Abstract
A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and \({X \not\leq_T Y}\). A real X is relatively simple and above if there is a real Y < T X such that X is c.e.(Y) and there is no infinite set \({Z \subseteq \overline{X}}\) such that Z is c.e.(Y). We prove that every nonempty \({\Pi^0_1}\) class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.
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Anderson, B.A. Relatively computably enumerable reals. Arch. Math. Logic 50, 361–365 (2011). https://doi.org/10.1007/s00153-010-0219-2
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DOI: https://doi.org/10.1007/s00153-010-0219-2