Date: Mar 28, 2013 2:54 PM
Author: Herman Rubin
Subject: Re: Using classes instead of sets

On 2013-03-28, Frederick Williams <freddywilliams@btinternet.com> wrote:
> pepstein5@gmail.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.



The best my late wife and I were able to do in our book,
_Equivalents of the Axiom of Choice II_, was to use relations.
A rerlation is a class of ordered pairs, so if R is a relation,
{y: <x,y> \in R} can be considered the R-class whose indes is x.
This does give a way if looking at a class of classes, but it is
restrictive; one cannot get a larger index class than the universe,
which is the class of all sets in NBG, or the class of all elements
in NBGU. One cannot have the class of all classes as indices.


--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558