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July 8th, 2014, 04:24 AM   #1
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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.
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July 8th, 2014, 05:08 AM   #2
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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.
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July 9th, 2014, 11:41 AM   #3
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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.
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July 9th, 2014, 12:09 PM   #4
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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.
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July 9th, 2014, 03:31 PM   #5
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Well personally I prefer "the set of all non-empty Russell sets" but that is rather hard to use in practice, too.
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