Symmetric Set Theory: A General Theory of Isomorphism, Abstraction, and Representation

Abstract

It is possible to represent a finite set of points (atoms) by a finite sequence of points. However a finite set of points has no distinguished member and therefore it is impossible to define a function which takes a finite set of points and returns a "first" point in that set. Thus it is impossible to represent a finite sequence of points by a finite set of points. The theory of symmetric sets provides a framework in which the observation about sets and sequences can be proven. The theory of symmetric sets is similar to classical (Zermello-Fraenkel) set theory with the exception that the universe of symmetric sets includes points (ur-elements). Points provide a basis for general notions of isomorphism and symmetry. The general notions of isomorphism and symmetry in turn provide a basis for natural, simple, and universal definitions of abstractness, essential properties and functions, canonicality, and representations. It is expected that these notions will play an important role in the theory of data structures and in the construction of general techniques for reasoning about data structures.