A Short Note on Spector’s Proof of Consistency of Analysis

  • Fernando Ferreira
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


In 1962, Clifford Spector gave a consistency proof of analysis using so-called bar recursors. His paper extends an interpretation of arithmetic given by Kurt Gödel in 1958. Spector’s proof relies crucially on the interpretation of the so-called numerical double negation shift principle. The argument for the interpretation is ad hoc. On the other hand, William Howard gave in 1968 a very natural interpretation of bar induction by bar recursion. We show directly that, within the framework of Gödel’s interpretation, numerical double negation shift is a consequence of bar induction.


Intuitionistic Logic Proof Theory Peano Arithmetic Consistency Proof Characteristic Principle 
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  1. 1.
    Avigad, J., Feferman, S.: Gödel’s functional (“Dialectica”) interpretation. In: Buss, S.R. (ed.) Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 337–405. North Holland, Amsterdam (1998)CrossRefGoogle Scholar
  2. 2.
    Brouwer, L.E.J.: Über Definitionsbereiche von funktionen. Mathematische Annalen 93, 60–75 (1927); English translation in [13], pp. 457–463Google Scholar
  3. 3.
    Ferreira, F.: A most artistic package of a jumble of ideas. Dialectica 62, 205–222 (2008); Special Issue: Gödel’s dialectica Interpretation. Guest editor: Thomas StrahmMathSciNetCrossRefGoogle Scholar
  4. 4.
    Gödel, K.: Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12, 280–287 (1958); Reprinted with an English translation in [5], pp. 240–251MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gödel, K., Feferman, S., et al. (eds.): Collected Works, vol. II. Oxford University Press, Oxford (1990)zbMATHGoogle Scholar
  6. 6.
    Howard, W.A.: Functional interpretation of bar induction by bar recursion. Compositio Mathematica 20, 107–124 (1968)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin (2008)zbMATHGoogle Scholar
  8. 8.
    Kreisel, G.: Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 101–128. North Holland, Amsterdam (1959)Google Scholar
  9. 9.
    Oliva, P.: Understanding and Using Spector’s Bar Recursive Interpretation of Classical Analysis. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 423–434. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Dekker, F.D.E. (ed.) Recursive Function Theory: Proceedings of Symposia in Pure Mathematics, vol. 5, pp. 1–27. American Mathematical Society, Providence (1962)Google Scholar
  11. 11.
    Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. Springer, Berlin (1973)zbMATHGoogle Scholar
  12. 12.
    van Atten, M.: On Brouwer. Wadsworth (2004)Google Scholar
  13. 13.
    van Heijenoort, J. (ed.): From Frege to Gödel. Harvard University Press (1967)Google Scholar
  14. 14.
    Yasugi, M.: Intuitionistic analysis and Gödel’s interpretation. Journal of the Mathematical Society of Japan 15, 101–112 (1963)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fernando Ferreira
    • 1
  1. 1.Universidade de LisboaLisboaPortugal

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