Journal of Symbolic Logic

On the Structures Inside Truth-Table Degrees

Frank Stephan

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.

Abstract

The following theorems on the structure inside nonrecursive truth-table degrees are established: Degtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and therefore do not contain any objective set. Furthermore, a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enumerable semirecursive sets, one of coenumerable semirecursive sets and one of sets, which are neither enumerable nor coenumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integer as for example nineteen, but it is never an even finite number.

Article information

Source
J. Symbolic Logic, Volume 66, Issue 2 (2001), 731-770.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183746471

Mathematical Reviews number (MathSciNet)
MR1833476

Zentralblatt MATH identifier
1004.03035

JSTOR
links.jstor.org

Citation

Stephan, Frank. On the Structures Inside Truth-Table Degrees. J. Symbolic Logic 66 (2001), no. 2, 731--770. https://projecteuclid.org/euclid.jsl/1183746471


Export citation