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

Abstract

In [Countable thin [Formula: see text] classes, Ann. Pure Appl. Logic 59 (1993) 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees (not necessarily c.e.) 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 (See [P. Fejer, The density of the nonbranching degrees, Ann. Pure Appl. Logic 24 (1983) 113–130] for nonbranching degrees and [T. A. Slaman, The density of infima in the recursively enumerable degrees, Ann. Pure Appl. Logic 52 (1991) 155–179] for branching degrees).