Date: Mar 30, 2013 4:26 PM
Author: Guest
Subject: Re: Using classes instead of sets
On 2013-03-29, Shmuel Metz <spamtrap@library.lspace.org.invalid> wrote:

> In <f8ral8lndkanp4gd70pu1fmblq4g912o74@4ax.com>, on 03/29/2013

> at 06:07 AM, quasi <quasi@null.set> said:

>>There are certain concepts for which sets are inadequate and classes

>>come to the rescue. For example, we need the class concept if we

>>want to define an equivalence on the collection of all groups, since

>>that collection is not a set.

> There's no such set in ZFC, but there are set theories in which it

> exists aqnd is a set.

I do not see how that can be the case, excluding Quine's

_New Foundations_, which has lots of its problems.

There must be as many groups equivalent to a given group

as there are sets, as {<x,y>: y \in g} for a given group g

has an obvious operation which makes the class of elements

a group isomorphic to g. This even holds if the group is

a proper class. So the collection of all groups is at least

as large as the collection of all sets.

--

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