Non-circular definition of final club with pc's addendums included

[Steven Belknap <sbelknap@UIC.EDU>]

A student at Yale may belong to zero or more clubs. Some clubs are 
final clubs. A final club is defined as "a club such that membership 
in it precludes membership in any other final club".

0. There are at least two distinct clubs. (If there were not, there 
would not be a solution, and we are given that there is a solution.)

1. Call a set, s, of clubs preclusive if it contains more than one 
member and if being a member of any one of the clubs in s precludes 
being a member of any other club in s.

2. Call a set, m, of clubs maximally preclusive if it is preclusive 
and every proper superset of m is not preclusive.

3. There is one and only one nonempty set, m. (We are told that some 
clubs are final clubs. If there were no nonempty sets m, that would 
violate the conditions of the problem. If there were more than one 
set m, than it would not be possible to know which set m contained 
the final clubs, and a definition of final club would be impossible. 
Since the problem contains a definition, albeit a circular one, a 
definition must be possible and there must be only one nonempty set, 
m.)

4. Call a club, c, a final club iff it is a member of m.

QED

There are some singularities to consider. For example, what if no 
Yale student is a member of any club? Can there be a final club? I 
think that there can not be, but it is not clear. I interpret the 
question as requiring that at least two Yale students are members of 
clubs, and that each student would then be a member of a final club. 
But the question is vague about how to consider clubs which have 
rules and no members. Also, pc points out that the question does not 
exclude the possibility that some clubs are not distinct from other 
clubs. Also, do we base our understanding of what preclusive means on 
what the rules of a club are or what the behavior of a club is?

co'o mi'e stivn
Steven Belknap, M.D.
Assistant Professor of Clinical Pharmacology and Medicine
University of Illinois College of Medicine at Peoria