Jumps of orderings

Ash, C. J.; Jockusch, C. G.; Knight, J. F.

doi:10.1090/S0002-9947-1990-0955487-0

Jumps of orderings
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by C. J. Ash, C. G. Jockusch and J. F. Knight

Trans. Amer. Math. Soc. 319 (1990), 573-599

DOI: https://doi.org/10.1090/S0002-9947-1990-0955487-0

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Abstract:

Here it is shown that for each recursive ordinal $\alpha \geqslant 2$ and each Turing degree ${\mathbf {d}} > {{\mathbf {0}}^{(\alpha )}}$, 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 $\beta$th jumps of degrees of copies of ${\mathbf {A}}$ has no least element.

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