(untitled) (Contemporary Mathematics, 2025)
Classical sets are characterized by ’well-defined’ objects or elements, but this property is relative and linguistic. Classical sets, like the set of natural numbers, are primarily used for counting and measuring quantitative data. However, the literature does not address the differences in the representation of cardinalities between infinite sets and their infinite subsets. This study investigates the cardinalities of infinite sets and their finite versions (cut cardinalities) within the finite approximation of the real number system. Traditional fuzzy sets measure only belongingness or membership values, thus capturing qualitative data. In the second part, this study introduces nature sets, which describe both the quantitative and qualitative aspects of elements. Quality measurement is provided by various techniques. Several properties of nature sets are examined, and potential applications are highlighted.