Are there paradoxes of the set of all sets?

Abstract

This paper contains a deliberately “naive” exploration of the properties of sethood and membership, with especial reference to “very large” sets. No formal paradoxes are generated, but a variety of curious properties is exhibited, whose study and proof would form useful exercises for beginners in set theory. Further, when the set of “all” sets is considered in connection with Cantor's diagonal argument, paradoxes are produced. Indeed, it is argued that these paradoxes are independent of the usual paradoxes of set theory, and that Russell missed spotting them when discovering his own paradox.