A basis result for Σ⁰₃ sets of reals with an application to minimal covers
Leo A. Harrington, Alexander S. Kechris · Proceedings of the American Mathematical Society · 1975
It is shown that every Σ 3 0 \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 O \mathcal {O} . As an application it is shown that every Turing degree above the Turing degree of Kleene’s O \mathcal {O} is a minimal cover.