Degrees containing members of thin Π10 classes are dense and co-dense

Journal of Mathematical Logic 18 (1):1850001 (2018)
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Abstract

In [Countable thin [Formula: see text] classes, Ann. Pure Appl. Logic 59 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees containing members of countable thin [Formula: see text] classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin [Formula: see text] classes. We will prove in this paper that the c.e. degrees containing no members of thin [Formula: see text] classes are dense in the c.e. degrees. We will also prove that the c.e. degrees containing members of thin [Formula: see text] classes are dense in the c.e. degrees, improving the result of Cenzer et al. mentioned above. Thus, we obtain a new natural subclass of c.e. degrees which are both dense and co-dense in the c.e. degrees, while the other such class is the class of branching c.e. degrees 113–130] for nonbranching degrees and [T. A. Slaman, The density of infima in the recursively enumerable degrees, Ann. Pure Appl. Logic 52 155–179] for branching degrees).

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References found in this work

Countable algebra and set existence axioms.Harvey M. Friedman - 1983 - Annals of Pure and Applied Logic 25 (2):141.
The density of infima in the recursively enumerable degrees.Theodore A. Slaman - 1991 - Annals of Pure and Applied Logic 52 (1-2):155-179.
The density of the nonbranching degrees.Peter A. Fejer - 1983 - Annals of Pure and Applied Logic 24 (2):113-130.

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