  The uncountability of $\mathbb{R}$ is one of its most basic properties, known
far outside of mathematics. Cantor's 1874 proof of the uncountability of
$\mathbb{R}$ even appears in the very first paper on set theory, i.e. a
historical milestone. In this paper, we study the uncountability of
$\mathbb{R}$ in Kohlenbach's higher-order Reverse Mathematics (RM for short),
in the guise of the following principle: $$\hbox{for a countable set $A\subset
\mathbb{R}$, there exists $y\in \mathbb{R}\setminus A$.}$$ An important
conceptual observation is that the usual definition of countable set -- based
on injections or bijections to $\mathbb{N}$ -- does not seem suitable for the
RM-study of mainstream mathematics; we also propose a suitable (equivalent over
strong systems) alternative definition of countable set, namely union over
$\mathbb{N}$ of finite sets; the latter is known from the literature and closer
to how countable sets occur 'in the wild'. We identify a considerable number of
theorems that are equivalent to the centred theorem based on our alternative
definition. Perhaps surprisingly, our equivalent theorems involve most basic
properties of the Riemann integral, regulated or bounded variation functions,
Blumberg's theorem, and Volterra's early work circa 1881. Our equivalences are
also robust, promoting the uncountability of $\mathbb{R}$ to the status of
'big' system in RM.
