What sets could not be

Sets are often taken to be collections, or at least akin to them. In contrast, this paper argues that. although we cannot be sure what sets are (and the question, perhaps, does not even make sense), what we can be entirely sure of is that they are not collections of any kind. The central argument will be that being an element of a set and being a member in a collection are governed by quite different axioms. For this purpose, a brief logical investigation into how set theory and collection theory are related is offered.

The latter part of the paper concerns attempts to modify the `sets are collections' credo by use of idealization and abstraction, as well as the Fregean notion of sets as the extensions of concepts. These are all shown to be either unmotivated or unable to provide the desired support. We finish on a more positive note with some ideas on what can be said of sets. The main thesis here is that (i) sets are points in a set structure, (ii) a set structure is a model of a set theory, and (iii) set theories constitute a family of formal and informal theories, loosely defined by their axioms.