Paul
Posts:
676
Registered:
7/12/10


Re: Using classes instead of sets
Posted:
Mar 28, 2013 10:43 AM


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.
Paul Epstein

