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Recursive Functions and Arithmetization of Theories

  • Shashi Mohan Srivastava
Chapter
Part of the Universitext book series (UTX)

Abstract

Let us ask the following question: is there an algorithm to decide whether an arbitrary sentence of the language of N is true in \(\mathbb{N}\)? Many important mathematical problems are of this type.

References

  1. 1.
    Ax, J.: The elementary theory of finite fields. Ann. Math. 88, 103–115 (1968)Google Scholar
  2. 2.
    Bochnak, J., Coste, M., Roy, M-F.: Real Algebraic Geometry, vol. 36, A Series of Modern Surveys in Mathematics. Springer, New York (1998)Google Scholar
  3. 3.
    Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. North-Holland, London (1990)Google Scholar
  4. 4.
    Flath, D., Wagon, S.: How to pick out integers in the rationals: An application of number theory to logic. Am. Math. Mon. 98, 812–823 (1991)Google Scholar
  5. 5.
    Hinman, P.: Fundamentals of Mathematical Logic. A. K. Peters (2005)Google Scholar
  6. 6.
    Hofstadter, D.R.: Gödel, Escher, Bach: An Eternal Golden Braid. Vintage Books, New York (1989)Google Scholar
  7. 7.
    Hrushovski, E.: The Mordell–Lang conjecture for function fields. J. Am. Math. Soc. 9(3), 667–690 (1996)Google Scholar
  8. 8.
    Jech, T.: Set Theory, Springer Monographs in Mathematics, 3rd edn. Springer, New York (2002)Google Scholar
  9. 9.
    Kunen, K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)Google Scholar
  10. 10.
    Lang, S.: Algebra, 3rd edn. Addison-Wesley (1999)Google Scholar
  11. 11.
    Marker, D.: Model Theory: An Introduction, GTM 217. Springer, New York (2002)Google Scholar
  12. 12.
    Rogers, H.J.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)Google Scholar
  13. 13.
    Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1990)Google Scholar
  14. 14.
    Pila, J.: O-minimality and André-Oort conjecture for C n. Ann. Math. (2) 172(3), 1779–1840 (2011)Google Scholar
  15. 15.
    Pila, J., Zannier, U.: Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei(9). Mat. Appl. 19(2), 149–162 (2008)Google Scholar
  16. 16.
    Shoenfield, J.R.: Mathematical Logic. A. K. Peters (2001)Google Scholar
  17. 17.
    Srivastava, S.M.: A Course on Borel Sets, GTM 180. Springer, New York (1998)Google Scholar
  18. 18.
    Swan, R.G.: Tarski’s principle and the elimination of quantifiers (preprint)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Shashi Mohan Srivastava
    • 1
  1. 1.Indian Statistical InstituteKolkataIndia

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