On the orbits of computably enumerable sets

Abstract

The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, E \mathcal {E} , such that the question of membership in this orbit is Σ 1 1 \Sigma ^1_1 -complete. This result and proof have a number of nice corollaries: the Scott rank of E \mathcal {E} is ω 1 C K + 1 \omega _1^{ {CK}}+1 ; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E \mathcal {E} ; for all finite α ≥ 9 \alpha \geq 9 , there is a properly Δ α 0 \Delta ^0_\alpha orbit (from the proof).