The Complexity of Orbits of Computably Enumerable Sets

Abstract

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

To appear in the Bulletion of Symbolic Logic

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