Re: final clubs

[Carl Burke <cburke@MITRE.ORG>]

Jorge Llambias wrote:
> 
> From: "Jorge Llambias" <jjllambias@hotmail.com>
> 
> > >        Definition: Every club is a final club.
> > >
> > > To disprove it all you have to do is find a configuration
> > > where some clubs are not final and yet final clubs are
> > > well defined. I don't think there is one.
> >
> >Suppose we have the following situation:
> >Club A imposes no conditions on its members
> >Club B requires members to swear a loyalty oath,
> >        and to swear no other oaths
> >Club C requires members to swear a loyalty oath,
> >        and to swear no other oaths
> >
> >The set of final clubs is well-defined, and is composed of
> >clubs B and C (the set of clubs which require loyalty oaths);
> >membership in club A neither precludes nor is precluded by
> >membership in clubs B or C.  This doesn't help us find a
> >non-recursive definition of 'final club', but it does
> >illustrate that not all clubs are necessarily final.
> >
> >--
> >Carl Burke
> 
> How can you tell in that example that it is B and C that
> are final, and not that A is the only final club?

B and C are final because they fit the recursive definition
of 'final club': membership in one member of the set precludes
membership in any other member of that set.  The set of
final clubs is a mutually exclusive set.

A is not final because A does not preclude membership in
any other club, and therefore does not meet the definition.
(Also, the definition of 'final club' implies that there must
be more than one final club so that there is something to
preclude!)

--
Carl Burke
cburke@mitre.org