Cantor's theorem

In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set ${\displaystyle A}$, the set of all subsets of ${\displaystyle A,}$ known as the power set of ${\displaystyle A,}$ has a strictly greater cardinality than ${\displaystyle A}$ 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 ${\displaystyle n}$ elements has a total of ${\displaystyle 2^{n}}$ subsets, and the theorem holds because ${\displaystyle 2^{n}>n}$ 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 sets also. As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, 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").