## Bulletin of Symbolic Logic

### 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, $\varepsilon$, 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).

#### Article information

Source
Bull. Symbolic Logic Volume 14, Issue 1 (2008), 69-87.

Dates
First available in Project Euclid: 16 April 2008

https://projecteuclid.org/euclid.bsl/1208358844

Digital Object Identifier
doi:10.2178/bsl/1208358844

Mathematical Reviews number (MathSciNet)
MR2395047

Zentralblatt MATH identifier
1142.03022

Subjects
Primary: 03D25: Recursively (computably) enumerable sets and degrees

#### Citation

Cholak, Peter A.; Downey, Rodney; Harrington, Leo A. The complexity of orbits of computably enumerable sets. Bull. Symbolic Logic 14 (2008), no. 1, 69--87. doi:10.2178/bsl/1208358844. https://projecteuclid.org/euclid.bsl/1208358844