Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 997–1013 | Cite as

Division by zero

Article

Abstract

For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine equations provably unsolvable in Robinson’s arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process.

Keywords

Robinson arithmetic Diophantine equation Decidability Universal theory 

Mathematics Subject Classification

03F30 

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References

  1. 1.
    Doyle, P.G., Conway, J.H.: Division by three (1994). arXiv:math/0605779 [math.LO]
  2. 2.
    Doyle, P.G., Qiu, C.: Division by four (2015). arXiv:1504.01402 [math.LO]
  3. 3.
    Eijck, J., Iemhoff, R., Joosten, J.J. (eds.): Liber Amicorum Alberti: A Tribute to Albert Visser, Tributes, vol. 30. College Publications, London (2016)Google Scholar
  4. 4.
    Gaifman, H., Dimitracopoulos, C.: Fragments of Peano’s arithmetic and the MRDP theorem. In: Logic and Algorithmic, no. 30 in Monographie de L’Enseignement Mathématique, pp. 187–206. Université de Genève (1982)Google Scholar
  5. 5.
    Kaye, R.: Diophantine induction. Ann. Pure Appl. Log. 46(1), 1–40 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kaye, R.: Hilbert’s tenth problem for weak theories of arithmetic. Ann. Pure Appl. Log. 61(1–2), 63–73 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory, Encyclopedia of Mathematics and Its Applications, vol. 60. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  8. 8.
    Manders, K.L., Adleman, L.M.: \(\mathit{NP}\)-complete decision problems for binary quadratics. J. Comput. Syst. Sci. 16(2), 168–184 (1978)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Otero, M.: On Diophantine equations solvable in models of open induction. J. Symb. Log. 55(2), 779–786 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    rainmaker: Decidability of diophantine equation in a theory. MathOverflow (2015). http://mathoverflow.net/q/194491
  11. 11.
    Shepherdson, J.C.: A nonstandard model for a free variable fragment of number theory. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 12(2), 79–86 (1964)MathSciNetMATHGoogle Scholar
  12. 12.
    van den Dries, L.: Which curves over \(\mathbf{Z}\) have points with coordinates in a discrete ordered ring? Trans. Am. Math. Soc. 264(1), 181–189 (1981)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wilkie, A.J.: Some results and problems on weak systems of arithmetic. In: Macintyre, A. (ed.) Logic Colloquium ’77, pp. 285–296. North-Holland, Amsterdam (1978)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic

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