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The -spectrum of a linear order

  • Russell Miller (a1)

Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable degree, but not 0. Since our argument requires the technique of permitting below a set, we include a detailed explantion of the mechanics and intuition behind this type of permitting.

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[1] C. J. Ash , C. G. Jockusch , and J. F. Knight , Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), pp. 573599.

[2] J. C. E. Dekker , A theorem on hypersimple sets, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 791796.

[5] R. Downey and C. G. Jockusch , Every low boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871880.

[6] R. Downey and J. F. Knight , Orderings with α-th jump degree 0(α), Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545552.

[8] C. G. Jockusch and R. I. Soare , Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 3964.

[14] T. Slaman , Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21172122.

[15] R. I. Soare , Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.

[17] S. Wehner , Enumerations, countable structures, and turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21312139.

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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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