On the degree spectrum of a $\Pi ^0_1$ class
Thomas F. Kent, Andrew E. M. Lewis · Transactions of the American Mathematical Society · 2010
For any $\mathcal {P} \subseteq 2^{\omega }$, define $S(\mathcal {P})$, the degree spectrum of $\mathcal {P}$, to be the set of all Turing degrees $\bf {a}$ such that there exists $A\in \mathcal {P}$ of degree $\bf {a}$. We prove a number of basic properties of the structure which is the degree spectra of $\Pi ^0_1$ classes ordered by inclusion and also study in detail some other phenomena relating to the study of $\Pi ^0_1$ classes from a degree theoretic point of view, which are brought to light as a result of this analysis.