A basis result for Σ⁰₃ sets of reals with an application to minimal covers

Abstract

It is shown that every $\Sigma _3^0$ set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene’s $\mathcal {O}$ . As an application it is shown that every Turing degree above the Turing degree of Kleene’s $\mathcal {O}$ is a minimal cover.