Cantor's theorem

The set of Cantor says that a lot is less powerful than its power set ( the set of all subsets ), so that the following holds. He comes from the mathematician Georg Cantor and is a generalization of Cantor's second diagonal argument. The theorem is valid in all models that satisfy the axiom.

Note: The rate applies to all quantities, especially for the empty set, because is a singleton. In general, for finite sets, that the power set of a - element set has elements. As always, the set of Cantor for finite sets is clear, but it applies just as well for infinite sets.

Evidence

Obviously true, since an injective mapping is.

We now want to show that there can be no surjective mapping.

To obtain a contradiction, we assume that there still is a surjective mapping.

We now define. Due to the separation axiom is a quantity and thus. Because of the assumption that onto, there is a with. But then by the definition of:

This contradiction shows that the assumption is wrong and there can be no surjective mapping - then it can certainly be no bijective mapping, which excludes the case, and we know that.

Historical

Cantor provided a first evidence in his treatise On an elementary question of the theory of manifolds of 1892. For this he showed that the set of all functions is more powerful than itself, and the set of functions has the same cardinality as the power set of (see power set # Characteristic Functions ). Further evidence comes from Felix Hausdorff in Broad set theory ( 1914) and by Ernst Zermelo in studies on the foundations of set theory (1908 ).

Connection with Cantor's other works

One can prove the second diagonal argument of Cantor also on the set of Cantor, if we know that. Because by then.

Furthermore, it can be shown the second Cantor antinomy with the set of Cantor. This states that the universal class is not a set but a proper class. Because, by definition, the power set of the universal class would be a subset of them, which contradicts the theorem of Cantor.

Swell

  • Introduction to set theory by Oliver Deiser. Springer, Berlin Heidelberg 2004, 2nd edition. ISBN 978-3-540-20401-5
  • Set ( mathematics)
  • Set theory
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