Abstract: This chapter presents the theory of quasi-sets, and argues that this offers an appropriate set-theoretic formalism for non-individual quantum objects. The basic idea is that in a quasi-set, there may exist elements for which the traditional concept of identity does not apply — these elements are called the ‘m-atoms’. Thus, a quasi-set may have a cardinal but not an associated ordinal. The other elements can be regarded as standard elements of a set. Hence, the theory encompasses a copy of Zermelo-Frankel set theory with Urelement. A number of alternatives are also presented with a view to possible physical applications, such as allowing the cardinality of a quasi-set to vary in time and thereby accommodating aspects of relativistic quantum theory. The formal framework is also applied to the development of quantum statistics, which can be naturally captured by the theory.

Keywords: quasi-set theory, unobservability of permutations, cardinality, ordinality,