On computable automorphisms of the rational numbers (Q2758070)

The authors study the relationship between ideals \textbf{I} of the Turing degrees and groups of \textbf{I}-recursive automorphisms of orderings on the rational numbers. For any ideal \textbf{I} of the Turing degrees the group of \textbf{I}-recursive automorphisms consists of all automorphisms of the orderings on the rational numbers which have (under a suitable coding) a Turing degree lying in \textbf{I}. One of the main results of the paper is that the isomorphism type of such an \textbf{I}-recursive group defines the ideal \textbf{I}, or in other words, the ideal \textbf{I} can be `recovered' from the group. The authors also give a general correspondence between principal ideals \textbf{d} of the Turing degrees and the first-order properties of such \textbf{d}-recursive groups. The key idea involved in the proofs is to reconstruct the ideal by interpreting various concepts inside the group of \textbf{I}-recursive automorphisms, specifically the natural numbers and the first-order arithmetic.