28.3.2013 18:24, email@example.com wrote: > On Thursday, March 28, 2013 3:11:31 PM UTC, Frederick Williams wrote: > >> ... >> If groups could have classes for the collection of their elements, and >> >> if we call such groups "Groups", then we couldn't call the collection of >> >> Groups a set or a class, could we? > ... > > I don't see why not. Without further restrictions, the collection of Groups would seem to be too big to be a set, but your Groups could form a class, I would think. Classes are allowed to contain other classes after all. Of course, we get Russell-type paradoxes if we allow entities to contain themselves, whether the entities be sets or classes.
In general you have level-0 collections, which are the ordinary sets, level-1 collections (classes), which are collections of level-0 collections, and so on level-n collections are collections of level-(n-1) collections. There is an infinity of different kinds of collections. No paradoxes in that:) Classes represent a single step in this generalization.
Such ideas are needed and used in higher category theory: