Pseudojump Operators and $\Pi^0_1$ Classes

Abstract

For a pseudojump operator V ^X and a $\Pi^0_1$ class P, we consider properties of the set {V ^X: 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 V ^X is Turing incomparable with V ^Y 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.