Relatively recursively enumerable reals
نویسنده
چکیده
We say that a real X is relatively r.e. if there exists a real Y such that X is r.e. (Y ) and X 6≤T Y . We say X is relatively REA if there exists such a Y ≤T X. We define A ≤e1 B if there exists a Σ1 set C such that n ∈ A if and only if there is a finite E ⊆ B with (n, E) ∈ C. In this paper we show that a real X is relatively r.e. if and only if X 6≤e1 X. We prove that every nonempty Π01 class contains a real which is not relatively r.e. We also construct a real which is relatively r.e. but not relatively REA. We say that a real X is relatively simple and above if there exists a real Y such that X is r.e. (Y ) and there is no infinite Z ⊆ X such that Z is r.e. (Y ). We prove that every 1-generic real is relatively simple and above.
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