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The Complexity of Orbits of Computably Enumerable Sets

Peter A. Cholak, Rodney Downey and Leo A. Harrington
The Bulletin of Symbolic Logic
Vol. 14, No. 1 (Mar., 2008), pp. 69-87
Stable URL: http://www.jstor.org/stable/40039610
Page Count: 19
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The Complexity of Orbits of Computably Enumerable Sets
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Abstract

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, 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 ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta _\alpha ^0 $ orbit (from the proof).

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