Skip to main content
    • Aa
    • Aa

Maximal contiguous degrees

  • Peter Cholak (a1), Rod Downey (a2) and Stephen Walk (a3)

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Klaus Ambos-Spies [1984], Contiguous r.e. degrees, Computation and proof theory (Aachen, 1983), Springer, Berlin, pp. 137.

R. G. Downey [1987], Localization of a theorem of Ambos-Spies and the strong antisplitting property, Archiv für Mathematiscke Logik and Grundlagenforschung, vol. 26, no. 3-4. pp. 127127.

R. G. Downey and M. Stob [1986], Structural interactions of the recursively enumerable T- and w-degrees, Annals of Pure and Applied Logic, vol. 31, no. 2-3, pp. 205236, Special issue: second Southeast Asian logic conference (Bangkok, 1984).

Rodney G. Downey [1990], Lattice nonembeddings and initial segments of the recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 49, pp. 97119.

Rodney G. Downey , C. G. Jockusch , and Michael Stob [1990], Array nonrecursive sets and multiple permitting arguments, Recursion theory week (Proceedings, Oberwolfach 1989) ( K. Ambos-Spies , G. H. Müller , and G. E. Sacks , editors). Lecture Notes in Mathematics, no. 1432. Springer-Verlag, pp. 141173.

Rodney G. Downey , Steffen Lempp , and Richard A. Shore [1993], Highness and bounding minimal pairs, Mathematical Logic Quarterly, vol. 39, no. 4, pp. 475491.

Rodney G. Downey and R. A. Shore [1996], Lattice embeddings below a nonlowi recursively enumerable degree, Israel Journal of Mathematics, vol. 94, pp. 221246.

Rodney G. Downey and Michael Stob [1993], Splitting theorems in recursion theory, Annals of Pure and Applied Logic, vol. 65, pp. 1106.

R. E. Ladner and L. P. Sasso [1975], The weak truth table degrees of recursively enumerable sets, Annals of Mathematical Logic, vol. 8, pp. 429448.

Steffen Lempp , André Nies , and Theodore A. Slaman [1998], The Π3-theory of the computably enumerable Turing degrees is undecidable, Transactions of the American Mathematical Society, vol. 350, no. 7, pp. 27192736.

André Nies [2000], Definability in the c.e. degrees: questions and results, Computability theory and its applications (Boulder, CO, 1999) ( Peter Cholak , Steffen Lempp , Manny Lerman , and Richard Shore , editors). American Mathematical Society, Providence, RI, pp. 207213.

André Nies , Richard A. Shore , and Theodore A. Slaman [1998], Interpretability and definability in the recursively enumerable degrees, Proceedings of the London Mathematical Society. Third Series, vol. 77, no. 2, pp. 241291.

Richard A. Shore [2000], Natural definability in degree structures, Computability theory and its applications (Boulder, CO, 1999) ( Peter Cholak , Steffen Lempp , Manny Lerman , and Richard Shore , editors), American Mathematical Society, Providence, RI, pp. 255271.

R. I. Soare [1987], Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, New York.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 46 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th August 2017. This data will be updated every 24 hours.