Pseudojump Operators and $\Pi^0_1$ Classes.

For a pseudojump operator VX and a $\Pi^0_1$ class P, we consider properties of the set {VX: X ∈ P}. We show that there always exists X ∈ P with $V^X \leq_T {\mathbf 0'}$ and that if P is Medvedev complete, then there exists X ∈ P with $ V^X \equiv_T {\mathbf 0'}$. We examine the consequences when VX is Turing incomparable with VY for X ≠ Y in P and when $W_e^X = W_e^Y$ for all X,Y ∈ P. Finally, we give a characterization of the jump in terms of $\Pi^0_1$ classes. [ABSTRACT FROM AUTHOR]