### The complexity of orbits of computably enumerable sets

Peter A. Cholak, Rodney Downey, and Leo A. Harrington
Source: Bull. Symbolic Logic Volume 14, Issue 1 (2008), 69-87.

#### Abstract

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\varepsilon$, 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 $\varepsilon$ is $\omega_{1}^{CK}+1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\varepsilon$; for all finite $\alpha \geq$ 9, there is a properly $\Delta_{\alpha}^{0}$ orbit (from the proof).

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Primary Subjects: 03D25