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Suppose that $A \leq_a 0^\omega$ (i.e. $A$ is arithmetic in $0^\omega$) does there exist $\widehat{A} \equiv_a A$ with $\widehat{A} \leq_T 0^\omega$ [1]? More generally, say that a set $X$ is aT-...

A degree is $\alpha$-generic if it has representative that is $\alpha$-generic. Are the $\omega$-generic arithmetic degrees (i.e. the degree structure induced by arithmetic reproducibility) downward ...