The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\text{E}$, such that the question of membership in this orbit is ${\mathrm{\Sigma }}_1^1$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\text{E}$ is $\text{wock}+1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\text{E}$; for all finite $\alpha \ge 9$, there is a properly ${\mathrm{\Delta }}_{\alpha }^0$ orbit (from the proof). A few small corrections made in this version