On the Lattices of Effectively Open Sets

  • Oleg V. Kudinov
  • Victor L. SelivanovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)


We show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is hereditarily undecidable. Moreover, for several important spaces (e.g., finite-dimensional Euclidean spaces and the domain \(P\omega \)) this theory is m-equivalent to the first-order arithmetic.


Lattice Effectively open set First-order theory Interpretation 



We thank André Nies for a discussion of the lower bound problem for \(Th(\varSigma ^0_1(\mathcal {N}))\), and the referees for valuable comments.


  1. 1.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford University Press, Oxford (1994)Google Scholar
  2. 2.
    Ershov, Y.: Computable functionals of finite types. Algebra Log. 11(4), 367–433 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ershov, Y.: Theory of domains and nearby. In: Bjørner, D., Broy, M., Pottosin, I.V. (eds.) Formal Methods in Programming and Their Applications. LNCS, vol. 735, pp. 1–7. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  4. 4.
    Grzegorczyk, A.: Undecidability of some topological theories. Fundamenta Mathematicae 38, 137–152 (1951)MathSciNetGoogle Scholar
  5. 5.
    Herrmann, E.: Definable boolean pairs in the lattice of recursively enumerable sets. In: Proceedings of 1st Easter Conference in Model Theory, Diedrichshagen, pp. 42–67 (1983)Google Scholar
  6. 6.
    Herrmann, E.: The undecidability of the elementary theory of the lattice of recursively enumerable sets. In: Proceedings of 2nd Frege Conference at Schwerin, GDR, vol. 20, pp. 66–72 (1984)Google Scholar
  7. 7.
    Harrington, L., Nies, A.: Coding in the lattice of enumerable sets. Adv. Math. 133, 133–162 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Korovina, M.V., Kudinov, O.V.: The uniformity principle for sigma-definability. J. Log. Comput. 19(1), 159–174 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Moschovakis, Y.N.: Descriptive Set Theory. North Holland, Amsterdam (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Nies, A.: Effectively dense Boolean algebras and their applications. Trans. Am. Math. Soc. 352(11), 4989–5012 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Selivanov, V.L.: Towards a descriptive set theory for domain-like structures. Theoret. Comput. Sci. 365, 258–282 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.S.L. Sobolev Institute of Mathematics SB RASNovosibirskRussia
  2. 2.A.P. Ershov Institute of Informatics Systems SB RASNovosibirsk State UniversityNovosibirskRussia

Personalised recommendations