In this talk, we consider the structure of Turing degrees below \(0’\) in the theory that is a fragment of Peano arithmetic without \(\Sigma_1\) induction, with special focus on proper \(d\)-r.e. degrees and non-r.e. degrees. We prove
(1) \(P^-+ B\Sigma_1+ \mathrm{Exp}\) implies the existence of a proper \(d\)-r.e. degree.
(2) Over \(P^-+ B\Sigma_1+ \mathrm{Exp}\), the existence of a proper \(d\)-r.e. degree below \(0’\) is equivalent with \(I\Sigma_1\).
(3) \(P^-+ B\Sigma_1+ \mathrm{Exp}\) is not enough to show there is a non-r.e. degree below \(0’\).