Editing Cantor's theorem (section)

{{Short description|Every set is smaller than its power set}}
{{For|other theorems bearing Cantor's name}}
[[File:Hasse diagram of powerset of 3.svg|thumb|The cardinality of the set {''x'', ''y'', ''z''}, is three, while there are eight elements in its power set (3 < 2<sup title="Order theory">3</sup> = 8), here [[order theory|ordered]] by [[Inclusion (set theory)|inclusion]].]]
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In mathematical [[set theory]], '''Cantor's theorem''' is a fundamental result which states that, for any [[Set (mathematics)|set]] <math>A</math>, the set of all [[subset]]s of <math>A,</math> known as the [[power set]] of <math>A,</math> has a strictly greater [[cardinality]] than <math>A</math> itself. 

For [[finite set]]s, Cantor's theorem can be seen to be true by simple [[enumeration]] of the number of subsets. Counting the [[empty set]] as a subset, a set with <math>n</math> elements has a total of <math>2^n</math> subsets, and the theorem holds because <math>2^n > n</math> for all [[non-negative integers]].

Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for [[infinite set|infinite]] sets also.  As a consequence, the cardinality of the [[real number]]s, which is the same as that of the power set of the [[integer]]s, is strictly larger than the cardinality of the integers; see [[Cardinality of the continuum]] for details.

The theorem is named for [[Georg Cantor]], who first stated and proved it at the end of the 19th century.  Cantor's theorem had immediate and important consequences for the [[philosophy of mathematics]].  For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it.  Consequently, the theorem implies that there is no largest [[cardinal number]] (colloquially, "there's no largest infinity").