Jumps of orderings

Abstract

Here it is shown that for each recursive ordinal α ⩾ 2 \alpha \geqslant 2 and each Turing degree d > 0 ( α ) {\mathbf {d}} > {{\mathbf {0}}^{(\alpha )}} , there is a linear ordering A {\mathbf {A}} such that d {\mathbf {d}} is least among the α \alpha th jumps of degrees of (open diagrams of) isomorphic copies of A {\mathbf {A}} and for β > α \beta > \alpha , the set of β \beta th jumps of degrees of copies of A {\mathbf {A}} has no least element.