Peter Apostoli

University of Toronto

"Metrizable domains of hyper-continuous functions: towards a geometry of abstract sets"

Abstract:

Domain Theory [Scott] provides a complete partial order D_infinity, called a reflexive domain, satisfying the equation D isomorphic to [D ->T], where [D ->T] is the family of all continuous functions from D to T, the complete partial order of truth values true, false, bottom under the information ordering. Finding this solution D_infinity in the category of Sequences of Finite Projections [Plotkin] provides our starting point for a new reflexive domain of partial characteristic functions. This domain admits such infinitary operations as quantification and thus (when attention is restricted to maximal elements) provides an extensional model of classical naive set theory. The model features a non T_0 topology as studied in both Synthetic Differential Geometry and Rough Set Theory. More simply put, the model maintains consistency in the presence of "naive comprehension" by rejecting the principle of the identity of indiscernibles. We call this non T_0 universe of sets the manifold of abstract rough sets. Deploying chains of graded indiscernibility relations (from the study of information systems in rough set theory) in the context of the SFP construction, the distance between two abstract sets may be defined as the minimum height in ascending sequence of finite projections at which the sets become discernible. This is a measure of the finite complexity of the computation required to discern abstract sets. In this way the indiscernibility relation is characterized as a relation of infinitesimal proximity with respect to the fundamental *metric* of this universe. Thus arises the fundamental question: what is the *geometry* of abstract rough sets?

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