## 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)].

### MSC:

 03D30 Other degrees and reducibilities in computability and recursion theory

### Citations:

Zbl 1131.03019; Zbl 0707.03034
Full Text:

### References:

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