Re: Applications of proper classes?

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 11/20/04 

Date: 20 Nov 2004 15:58:12 -0500

In article <716e06f5.0411182009.fbf9655@posting.google.com>,
Bill Taylor <w.taylor@math.canterbury.ac.nz> wrote:
>> >>Has the notion of a proper class been necessary to solve any problems in
>> >>number theory or real analysis?

>> > No. ZFC has no proper classes, and (so far) all advances in
>> > number theory and real analysis can be formalized in ZFC
>> > (plus, perhaps, some large cardinal axioms).

>> The class of all sets would be a proper class. It's just that sets are
>> more interesting.

>Furthermore, it can be shown that anything you can do with proper classes
>(including super-classes of THESE, etc), can in a suitable sense already
>be done with sets plus large cardinals.

NBG is a conservative extension of ZF with proper classes.
Any theorem of NBG not explicitly involving proper classes
is also a theorem of ZF.

Large cardinals give MORE than proper classes. Also, it
cannot be proved that large cardinals are consistent, with
or without choice.

The main advantages of proper classes are those of convenience. 

-- 
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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