July 8th, 2014, 03:24 AM  #1 
Newbie Joined: Dec 2012 Posts: 5 Thanks: 1  Does Empty Set Exist?
Assume empty set exists: The empty set is a subset of every set. Definition of subset. Therefore the empty set is a subset of the empty set. Contradiction. Therefore the empty set does not exist. 
July 8th, 2014, 04:08 AM  #2 
Newbie Joined: Dec 2012 Posts: 5 Thanks: 1 
made fundamental error: a subset of a set is not necessarily a member of the set. there is no contradiction. conclusion of OP is false. my apologies. 
July 9th, 2014, 10:41 AM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88 
Given that any OTHER set (say $A$) exists, one can define: $\emptyset = \{a \in A: a \in a\}$ The predicate $P(a) = a \in a$ is clearly a wellformed formula given any meaningful definition of $\in$, so if ANY sets exist, the empty set is surely one of them. 
July 9th, 2014, 11:09 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
That definition is valid in ZF, but I would never use it. Much better, from my biased perspective, would be $$ \emptyset=\{a\in A:a\ne a\} $$ which does not rely on the nonexistance of the Quine atoms and related sets. 
July 9th, 2014, 02:31 PM  #5 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88 
Well personally I prefer "the set of all nonempty Russell sets" but that is rather hard to use in practice, too.


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