This paper deals with alternative set theory which substantially departs from classical set theory. The main notion of it -a semiset-is proposed to model the same intuitive notion as fuzzy set but it is more general. The reasonable tool for ‘grasping’ semisets could be provided by fuzzy sets whose applicability is beyond discussion. In the paper reasons for such an approximation are given and two ways how to provide it are proposed. The fact that the membership function is modeled within the theory should be stressed. At the end, some problems and an indication for further investigation are discussed.