Cantorian set theory

Abstract

This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor’s idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. But for those who prefer to retain the anomalous sets, the chapter concludes by explaining how a conventional iterative set theory can be based on full plural logic.