  In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall
x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists
y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists
y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik
reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and
$A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.
