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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.

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

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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.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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