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Cantor's theorem holds in ZF but not in New Foundations |
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''Note: in order to fully understand this article you may want to refer to the set theory portion of the [[Table of mathematical symbols]].''
In Zermelo-Fränkel [[set theory]], '''Cantor's theorem''' states that the [[power set]] ([[set]] of all [[subset]]s) of any set ''A'' has a strictly greater [[cardinality]] than that of ''A''. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the [[power set]] of a [[countable set|countably infinite]] set is '''un'''countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below.
==The proof==
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