  We construct a topos in which the Dedekind reals are countable.
  To accomplish this, we first define a new kind of toposes that we call
parameterized realizability toposes. They are built from partial combinatory
algebras whose application operation depends on a parameter, and in which
realizers operate uniformly with respect to a given parameter set. Our topos is
the parameterized realizability topos whose realizers are oracle-computable
partial maps, with oracles serving as parameters and ranging over the
representations of a non-diagonalizable sequence, discovered by Joseph Miller.
It is a sequence of reals in $[0,1]$ that is non-diagonalizable in the sense
that any real in $[0,1]$ that is oracle-computable, uniformly in oracles
representing the sequence, must already appear in the sequence. The Dedekind
reals are countable in the topos because the non-diagonalizable sequence
appears in it as an epimorphism.
  The topos is intuitionistic, as it invalidates both the law of excluded
middle and the axiom of countable choice. The Cauchy reals are uncountable. The
Hilbert cube is countable, from which Brouwer's fixed-point theorem follows as
an easy corollary of Lawvere's fixed-point theorem. From the 1-dimensional
Brouwer's fixed-point theorem we obtain the intermediate value theorem and the
lesser limited principle of omniscience. The Kreisel-Lacombe-Shoenfield-Tseitin
theorem stating that all real-valued maps are continuous is valid, because the
usual proof is uniform with respect to oracles. Lastly, the closed interval
$[0,1]$, being countable, can trivially be covered by a sequence of open
intervals whose lengths add up to any prescribed $0 < \epsilon < 1$, and such a
cover has no finite subcover. However, we show that any sequence of open
intervals with rational endpoints covering $[0,1]$ must has a finite subcover.
