Cantor's theorem

Every set is smaller than its power set

In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle A} , the set of all subsets of A , {\displaystyle A,} known as the power set of A , {\displaystyle A,} has a strictly greater cardinality than A {\displaystyle A} itself. For finite sets, 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 n {\displaystyle n} elements has a total of 2 n {\displaystyle 2^{n}} subsets, and the theorem holds because 2 n > n {\displaystyle 2^{n}>n} for all non-negative integers.

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