Notre Dame Journal of Formal Logic

Embedding and Coding below a 1-Generic Degree

Noam Greenberg and Antonio Montalbán

Abstract

We show that the theory of 𝒟(≤ g), where g is a 2-generic or a 1-generic degree below 0ʹ, interprets true first-order arithmetic. To this end we show that 1-genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin's method of coding in Turing degrees. We also prove that any recursive lattice can be embedded below a 1-generic degree preserving top and bottom.

Article information

Source
Notre Dame J. Formal Logic, Volume 44, Number 4 (2003), 200-216.

Dates
First available in Project Euclid: 29 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1091122498

Digital Object Identifier
doi:10.1305/ndjfl/1091122498

Mathematical Reviews number (MathSciNet)
MR2130306

Zentralblatt MATH identifier
1066.03045

Subjects
Primary: 03D28: Other Turing degree structures 03D35: Undecidability and degrees of sets of sentences

Keywords
generic degree elementary theory lattice embedding

Citation

Greenberg, Noam; Montalbán, Antonio. Embedding and Coding below a 1-Generic Degree. Notre Dame J. Formal Logic 44 (2003), no. 4, 200--216. doi:10.1305/ndjfl/1091122498. https://projecteuclid.org/euclid.ndjfl/1091122498


Export citation

References

  • Chong, C. T., and R. G. Downey, "Minimal degrees recursive in $1$"-generic degrees, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 215–25.
  • Chong, C. T., and C. G. Jockusch, "Minimal degrees and $1$"-generic sets below ${\bf 0}'$, pp. 63–77 in Computation and Proof Theory (Aachen, 1983), vol. 1104 of Lecture Notes in Mathematics, Springer, Berlin, 1984.
  • Downey, R., C. G. Jockusch, and M. Stob, "Array nonrecursive degrees and genericity", pp. 93–104 in Computability, Enumerability, Unsolvability, vol. 224 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1996.
  • Harrington, L., and S. Shelah, "The undecidability of the recursively enumerable degrees", American Mathematical Society Bulletin. New Series, vol. 6 (1982), pp. 79–80.
  • Haught, C. A., "The degrees below a $1$"-generic degree $<{\bf 0}'$, The Journal of Symbolic Logic, vol. 51 (1986), pp. 770–77.
  • Jockusch, C. G., Jr., "Degrees of generic sets", pp. 110–39 in Recursion Theory: Its Generalisation and Applications (Proceedings of Logic Colloquium, University of Leeds, Leeds, 1979), edited by F. R. Drake and S. S. Wainer, vol. 45 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1980.
  • Jockusch, C. G., Jr., and R. A. Shore, "Pseudojump operators. II". Transfinite iterations, hierarchies and minimal covers, The Journal of Symbolic Logic, vol. 49 (1984), pp. 1205–36.
  • Jónsson, B., "On the representation of lattices", Mathematica Scandinavica, vol. 1 (1953), pp. 193–206.
  • Kanamori, A., The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings, 2d edition, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
  • Kumabe, M., "A $1$"-generic degree which bounds a minimal degree, The Journal of Symbolic Logic, vol. 55 (1990), pp. 733–43.
  • Kumabe, M., "A $1$"-generic degree with a strong minimal cover, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1395–1442.
  • Kunen, K., Set Theory. An Introduction to Independence Proofs, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1983.
  • Lachlan, A. H., "Distributive initial segments of the degrees of unsolvability", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 457–72.
  • Lerman, M., "Initial segments of the degrees below $0'$", Notices of the American Mathematical Society, vol. 25 (1978), pp. A–506.
  • Lerman, M., Degrees of Unsolvability. Local and Global Theory, Perspective in Mathematical Logic, Omega Series. Springer-Verlag, Berlin, 1983.
  • Nerode, A., and R. A. Shore, "Reducibility orderings: T"heories, definability and automorphisms, Annals of Mathematical Logic, vol. 18 (1980), pp. 61–89.
  • Nerode, A., and R. A. Shore, "Second order logic and first order theories of reducibility orderings", pp. 181–200 in The Kleene Symposium (Proceedings, University of Wisconsin, Madison, 1978), vol. 101 of Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam, 1980.
  • Nies, A., R. A. Shore, and T. A. Slaman, "Interpretability and definability in the recursively enumerable degrees", Proceedings of the London Mathematical Society. 3d Series, vol. 77 (1998), pp. 241–91.
  • Odifreddi, P., and R. Shore, "Global properties of local structures of degrees", Unione Matematica Italiana. Bollettino. B. Serie VII, vol. 5 (1991), pp. 97–120.
  • Shoenfield, J. R., Mathematical Logic, Addison-Wesley Publishing Co., Reading, 1967.
  • Shore, R. A., "The theory of the degrees below ${\bf 0}\sp{\prime} $", The Journal of the London Mathematical Society. 2d Series, vol. 24 (1981), pp. 1–14.
  • Shore, R. A., "Finitely generated codings and the degrees r.e. in a degree ${\bf d}$", Proceedings of the American Mathematical Society, vol. 84 (1982), pp. 256–63.
  • Simpson, S. G., "First-order theory of the degrees of recursive unsolvability", Annals of Mathematics. 2d Series, vol. 105 (1977), pp. 121–39.
  • Slaman, T. A., and W. H. Woodin, "Decidability in degree structures", In Preparation.
  • Slaman, T. A., and W. H. Woodin, "Definability in the T"uring degrees, Illinois Journal of Mathematics, vol. 30 (1986), pp. 320–34.
  • Soare, R. I., Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987.
  • Whitman, P. M., "Lattices, equivalence relations, and subgroups", Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 507–22.