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 hyperdoctrine (...) of mass problems. We study the intermediate logic that the hyperdoctrine of mass problems gives us, and we study the theories of subintervals of the hyperdoctrine of mass problems 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 hyperdoctrine of mass problems and prove an analogue of Tennenbaum’s theorem on computable models of arithmetic. (shrink)

We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.