On 2013-03-28, firstname.lastname@example.org <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.
Other than with types, which are usually avoided because of difficulties, proper classes are not allowed to be elements. ZF has no proper classes; the essentially equivalent NBG avoids the known paradexes by not allowing proper classes to be elements. With the strongest forms of the Axiom of Choice, proper classes are all of the same size as the universe.
> Paul Epstein
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558