## Countable thin Π0 1 classes (1993)

Venue: | Annals of Pure and Applied Logic |

Citations: | 5 - 4 self |

### BibTeX

@ARTICLE{Cenzer93countablethin,

author = {Douglas Cenzer and Rodney Downey and Carl Jockusch and Richard Shore},

title = {Countable thin Π0 1 classes},

journal = {Annals of Pure and Applied Logic},

year = {1993},

volume = {59},

pages = {79--139}

}

### OpenURL

### Abstract

Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive Cantor-Bendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper non-principal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.