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

Author:

Downey Rodney G.1,Wu Guohua2,Yang Yue3

Affiliation:

1. School of Mathematics, Statistics and Operational Research, Victoria University of Wellington, Wellington, New Zealand

2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

3. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

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).

Publisher

World Scientific Pub Co Pte Lt

Subject

Logic

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