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A jump class of noncappable degrees

  • S. B. Cooper (a1)

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Friedberg [3] showed that every degree of unsolvability above 0′ is the jump of some degree, and Sacks [9] showed that the degrees above 0′ which are recursively enumerable (r.e.) in 0′ are the jumps of the r.e. degrees.

In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree c > 0′ which is r.e. in 0′ but which is not the jump of half a minimal pair of r.e. degrees.

This extends Yates' result [13] proving the existence of noncappable degrees (that is, r.e. degrees a < 0′ for which there is no corresponding r.e. b > 0 with ab = 0).

It also throws more light on the class PS of promptly simple degrees. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that PS coincides with the class NC of noncappable degrees, and with the class LC of all low-cuppable degrees, and (using earlier work of Maass, Shore and Stob [5]) that PS splits every class Hn or Ln, n ≥ 0, in the high-low hierarchy of r.e. degrees.

If c > 0′, with c r.e. in 0′, let

and call c−1 the jump class for c. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low aLC, where of course [a, 0′] (= {br.e. ∣ab0′}) ⊆ LC = PS. But by Robinson [7] [a, 0′] intersects with every jump class.

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References

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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]Bickford, M., The jump operator in strong reducibilities, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin, 1983.
[3]Friedberg, R. M., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1957), pp. 159160.
[4]Lachlan, A. H., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, ser. 2, vol. 16 (1966), pp. 537569.
[5]Maass, W., Shore, R. A. and Stob, M., Splitting properties and jump classes, Israel Journal of Mathematics, vol. 39 (1981), pp. 210224.
[6]Miller, D., High recursively enumerable degrees and the anti-cupping property, Logic year 1979–80 (Lerman, M.et al., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 230267.
[7]Robinson, R. W., Jump restricted interpolation in the r.e. degrees, Annals of Mathematics, ser. 2, vol. 93 (1971), pp. 586596.
[8]Sacks, G. E., Degrees of unsolvability, rev. ed., Princeton University Press, Princeton, New Jersey, 1966.
[9]Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223239.
[10]Shore, R. A., A non-inversion theorem for the jump operator, Annals of Pure and Applied Logic (to appear).
[11]Soare, R. I. and Stob, M., Relative recursive enumerability, Proceedings of the Herbrand symposium (Stern, J., editor), North-Holland, Amsterdam, 1982, pp. 299324.
[12]Soare, R. I., Tree arguments in recursion theory and the 0‴ priority method, Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 53106.
[13]Yates, C. E. M., A minimal pair of recursively enumerable degrees, this Journal, vol. 31 (1966), pp. 159168.

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A jump class of noncappable degrees

  • S. B. Cooper (a1)

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