The $Delta^0_2$-Spectrum of a Linear Order

Journal of Symbolic Logic 66 (2):470-486 (2001)

Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable $\Delta^0_2$ degree, but not 0. Since our argument requires the technique of permitting below a $\Delta^0_2$ set, we include a detailed explanation of the mechanics and intuition behind this type of permitting
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