Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!uunet!seismo!nbires!ico!cadnetix.UUCP!pem 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)