Multiset theory. (English) Zbl 0668.03027

Blizard, Wayne D.

Notre Dame J. Formal Logic 30, No. 1, 36-66 (1989).

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 \(x\in^ ny\) 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 \(x\in^ ny\) as \(y(x)=n\) (multisets are modeled by positive integer-valued functions).

Reviewer: W.D.Blizard

MSC:

03E70	Nonclassical and second-order set theories
03E99	Set theory

Keywords:

bags; repeated elements; consistency; first-order two-sorted theory MST for multisets; model of MST in ZFC