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Elementary Differences Among Jump Hierarchies

  • Angsheng Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4484)

Abstract

It is shown that Th(H 1) ≠ Th (H n ) holds for every n > 1, where H m is the upper semi-lattice of all high m computably enumerable (c.e.) degrees for m > 0, giving a first elementary difference among the highness hierarchies of the c.e. degrees.

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References

  1. 1.
    Cooper, S.B.: On a theorem of C.E.M. Yates (handwritten notes) (1974a)Google Scholar
  2. 2.
    Cooper, S.B.: Minimal pairs and high recursively enumerable degrees. J. Symbolic Logic 39, 655–660 (1974b)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cooper, S.B., Li, A.: Splitting and nonsplitting, II: A low2 c.e. degree above which 0′ is not splittable. The Journal of Symbolic Logic 67(4) (2002)Google Scholar
  4. 4.
    Downey, R.G., Lempp, S., Shore, R.A.: Highness and bounding minimal pairs. Math. Logic Quarterly 39, 475–491 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Harrington, L.: On Cooper’s proof of a theorem of Yates, Parts I and II (handwritten notes) (1976)Google Scholar
  6. 6.
    Carl, G., et al.: A join theorem for the computably enumerable degrees. Transactions of the American Mathematical Society 356(7), 2557–2568 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harrington, L.: Plus cupping in the recursively enumerable degrees (handwritten notes) (1978)Google Scholar
  8. 8.
    Lachlan, A.H.: On a problem of G.E. Sacks. Proc. Amer. Math. Soc. 16, 972–979 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lachlan, A.H.: A recursively enumerable degree which will not split over all lesser ones. Ann. Math. Logic 9, 307–365 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lachlan, A.H.: Bounding minimal pairs. J. Symbolic Logic 44, 626–642 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Martin, D.A.: On a question of G.E. Sacks. J. Symbolic Logic 31, 66–69 (1966a)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Martin, D.A.: Classes of recursively enumerable sets and degrees of unsolvability, 2. Math. Logik Grundlag. Math. 12, 295–310 (1966b)zbMATHCrossRefGoogle Scholar
  13. 13.
    Miller, D.: High recursively enumerable degrees and the anti-cupping property. In: Lerman, M., Schmerl, J.H., Soare, R.I. (eds.) Logic Year 1979–1980. Lecture Notes in Mathematics, vol. 859, Springer, Heidelberg (1981)CrossRefGoogle Scholar
  14. 14.
    Nies, A., Shore, R.A., Slaman, T.A.: Interpretability and definability in the recursively enumerable degrees. Proc. London Math. (3) 77(2), 241–291 (1998)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Robinson, R.W.: Interpolation and embedding in the recursively enumerable degrees. Ann. of Math. (2) 93, 285–314 (1971a)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Robinson, R.W.: Jump restricted interpolation in the recursively enumerable degrees. Ann. of Math. (2) 93, 586–596 (1971b)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Sacks, G.E.: Recursive enumerability and the jump operator. Tran. Amer. Math. Soc. 108, 223–239 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sacks, G.E.: The recursively enumerable degrees are dense. Ann. of Math. (2) 80, 300–312 (1964)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Sacks, G.E.: On a theorem of Lachlan and Martin. Proc. Amer. Math. Soc. 18, 140–141 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shore, R.A.: The lowm and lown r.e. degrees are not elementarily equivalent. Science in China, Series A (2004)Google Scholar
  21. 21.
    Shore, R.A., Slaman, T.A.: Working below a low2 recursively enumerable degree. Archive for Math. Logic 29, 201–211 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Shore, R.A., Slaman, T.A.: Working below a high recursively enumerable degree. J. Symbolic Logic 58(3), 824–859 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Angsheng Li
    • 1
  1. 1.State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing 100080P.R. China

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