Π10 classes and strong degree spectra of relations

Abstract

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable Π₁⁰ subsets of 2^ω and Kolmogorov complexity play a major role in the proof.