Skip to main content
SpringerLink
Log in
Book cover

Conference on Computability in Europe

CiE 2012: How the World Computes pp 129–138Cite as

On Mathias Generic Sets

  • Conference paper

  • 1654 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7318))

Abstract

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 3 then it satisfies the jump property G (n − 1) = G′ ⊕ ∅ (n). We prove that every such G has generalized high degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that G, together with any bi-immune A ≤  T  ∅ (n − 1), computes a Cohen n-generic.

This is a preview of subscription content, 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 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Binns, S., Kjos-Hanssen, B., Lerman, M., Solomon, R.: On a conjecture of Dobrinen and Simpson concerning almost everywhere domination. J. Symbolic Logic 71, 119–136 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cholak, P.A., Jockusch Jr., C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symbolic Logic 66, 1–55 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Downey, R.G., Hirschfeldt, D.R.: Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York (2010)

    Book  MATH  Google Scholar 

  4. Dzhafarov, D.D., Jockusch Jr., C.G.: Ramsey’s theorem and cone avoidance. J. Symbolic Logic 74, 557–578 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jockusch Jr., C.G.: The degrees of bi-immune sets. Z. Math. Logik Grundlagen Math. 15, 135–140 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jockusch Jr., C.G.: Degrees of generic sets. In: Drake, F.R., Wainer, S.S. (eds.) Recursion Theory: its Generalisation and Applications. London Math. Soc. Lecture Note Ser., vol. 45, pp. 110–139. Cambridge University Press, Cambridge (1980)

    Chapter  Google Scholar 

  7. Jockusch Jr., C.G., Posner, D.B.: Double jumps of minimal degrees. J. Symbolic Logic 43, 715–724 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kurtz, S.A.: Notions of weak genericity. J. Symbolic Logic 48, 764–770 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Seetapun, D., Slaman, T.A.: On the strength of Ramsey’s theorem. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36, 570–582 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Soare, R.I.: Computability theory and applications. Theory and Applications of Computability. Springer, New York (2012)

    Google Scholar 

  11. Soare, R.I.: Sets with no subset of higher degree. J. Symbolic Logic 34, 53–56 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yu, L.: Lowness for genericity. Arch. Math. Logic 45, 233–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556-4618, United States of America

    Peter A. Cholak & Damir D. Dzhafarov

  2. Department of Mathematical Sciences, Appalachian State University, 322 Walker Hall, Boone, NC, 28608-2091, United States of America

    Jeffry L. Hirst

Authors
  1. Peter A. Cholak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Damir D. Dzhafarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jeffry L. Hirst
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Computer Laboratory, University of Cambridge, Willialm Gates Building, J.J. Thomson Avenue, CB3 0FD, Cambridge, UK

    Anuj Dawar

  3. Institute for Logic, Language and Computation, University of Amsterdam, 1090 GE, Amsterdam, The Netherlands

    Benedikt Löwe

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cholak, P.A., Dzhafarov, D.D., Hirst, J.L. (2012). On Mathias Generic Sets. In: Cooper, S.B., Dawar, A., Löwe, B. (eds) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol 7318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30870-3_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-30870-3_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30869-7

  • Online ISBN: 978-3-642-30870-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation