Past exam of the logic and set theory course of the University of Oxford / 2007 / B / 7 / i / Solution

The Cantor-Schröder-Bernstein theorem says that injections $X\to Y$ and $Y\to X$ imply a bijection $X\to Y$. Thus mutual cardinal inequalities imply equal cardinalities, without a choice assumption.

Write $\mathfrak{c}=2^{{\aleph }_0}=|\mathbb{R}|$. Standard cardinal arithmetic gives $\mathfrak{c}\cdot \mathfrak{c}=\mathfrak{c}$ and ${\mathfrak{c}}^{{\aleph }_0}=\mathfrak{c}$. The correspondence $x+iy\leftrightarrow (x,y)$ is a bijection $\mathbb{C}\leftrightarrow {\mathbb{R}}^2$. HenceEquivalently, the real axis supplies the lower inequality and a coding of two real coordinates supplies the upper one; Cantor-Schröder-Bernstein then gives the equality.