives:
ο (A & B) ⇔ ο A & ο B,
ο ∼ A ⇔ ∼ ο A.
Completeness of S4C for the class of all topological spaces has been proved, in particular for finite spaces derived from Kripke models. These spaces do not satisfy topological separability axioms and are not very natural mathematically. P. Kremer pointed out that the real line is not complete for S4C. In this paper we prove completeness of S4C for Cantor space, a space that is very popular in the theory of dynamical systems.
