In this paper, we study the existence of minimal covers and strong minimal covers in the Weihrauch degrees. We characterize when a problem f f is a minimal cover or strong minimal cover of a problem h h . We show that strong minimal covers only exist in the cone below i d id and that the Weihrauch lattice above i d id is dense. From this, we conclude that the degree of i d 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.
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Lempp et al. (2024) studied this question.
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