From pycyn@aol.com Wed Apr 12 09:23:02 2000 Return-Path: Received: (qmail 1081 invoked from network); 12 Apr 2000 16:16:28 -0000 Received: from unknown (10.1.10.27) by m2.onelist.org with QMQP; 12 Apr 2000 16:16:28 -0000 Received: from unknown (HELO imo21.mx.aol.com) (152.163.225.65) by mta2 with SMTP; 12 Apr 2000 16:16:27 -0000 Received: from Pycyn@aol.com by imo21.mx.aol.com (mail_out_v25.3.) id h.8.38cab07 (4357) for ; Wed, 12 Apr 2000 12:16:20 -0400 (EDT) Message-ID: <8.38cab07.2625fb53@aol.com> Date: Wed, 12 Apr 2000 12:16:19 EDT Subject: RECORD: final clubs To: lojban@onelist.com MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: AOL 4.0 for Windows sub 33 X-eGroups-From: Pycyn@aol.com From: pycyn@aol.com X-Yahoo-Message-Num: 2360 Summary on Final Clubs. (Final Sets > Sets, etc. < Quine Challenge) [pc finally zig-zags to where xorxes has been from the beginning] A final club is defined as a club whose members may not be members of any other final club. A preclusive set of clubs is a set of clubs, with more than one member, such that any member of one club in the set may not also be a member of any other club in the set (such that any pair of clubs in the set may have no members in common) A maximally preclusive set is a preclusive set such that adding any other club to the set will result in a set of clubs that is not preclusive (any non-member club may share a member with at least one member club). The definition picks out a unique set of final clubs just in case there is exactly one maximally preclusive set of clubs. And this happens only if every club that precludes membership in another club precludes membership in EVERY other club that precludes membership in some club (i.e., preclusion is transitive as well as symmetric, indeed stronger: if x precludes y and z precludes w, then x precludes z and w as well, assuming x, z, w are distinct). The set of final clubs is thus either empty or is a set of at least two members which includes all clubs subject to membership restriction in terms of clubs. In the original tale, about Yale, we are told that the set is not empty. Failing this condition, any maximally preclusive set might be final or none. If one is final, then it is selected by some criterion not mentioned in the supposed definition. This problem was introduced on analogy with the case of sets, which are often specified as classes that can be members of sets. But this is not a definition, only a handy test -- the set theories that have the class/set distinction also have a number of rules that specify some initial sets and some ways of building sure'nough sets out of other sets, so that the possibility of a variety of classes which might be the class of sets does not arise. (And lots of set theories just have nothing to do with classes at all and deal only with sets from the get-go. Although they still have to go through the same rigamarole to prove some description is that of a set -- "exists," as they would say).