Archive for Mathematical Logic

, Volume 45, Issue 6, pp 769–781 | Cite as

\(\Pi^0_1\)-Presentations of Algebras

  • Bakhadyr Khoussainov
  • Theodore Slaman
  • Pavel Semukhin


In this paper we study the question as to which computable algebras are isomorphic to non-computable \(\Pi_{1}^{0}\)-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\(\Pi_{1}^{0}\)-presentations. On the other hand, many of this structures fail to have non-computable \(\Sigma_{1}^{0}\)-presentation.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Bakhadyr Khoussainov
    • 1
  • Theodore Slaman
    • 2
  • Pavel Semukhin
    • 1
    • 3
  1. 1.Department of Computer ScienceThe University of AucklandAucklandNew Zealand
  2. 2.Department of MathematicsThe University of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsNovosibirsk State UniversityNovosibiriskRussia

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