A note on the enumeration degrees of 1-generic sets. (English) Zbl 1388.03041

The authors use a priority argument to prove that if \(A\) is a \(\Delta^0_2\) set of nonzero e-degree, then there is a set \(B \leq_e A\) such that \(B\) is \(1\)-generic. Using results from T. F. Kent and A. Sorbi [J. Symb. Log. 72, No. 4, 1405–1417 (2007; Zbl 1131.03019)], they show that that the \(1\)-generic e-degrees below \(0^\prime_e\) are not downwards closed, answering Question 4.13 of S. B. Cooper [Lect. Notes Math. 1432, 57–110 (1990; Zbl 0707.03034)].


03D30 Other degrees and reducibilities in computability and recursion theory
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