Cantor's theorem

The cardinality of the set {x, y, z}, is three, while there are eight elements in its power set (3 < 2³ = 8), here ordered by inclusion.

In elementary set theory, Cantor's theorem is a fundamental result which states that, for any set ${\displaystyle A}$, the set of all subsets of ${\displaystyle A}$ (the power set of ${\displaystyle A}$, denoted by ${\displaystyle {\mathcal {P}}(A)}$) 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 ${\displaystyle n}$ members has a total of ${\displaystyle 2^{n}}$ subsets, so that if ${\displaystyle \operatorname {card} (A)=n,}$ then ${\displaystyle \operatorname {card} ({\mathcal {P}}(A))=2^{n}}$, 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, which showed that the theorem holds for infinite sets, countable or uncountable, as well as finite ones. As a particularly important consequence, the power set of the set of natural numbers, a countably infinite set with cardinality ℵ₀ = card(N), is uncountably infinite and has the same size as the set of real numbers, a cardinality larger than that of the set of natural numbers that is often referred to as the cardinality of the continuum: