email@example.com wrote: > > On Thursday, March 28, 2013 1:45:34 PM UTC, Frederick Williams wrote: > ... > > > > Often one studies all groups, or all groups of a certain kind. Are > > > > those collections classes? > ... > > They are always classes and sometimes sets. A class is more general than a set so any collection which is a set is also a class. If we define isomorphic groups as being equal (as everyone does), then the collection of finite groups is a countably infinite set and we can talk about "the set of finite groups". > > However, the collection of groups is "too big" to be a set. Hence that collection is a class which is not a set. "Too big" because it contains a subcollection which corresponds to the class of all ordinals.
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 do not know if set theorists study (what I shall call) superclasses, supersuperclasses, and so on; where a superclass is a collection of classes in some theory, and a supersuperclass is a collection of superclasses in that theory or some other.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting