Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies

Abstract

This paper demonstrates two meta-mathematical propositions concerning the increasingly popular “intuitionistic” (= vague) approaches to fuzzy sets and fuzzy topology, as well as the closely related interval-valued (= grey) sets and interval-valued “intuitionistic” sets: (1) the term “intuitionistic” in these contexts is historically inappropriate given the standard mathematical usage of “intuitionistic”; and (2), at every level of existence—powerset level, topological fibre level, categorical level—interval-valued sets, interval-valued “intuitionistic” sets, and “intuitionistic” fuzzy sets and fuzzy topologies are redundant and represent unnecessarily complicated, strictly special subcases of standard fixed-basis set theory and topology. It therefore follows that theoretical workers should stop working in these restrictive and complicated programs and instead turn their efforts to substantial problems in the simpler and more general fixed-basis and variable-basis set theory and topology, while applied workers should carefully document the need or appropriateness of interval-valued or “intuitionistic” notions in applications.