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Date: Wed, 12 Apr 2000 12:16:19 EDT
Subject: RECORD: final clubs
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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).