Definition 4

A set is said to have the power of the continuum if it is equipotent to the set of all subsets of $\mathbf{N}$.

A set which has the power of the continuum is not countable (§ 3, no. 6, Theorem 2).

\* The name "power of the continuum" arises from the fact that the set of real numbers is equipotent to $\mathfrak{P}(\mathbf{N})$ (General Topology, Chapter IV, § 8). \* The continuum hypothesis is the assertion that every uncountable set contains a subset which has the power of the continuum; the generalized continuum hypothesis is the assertion that, for every infinite cardinal $\mathfrak{a}$, every cardinal $> \mathfrak{a}$ is $\geqslant 2^{\mathfrak{a}}$.