Foundations of fuzzy sets: A nonstandard approach

Abstract

The well known results on the inner limitations of classical mathematics (Löwenheim-Skolem Theorems, and Gödel's Theorems) and the introduction of a host of new set-like concepts, indicate the presence of something which is beyond the grasp of Cantorian set theory. In different ways each limitation expresses a basic uncertainty in Mathematics. Gödel's Theorems essentially express the following: Either we restrict ourselves to classical mathematical objects and their axiomatic description, and have some limitation on the amount of information that can be derived from these axioms systems or we consider nonclassical objects (fuzzy or variable objects, vague predicates, etc.) and lose the exactness associated with their description and possibly the law of excluded middle associated with their logic. On the other hand the Löwenheim-Skolem theorems express a limitation on the capability of axiom systems to describe models uniquely, and therefore a kind of relativity of models, according to the method of ‘observation’ of the mathematical reality. It is argued in this paper that the common element of all new set-like concepts introduced is variability and vagueness. We start the study of vagueness by indicating its aspects, first in the ‘intentional’ development of Nonstandard Analysis (NSA), and then passing to the case of B -fuzzy sets. The nonstandard approach to Fuzzy Sets is based on the idea that, if we introduce a ‘local observer’ into ZFC, who ‘sees’ the absolute ZFC framework with a local and non-Cantorian way, then we get a fuzzy deformation of ZFC, which constitues ‘A Theory of Fuzzy Sets’. The introduction of the observer is signified by the presence of the predicate ‘standard’, or of a probability space, etc. This is a first of a series of papers applying nonstandard methods to the study of vagueness and fuzzy sets.