XPost: rec.arts.books.tolkien   
   From: jcb@inf.ed.ac.uk   
      
   On 2014-07-23, Paul S  Person  wrote:   
   > On Wed, 23 Jul 2014 17:19:30 +0200, Taemon  wrote:   
   >   
   >>On 18-7-2014 18:57, Paul S. Person wrote:   
   >>> That sounds nice, and I would generally agree, but would you support a   
   >>> study of how to create the set of all objects not part of any set?   
   >>> While not the same sort of thing as determining the characteristics of   
   >>> the Deus absonditus, it shares with it the property that it actually   
   >>> is impossible.   
   >>   
   >>I would support the study, but I see your point.   
      
   You are both somewhat confused!   
      
   > The set of all objects not part of any set is a logical contradiction:   
      
   No, it's not. It is the empty set. In the usual set theoretic universe,   
   every set is a member of another set. The empty set has no members, so   
   every member of it is not an element of a set. However, in modern set   
   theory, the "set of all objects not part of any set" is not even a   
   definable object, so one doesn't need to talk about it - my assertion   
   that it is the empty set is made outside formal set theory. (Within   
   formal set theory, it is the empty class, not the empty set.)   
   You could consider the set of all sets, which was an early paradox:   
   modern set theory does not permit one to express the "set of all   
   sets" either, so solving the paradox.   
   Or you may have a confused recollection of the Russell Paradox:   
   the set of sets that are not members of themselves, which is resolved   
   by forbidden circular membership relations altogether. (There's a   
   niche industry in set theories that do allow circular membership, and   
   they use other techniques to avoid paradox.)   
      
   > if x is an element of no set, then it becomes a part of this set; but   
   > now it is a part of a set, and so cannot belong to this set. It is not   
   > even degenerate: the empty set, is degenerate, and the empty set is a   
   > subset (and so belongs to) every set -- except this one, except that   
   > it does, by its own definition. It is, then, less than nothing; it   
   > cannot even be said to not exist, not because it does exist, but   
   > because saying that it does not exist accords it too much reality.   
      
   Er, no. The empty set is a subset of every set, but not a   
   member of every set. Even in basic naive set theory, it is important   
   to know the different between members and subsets.   
      
   > It's only value is to show that, if you propose a non-empty set of   
   > objects, you must demonstrate that at least one object actually   
   > belongs to it, just to be sure it is not a logical contradiction.   
      
   That thesis depends a bit on what you mean by "demonstrating that at   
   least one object belongs to it". The working mathematician is quite   
   happy with sets where it is impossible to construct any actual member   
   of the set.   
      
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