Skip to main content
SpringerLink
Log in

On the Lattices of Effectively Open Sets

  • Conference paper
  • First Online:
Pursuit of the Universal (CiE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

Included in the following conference series:

  • 686 Accesses

Abstract

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.

O.V. Kudinov—Supported by RFBR projects 13-01-00015a and 14-01-00376.

V.L. Selivanov—Supported by RFBR project 13-01-00015a.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. Ershov, Y.: Computable functionals of finite types. Algebra Log. 11(4), 367–433 (1972)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  4. Grzegorczyk, A.: Undecidability of some topological theories. Fundamenta Mathematicae 38, 137–152 (1951)

    MathSciNet  Google Scholar 

  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. 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. Harrington, L., Nies, A.: Coding in the lattice of enumerable sets. Adv. Math. 133, 133–162 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Korovina, M.V., Kudinov, O.V.: The uniformity principle for sigma-definability. J. Log. Comput. 19(1), 159–174 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Moschovakis, Y.N.: Descriptive Set Theory. North Holland, Amsterdam (2009)

    Book  MATH  Google Scholar 

  10. Nies, A.: Effectively dense Boolean algebras and their applications. Trans. Am. Math. Soc. 352(11), 4989–5012 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Selivanov, V.L.: Towards a descriptive set theory for domain-like structures. Theoret. Comput. Sci. 365, 258–282 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  13. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

Download references

Acknowledgement

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.

Author information

Authors and Affiliations

  1. S.L. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Oleg V. Kudinov

  2. A.P. Ershov Institute of Informatics Systems SB RAS, Novosibirsk State University, Novosibirsk, Russia

    Victor L. Selivanov

Authors
  1. Oleg V. Kudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor L. Selivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor L. Selivanov .

Editor information

Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Kudinov, O.V., Selivanov, V.L. (2016). On the Lattices of Effectively Open Sets. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40189-8_31

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40188-1

  • Online ISBN: 978-3-319-40189-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation