Semiset

This article is about the set theory. If you are looking for the way of splitting sets, see Semialgebraic set.

In set theory, a semiset is a proper class that is contained in a set. The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972). It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is precluded by the axiom of separation. The concept of semisets opens the way for a formulation of an alternative set theory.

Semisets can be used to represent sets with imprecise boundaries. Vilém Novák (1984) studied approximation of semisets by fuzzy sets, which are often more suitable for practical applications of the modeling of imprecision.

References

• Vopěnka, P., and Hájek, P. The Theory of Semisets. Amsterdam: North-Holland, 1972.
• Novák, V. "Fuzzy sets—the approximation of semisets." Fuzzy Sets and Systems 14 (1984): 259–272.