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 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. Thanks from CRGreathouse
 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 well-formed 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. Thanks from Evgeny.Makarov
 July 9th, 2014, 02:31 PM #5 Senior Member   Joined: Mar 2012 Posts: 294 Thanks: 88 Well personally I prefer "the set of all non-empty Russell sets" but that is rather hard to use in practice, too. Thanks from CRGreathouse

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