Semiset

In set theory, a semiset is a proper class that is a subclass of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible due to the axiom schema of specification.

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. In particular, Vopěnka's Alternative Set Theory (1979) axiomatizes the concept of semiset, supplemented with several additional principles.

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

Vopěnka's alternative set theory

Vopěnka's "Alternative Set Theory" builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF) finite sets and they are called infinite in AST.

References

Vopěnka, P., and Hájek, P. The Theory of Semisets. Amsterdam: North-Holland, 1972.
Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.
Holmes, M.R. Alternative Axiomatic Set Theories, §9.2, Vopenka's alternative set theory. In E. N. Zalta (ed.): The Stanford Encyclopedia of Philosophy (Fall 2014 Edition).
Novák, V. "Fuzzy sets—the approximation of semisets." Fuzzy Sets and Systems 14 (1984): 259–272.
Proceedings of the 1st Symposium Mathematics in the Alternative Set Theory. JSMF, Bratislava, 1989.