Multiset theory.

A multiset is a collection of objects (called elements) in which elements may occur more than once. The number of times an element occurs in a multiset is called its multiplicity. The cardinality of a multiset is the sum of the multiplicities of its elements. Multisets are of interest in certain areas of mathematics, computer science, physics, and philosophy. Section 1 introduces multisets and surveys the relevant literature. Section 2 develops a first-order two-sorted theory MST for multisets that "contains" classical set theory. The intended interpretation of the atomic formula x E y is "x is an element of y with multiplicity n". In MST, one can extend the classical notion of a function. Section 3 constructs a model of MST in ZFC by interpreting x E n y as y(x) = n (multisets are modeled by positive integer-valued functions).