Orderings with $\alpha$th jump degree $\textbf {0}^ {(\alpha )}$

Abstract:

This paper completes an investigation of "jumps" of orderings. The last few cases are given in the proof that for each recursive ordinal $\alpha \geq 1$ and for each Turing degree ${\mathbf {d}} \geq {{\mathbf {0}}^{(\alpha )}}$, there is a linear ordering ${\mathbf {A}}$ such that ${\mathbf {d}}$ is least among the $\alpha$th jumps of degrees of (open diagrams of) isomorphic copies of ${\mathbf {A}}$, and for $\beta < \alpha$, the set of $\beta$ jumps of degrees of copies of ${\mathbf {A}}$ has no least element.