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From: ladkin@kestrel.ARPA (Peter Ladkin)
Newsgroups: sci.math,sci.math.symbolic,sci.philosophy.tech
Subject: Re: Russell's set of sets which... paradox
Message-ID: <25430@kestrel.ARPA>
Date: Wed, 29-Jul-87 19:58:06 EDT
Article-I.D.: kestrel.25430
Posted: Wed Jul 29 19:58:06 1987
Date-Received: Sat, 1-Aug-87 00:45:00 EDT
References: <1214@utx1.UUCP> <6678@reed.UUCP> <744@cadnetix.UUCP> <3830@garfield.UUCP>
Organization: Kestrel Institute, Palo Alto, CA
Lines: 21
Keywords: types,classes,Principia
Xref: mnetor sci.math:1688 sci.math.symbolic:113 sci.philosophy.tech:325

In article <3830@garfield.UUCP>, robertj@garfield.UUCP writes:
> [..] 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.

(Fraenkel's name is normally spelt with two e's)

I think you are talking about the von Neumann-Goedel-Bernays formulation
of set theory, otherwise known as NBG.

In ZF set theory, every object is a member of some other object, namely
its singleton (follows from pairing). 

ZF avoids the paradox also by using layering, except that the layering
is in the semantics rather than the syntax. The layering is obtained
by iterating the power set operation, starting with the empty set.
A set's rank is the index of the least layer in which it appears.
There are some subtleties if you don't include the axiom of choice.

peter ladkin
ladkin@kestrel.arpa