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From: obnoxio@BRAHMS.BERKELEY.EDU (Obnoxious Math Grad Student)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: Russell's set of sets which... paradox
Message-ID: <8707291241.AA19065@brahms.Berkeley.EDU>
Date: Wed, 29-Jul-87 08:41:54 EDT
Article-I.D.: brahms.8707291241.AA19065
Posted: Wed Jul 29 08:41:54 1987
Date-Received: Fri, 31-Jul-87 03:15:10 EDT
References: <1214@utx1.UUCP> <6678@reed.UUCP> <744@cadnetix.UUCP> <3830@garfield.UUCP>
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This topic never did have anything to do with sci.math.symbolic.  I am
posting a small technical correction:

In article <3830@garfield.UUCP>, robertj@garfield writes:
>			      the Frankel-Zermelo Axiomatization in which
>class and membership are the primitive (undefined) concepts (not sets!)
>and a set is defined as a class which is a *member* of another class.

In point of fact, in ZF sets are the primitive notion, and classes are
just a convenient way of talking about propositions.  For example, one
has in ZF that `{x|phi(x)}={x|psi(x)}' is just an "abbreviation" for
`for all x, phi(x) iff psi(x)', and similarly for some other basic no-
tions involving classes, like `y in {x|phi(x)}', meaning `phi(y)'.  One
does not define `{x|phi(x)} in y'.

In contrast, in GB (Goedel-Bernays), classes are primitive, and are much
as you describe.  ZF and GB are essentially the same.  There is a stronger
theory KM (Kelley-Morse) which gives more life to classes.

ucbvax!brahms!weemba	Matthew P Wiener/Brahms Gang/Berkeley CA 94720