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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 S - Set - Empty

 

Empty set : In set theory, the set without any members a.k.a. the void set. It's standard definition is {x: x≠x} i.e. the set of things that are not identical to themselves, and it is often written as Ø, so that Ø={x: x≠x}.

Accordingly, this is the way to speak of nothing in set-theory. Since in set theory two sets are equal iff they have the same elements, the empty set is unique (whence "the"), and since one set is included in another iff there is no element in the other set that is not in the one, the empty set is included in every set, and so also included in itself, though it is no element of itself.

Also, the powerset of Ø is {Ø, {Ø}}, which is one convenient technical way in set theory to generate something like the counterpart of truth-values.
 

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Literature:

Halmos, Quine, Suppes

 Original: Aug 24, 2004                                                Last edited: 12 December 2011.   Top