To establish the scientific-theoretical and philosophical- methodological foundations for the transition from classical set theory (Cantor set) to fuzzy set theory, the mathematical and ontological structures of the Cantor set are presented in a localized manner. This approach aligns with the research direction, incorporating philosophical reflections on the fundamental concepts of this theory, such as infinity, continuum, and the empty set. Particular emphasis is placed on infinity and infinite sets (countable and continuum), which are crucial to the development of set (...) theory as a scientific field and have been subjects of inquiry in both scientific and philosophical cognitive processes from ancient times to the present. In addition, a relatively concise explanation of the mathematical and ontological structures of both classical set theory and fuzzy set theory is provided. Through certain epistemological and cognitive approaches, the article identifies the scientific-theoretical and philosophical-methodological bases for the emergence of fuzzy set theory, framed within the context of the scientific-philosophical reflections on classical set theory and its core concepts. (shrink)