Non-Archimedean Fuzzy Reasoning

The informal sense of Archimedes' axiom is that anything can be measured by a ruler. The negation of this axiom allows to consider non-well founded phenomena (e.g. self- applicative programs, graph circularity, etc.). In this paper we propose non-Archimedean fuzziness for the first time, i.e. fuzziness that runs over the non-Archimedean number systems. We show that this fuzziness is multihierarchical (namely, it is regarded as omega-order vagueness) and it is constructed in the framework of the t-norm based approach as omega-order extension BLVforall_infin of the basic fuzzy logic BLVforall. We consider two cases of the non-Archimedean fuzziness: one with validity in the interval [0,1] of hyper numbers and one with validity in the ring of p-adic integers. This fuzziness has a lot of practical applications, e.g. it can be used in higher-order fuzzy clustering.