On Ralescu's cardinality of fuzzy sets

We provide a direct formula for Ralescu's scalar cardinality. Unlike the original, iterative definition, the formula reveals intuitive shortcomings of this concept of cardinality. These are apparent from examples and reflected formally in that, as we show, the concept violates one of the axioms of cardinality of fuzzy sets. In addition, we provide a relationship of this concept to Ralescu's concept of fuzzy cardinality which unveils a tight link between the two concepts and points out another counterintuitive property of the concept of scalar cardinality. We argue that the discussed concept of fuzzy cardinality represents an interesting proposition, suggest its geometric interpretation, and provide preliminary observations as a basis for future considerations.