PlanetMath: Cantor's theorem

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Cantor's theorem

(Theorem)

Let be any set and ${\mathcal P}(X)$ its power set. Then there is no bijection between and ${\mathcal P}(X)$ . Moreover, the cardinality of ${\mathcal P}(X)$ is strictly greater than that of ; that is, $\vert X\vert<\vert{\mathcal P}(X)\vert$ .

"Cantor's theorem" is owned by Wkbj79. [ full author list (2) | owner history (1) ]

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Attachments:: proof of Cantor's theorem (Proof) by Wkbj79

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Cross-references: strictly, cardinality, bijection, power set
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This is version 5 of Cantor's theorem, born on 2002-06-05, modified 2007-08-08.
Object id is 3051, canonical name is CantorsTheorem.
Accessed 9200 times total.

Classification:

AMS MSC:	03E17 (Mathematical logic and foundations :: Set theory :: Cardinal characteristics of the continuum)
	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

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