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Topic: Using classes instead of sets
Replies: 26   Last Post: Apr 1, 2013 8:04 PM

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Re: Using classes instead of sets
Posted: Mar 30, 2013 4:26 PM
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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



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