Skip to main content
Log in

On Lachlan’s major sub-degree problem

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, \({{\bf 0' = b \lor x}}\) if and only if \({{\bf 0' = a \lor x}}\) . In this paper, we show that every c.e. degree b0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambos-Spies K., Lachlan A.H., Soare R.I.: The continuity of cupping to 0′. Ann. Pure Appl. Log. 64, 195–209 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cooper, S.B.: On a theorem of C.E.M. Yates (handwritten notes) (1974a)

  3. Cooper S.B.: Minimal pairs and high recursively enumerable degrees. J. Symb. Log. 39, 655–660 (1974b)

    Article  Google Scholar 

  4. Cooper S.B.: Computability Theory. Chapman & Hall/CRC Press, London/West Palm Beach (2004)

    MATH  Google Scholar 

  5. Cooper S.B., Li A.: Splitting and nonsplitting, II: A Low2 c.e. degree above which 0′ is not splittable. J. Symb. Log. 67(4), 1391–1430 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Friedberg R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. USA 43, 236–238 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  7. Giorgi, M.B.: Continuity properties of degree structures. Ph.D. Dissertation, University of Leeds (2001)

  8. Harrington, L.: On Cooper’s Proof of a Theorem of Yates, Part I, (handwritten notes) (1976)

  9. Harrington, L.: Understanding Lachlan’s Monster Paper (handwritten notes) (1980)

  10. Harrington, L., Soare, R.I.: Games in recursion theory and continuity properties of capping degrees. In: Proceedings of the Workshop on Set Theory and the Continuum. Mathematical Sciences Research Institute, Berkeley, 16–20 Oct 1989

  11. Jockusch C.G. Jr., Shore R.A.: Pseudo jump operators I: the R.E. case. Trans. Am. Math. Soc. 275, 599–609 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lachlan A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. Lond. Math. Soc. 16, 537–569 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lachlan A.H.: On the lattice of recursively enumerable sets. Trans. Am. Math. Soc. 130, 1–37 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lachlan, A.H.: Embedding nondistributive lattices in the recursively enumerable degrees. In: Hodges, W. (ed.) Conference in mathematical logic, London, 1970. Lecture Notes in Mathematics, vol. 255. Springer, Berlin (1972)

  15. Lachlan A.H.: A recursively enumerable degree which will not split over all lesser ones. Ann. Math. Log. 9, 307–365 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lachlan A.H.: Bounding minimal pairs. J. Symb. Log. 44, 626–642 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. 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: University of Connecticut. Lecture Notes in Mathematics, vol. 859, pp. 230–245. Springer, Berlin (1981)

  18. Muchnik, A.A.: On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR N.S. 108, 194–197 (Russian) (1956)

    Google Scholar 

  19. Post E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284–316 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  20. Robinson R.W.: Interpolation and embedding in the recursively enumerable degrees. Ann. Math. 93(2), 285–314 (1971)

    Article  Google Scholar 

  21. Rogers H. Jr.: Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, New York (1967)

    MATH  Google Scholar 

  22. Sacks G.E.: On the degrees less than 0′. Ann. Math. 77(2), 211–231 (1963)

    Article  MathSciNet  Google Scholar 

  23. Sacks G.E.: The recursively enumerable degrees are dense. Ann. Math. 80(2), 300–312 (1964)

    Article  MathSciNet  Google Scholar 

  24. Seetapun, D.: Contributions to recursion theory. Ph.D. Thesis, Trinity College, Cambridge (1991)

  25. Seetapun, D.: Defeating Red, (handwritten notes) (1992)

  26. Shoenfield J.R.: Application of model theory to degrees of unsolvability. In: Addison, J.W., Henkin, L., Tarski, A.(eds) Symposium on the Theory of Models, North-Holland, Amsterdam (1965)

    Google Scholar 

  27. Shore R.A., Slaman T.A.: Working below low2 recursively enumerable degrees. Arch. Math. Log. 29, 201–211 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Slaman T.A., Soare R.I.: Algebraic aspects of the recursively enumerable degrees. Proc. Nat. Acad. Sci. USA 92, 617–621 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Soare R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Function and Computably Generated Sets. Springer, Berlin (1987)

    Google Scholar 

  30. Yates C.E.M.: A minimal pair of recursively enumerable degrees. J. Symb. Log. 31, 159–168 (1966)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Angsheng Li.

Additional information

Both authors were funded by EPSRC Research Grant no. GR/M 91419, “Turing Definability”, by INTAS-RFBR Research Grant no. 97-0139, “Computability and Models”, and by an NSFC Grand International Joint Project Grant no. 60310213, “New Directions in Theory and Applications of Models of Computation”. Both authors are grateful to Andrew Lewis for helpful suggestions regarding presentation, technical aspects of the proof, and verification. A. Li is partially supported by National Distinguished Young Investigator Award no. 60325206 (People’s Republic of China).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cooper, S.B., Li, A. On Lachlan’s major sub-degree problem. Arch. Math. Logic 47, 341–434 (2008). https://doi.org/10.1007/s00153-008-0083-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-008-0083-5

Keywords

Mathematics Subject Classification (2000)

Navigation