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The strong anticupping property for recursively enumerable degrees1

  • S. B. Cooper (a1)
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Following Soare [11] we say a recursively enumerable (r.e.) degree a has the anticupping (a.c.) property if there is a nonzero r.e. degree b < a such that for no r.e. c < a does a = bc.

Cooper [2] and Yates showed that 0′ has the a.c. property, while Harrington (see Miller [6]) proved that every high r.e. degree a has the a.c. property.

The recent paper by Ambos-Spies, Jockusch, Shore and Soare [1] describes a general theoretical framework for cupping and capping below 0′ which seems likely to be useful in a wider context.

Definition. (1) We say b is strongly noncuppable belowa if 0 < b < a and, for each d < a, b ∪ d ≠ a.

(2) We say an r.e. a has the strong anticupping property if there is an r.e. b which is strongly noncuppable below a.

The main results on cupping in (≤ 0′) are due to Epstein, Posner and Robinson. For instance it is known (Posner and Robinson [8]) that the s.a.c. property fails for 0′.

We prove below that r.e. degrees with the s.a.c. property do exist, hence obtaining a nonzero r.e. degree a such that (≤ a) ≢e(≤h) for any high r.e. degree h. This result, obtained by means of an infinite injury construction in (≤ 0′), extends Theorem 2 of [3], proved using a finite injury construction in (≤ 0′).

Our main source of notation and terminology is [3].

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1

We are grateful to the referee for a number of helpful suggestions and corrections. We are also grateful for valuable help from C. G. Jockusch and R. I. Soare in the preparation of the paper.

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[1]Ambos-Spies K.et al., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109128.
[2]Cooper S. B., On a theorem of C. E. M. Yates, handwritten notes, 1974.
[3]Cooper S. B. and Epstein R. L., Complementing below recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 1532.
[4]Epstein R. L., Initial segments of the degrees below 0′, Memoir no. 241, American Mathematical Society, Providence, Rhode Island, 1981.
[5]Lachlan A. H., Lower bounds for pairs of r.e. degrees, Proceedings of the London Mathematical Society, ser. 3, vol. 16 (1966), pp. 537569.
[6]Miller D., High recursively enumerable degrees and the anticupping property, Logic year 1979–80 (Lerman M.et al., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 230267.
[7]Posner D. B., High degrees, Ph.D. thesis, University of California, Berkeley, California, 1977.
[8]Posner D. and Robinson R. W., Degrees joining to 0′, this Journal, vol. 46 (1981), pp. 714721.
[9]Sacks G. E., Degrees of unsoltability, Annals of Mathematics Studies, vol. 55, Princeton University Press, Princeton, New Jersey, 1963.
[10]Shore R. A., The theory of the degrees below 0′, Journal of the London Mathematical Society, ser. 2, vol. 24 (1981), pp. 114.
[11]Soare R. I., Tree arguments in recursion theory and the 0′″-priority method, Recursion theory (Nerode A. and Shore R. A., editors), Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 53106.
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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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