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Logica Universalis

, Volume 4, Issue 1, pp 31–39 | Cite as

Diagonalization in Double Frames

  • Andrzej Wiśniewski
  • Jerzy Pogonowski
Article
  • 57 Downloads

Abstract

We consider structures of the form (Φ, Ψ, R), where Φ and Ψ are non-empty sets and \({R\subseteq \Psi\times \Phi}\) is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R-images of elements of Ψ. The proof is a very elementary one, without any reference even to e.g. the \({S_{n}^{m}}\)-theorem. Some consequences of the main result are also discussed.

Mathematics Subject Classification (2000)

03D80 

Keywords

Diagonalization double frames incompleteness 

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References

  1. 1.
    Cori R., Lascar D.: Mathematical Logic. A Course with Exercises. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  2. 2.
    Cutland N.: Computability. An Introduction to Recursive Function Theory. Cambridge University Press, Cambridge, MA (1980)zbMATHGoogle Scholar
  3. 3.
    Harrah D.: On Completeness in the logic of questions. Am. Philos. Q. 6(2), 158–164 (1969)Google Scholar
  4. 4.
    Hinman P.G.: Fundamentals of Mathematical Logic. A K Peters, Wellesley, MA (2005)zbMATHGoogle Scholar
  5. 5.
    Odifreddi P.G.: Classical Recursion Theory. North-Holland, Amsterdam (1989)zbMATHGoogle Scholar
  6. 6.
    Rogers H.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, MA (1972)zbMATHGoogle Scholar
  7. 7.
    Soare R.I.: Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets. Springer, Berlin (1987)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Chair of Logic and Cognitive ScienceAdam Mickiewicz UniversityPoznańPoland
  2. 2.Department of Applied LogicAdam Mickiewicz UniversityPoznańPoland

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