  Cantor's first set theory paper (1874) establishes the uncountability of
$\mathbb{R}$. We study this most basic mathematical fact formulated in the
language of higher-order arithmetic. In particular, we investigate the logical
and computational properties of NIN (resp. NBI), i.e. the third-order
statement: there is no injection (resp. bijection) from $[0,1]$ to
$\mathbb{N}$. Working in Kohlenbach's higher-order Reverse Mathematics, we show
that NIN and NBI are hard to prove in terms of (conventional) comprehension
axioms, while many basic theorems, like Arzela's convergence theorem for the
Riemann integral (1885), are shown to imply NIN and/or NBI. Working in Kleene's
higher-order computability theory based on S1-S9, we show that the following
fourth-order process based on NIN is similarly hard to compute: for a given
$[0,1]\rightarrow \mathbb{N}$-function, find reals in the unit interval that
map to the same natural number.
