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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]].
2^n can be seen with (n choose 0) + (n choose 1) ... or just turning each thing on and off
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.▼
▲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 German [[mathematician]] [[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").
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