The Complexity of Orbits of Computably Enumerable Sets
Peter A. Cholak, Rodney G. Downey, Leo A. Harrington · Bulletin of Symbolic Logic · 2008
Abstract The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).