  In this paper, we study the existence of minimal covers and strong minimal
covers in the Weihrauch degrees. We characterize when a problem $f$ is a
minimal cover or strong minimal cover of a problem $h$. We show that strong
minimal covers only exist in the cone below $\mathsf{id}$ and that the
Weihrauch lattice above $\mathsf{id}$ is dense. From this, we conclude that the
degree of $\mathsf{id}$ is first-order definable in the Weihrauch degrees and
that the first-order theory of the Weihrauch degrees is computably isomorphic
to third-order arithmetic.
