Corollary

The set of real numbers has a cardinal strictly greater than the cardinal of a countable set.

A set equipotent to $\mathbf{R}$ is said to have the power of the continuum. By Proposition 1 of § 4, no. 1, every interval which contains more than one point has the power of the continuum. Again, the complement of a countable subset of $\mathbf{R}$ has the power of the continuum in particular, the set of all irrational numbers has the power of the continuum.