  Kolmogorov introduced an informal calculus of problems in an attempt to
provide a classical semantics for intuitionistic logic. This was later
formalised by Medvedev and Muchnik as what has come to be called the Medvedev
and Muchnik lattices. However, they only formalised this for propositional
logic, while Kolmogorov also discussed the universal quantifier. We extend the
work of Medvedev to first-order logic using the notion of a first-order
hyperdoctrine from categorical logic to a structure which we will call the
Medvedev hyperdoctrine. We also study the intermediate logic that the Medvedev
hyperdoctrine gives us, where we focus in particular on subintervals of the
Medvedev hyperdoctrine in an attempt to obtain an analogue of Skvortsova's
result that there is a factor of the Medvedev lattice characterising
intuitionistic propositional logic. Finally, we consider Heyting arithmetic in
the Medvedev hyperdoctrine and prove an analogue of Tennenbaum's theorem on
computable models of arithmetic.
