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From: pem@cadnetix.UUCP (Paul Meyer)
Newsgroups: sci.math,sci.math.symbolic,sci.philosophy.tech
Subject: Re: Russell's set of sets which... paradox
Message-ID: <744@cadnetix.UUCP>
Date: Fri, 24-Jul-87 20:44:06 EDT
Article-I.D.: cadnetix.744
Posted: Fri Jul 24 20:44:06 1987
Date-Received: Sun, 26-Jul-87 03:03:15 EDT
References: <1214@utx1.UUCP> <6678@reed.UUCP>
Reply-To: pem@cadnetix.UUCP (Paul Meyer)
Organization: Cadnetix Corp., Boulder, CO
Lines: 32
Keywords: set theory, paradox, logic
Xref: mnetor sci.math:1659 sci.math.symbolic:105 sci.philosophy.tech:302

[]
	Actually, the paradox as stated leads to a very significant con-
   clusion:  not everything is a set.  Modern mathematics (at least as I
   was taught it) draws a distinction between "classes" (things which
   follow the intuitive idea that a set can be anything) and "sets" (which
   obey certain rules).

	A consequence of this distinction is that S is not a set.  Thus,
   we can say that "S is the class of all sets that do not include them-
   selves as members", and thus S does not include itself because it is not
   a set, or we can say "S is the class of all classes that do not include
   themselves as members" and then give up on S as a useful thing.

	The Z-F set theory is one consistent(*) axiomitization of what a
   "set" is.  It is also powerful enough to (given patience) produce the
   entirety of classical math--that is, it demonstrates that classes that
   are not sets are not required for classical mathematics.

	Unfortunately, the only text I can recommend for this stuff is the
   one I used, written by my professor, J. Malitz.  It is a good book, but
   quite terse.  In about 100 pages it covers 3 semesters' worth of upper-
   division math, covering set theory, computability, and formal logic.
   Without Dr. Malitz's lectures I wouldn't have been able to follow it
   very well, and even with them a good 50% of the classes tended to get
   very confused.

						pem

(*) - Of course, how do we prove that it is true?  Do we use it on itself?
   If not, how do we prove that whatever we use to prove Z-F is true?  In
   short, mathematics is in some ways as great a faith decision as theism--
   except that the consequences are no where near as far-reachese)