Lexicographic Sets

The purpose of this paper is to investigate whether a recently proposed infinite-valued logic can lead to a novel, non-classical set theory. As in fuzzy set theory, the members of a set in the proposed theory may have different degrees of participation, expressed by different levels of truth values. But unlike in fuzzy set theory, the subset relation as well as the union and intersection operations are defined in a lexicographic way with respect to the truth values of elements in the involved sets. That is why we call the sets of the proposed theory, lexicographic sets. We prove that many known properties that apply to the classical set theory still apply to lexicographic sets. In addition, we give indications that the proposed theory may have practical applications in areas of Computer Science where the lexicographic relation plays a key role. More specifically, using the proposed theory, we prove a generalized model intersection theorem for logic programs with negation.