The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof

  • Yvon Gauthier
Part of the Studies in Universal Logic book series (SUL)


The question of the consistency or non-contradiction of arithmetic is a philosophical question, that is the certainty of a mathematical theory and it has become a logical problem requiring a mathematical solution. It is Hilbert who has put the original question and who has attempted a first answer. Having demonstrated in 1899 the consistency of elementary geometry on the basis of a consistent arithmetic of real numbers, he turned to the question of that arithmetic (the second problem of his 1900 list) which included besides the axiom of elementary arithmetic an axiom of continuity, i.e. the Archimedean axiom with syntactic completeness. Hilbert introduces functionals, that is second-order functions or functions of functions, in order the use induction over the ordinals. Transfinite induction proved successful in the hands of Gentzen, and thereafter Gödel’s second incompleteness results which aimed to show that Peano’s arithmetic cannot contain its own consistency.


Homogeneous Polynomial Finite Type Double Negation General Arithmetic Consistency Proof 
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  1. E. Bishop (1967): Foundations of Constructive Analysis. New York, McGraw-Hill.zbMATHGoogle Scholar
  2. A. Davenport (1968): The Higher Arithmetic. Chap. VI. Hutchinson University Library, London.Google Scholar
  3. H. M. Edwards (1987a): Divisor Theory. Birkhaüser, Basel.Google Scholar
  4. H. M. Edwards (1987b): An appreciation of Kronecker. The Mathematical Intelligencer, 9(1):28–35.zbMATHMathSciNetCrossRefGoogle Scholar
  5. H. M. Edwards, O. Neumann and W. Purkert (1982): Dedekinds “Bunte Bemerkungen” zu Kroneckers “Grundzüge”. Archive for History of Exact Sciences, 27(1):49–85.zbMATHMathSciNetGoogle Scholar
  6. Y. Gauthier (1994): Hilbert and the Internal Logic of Mathematics. Synthese, 101:1–14.zbMATHMathSciNetGoogle Scholar
  7. Y. Gauthier (2013a): Kronecker in Contempory Mathematics. General Arithmetic as a Foundational Programme. Reports on Mathematical Logic, 48:37–65.Google Scholar
  8. K. Gödel (1990): Collected Works. S. Feferman, ed., volume II, 217–251. Oxford University Press, New York/Oxford.Google Scholar
  9. J. Herbrand (1968): Écrits logiques. J. van Heijenoort, ed. PUF, Paris.Google Scholar
  10. D. Hilbert (1904): Über die Grundlagen der Logik und Mathematik, 174–185. in: A. Krazer (Hrsg.) Verhandlungen des III. Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904. Leipzig, Teubner.Google Scholar
  11. D. Hilbert (1926): Über das Unendliche. Mathematische Annalen, 95: 161–190. trad. par A. Weil sous le titre “Sur l’infini”, Acta Mathematica 48: 91–122.Google Scholar
  12. D. Hilbert and P. Bernays (1968–1970): Grundlagen der Mathematik I et II. Springer, Berlin, 2 édition.Google Scholar
  13. K. Ireland and M. Rosen (1980): A Classical Introduction to Modern Number Theory. Springer, New York/Heidelberg/Berlin.Google Scholar
  14. U. Kohlenbach (1998): Arithmetising Proofs in Analysis. In Lascar D. Larrazabal, J.M. and G. Mints, editors. Logig Colloquium 96, volume 12 of Springer Lecture Notes in Logic.Google Scholar
  15. G. Kreisel (1961): Set-theoretic problems suggested by the notion of potential totality, 103–140. in: Infinitistic Methods. Pergamon Press, Oxford.Google Scholar
  16. L. Kronecker (1889): Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Werke, II:47–208.Google Scholar
  17. L. Kronecker (1968a): Über Systeme von Funktionen mehrerer Variablen. In K. Hensel, editor, Werke, volume I. Teubner, Leipzig.Google Scholar
  18. L. Kronecker (1968b): Werke, éd. K. Hensel, volume III, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, 245–387. Teubner, Leipzig.Google Scholar
  19. E. Nelson (1987a): Predicative Arithmetic. Number 32 of Mathematical Notes. Princeton University Press, Princeton, N.J.Google Scholar
  20. E. Nelson (1987b): A Radically Elementary Probability Theory. Princeton University Press, Princeton, N. J.Google Scholar
  21. B. A. Trakhtenbrot (1950): Nevozmozhnosty Algorifma dla Problemy Razreximosti Konechnyh Klassah, (The impossibility of an algorithm for the decision problem in finite classes). Dokl. Akad. Nauk, SSSR, 70:569–572.Google Scholar
  22. H. S. Vandiver (1936): Constructive Derivation of the Decomposition Field of a Polynomial. Annals of Mathematics, 37(1):1–6.MathSciNetCrossRefGoogle Scholar
  23. A. Weil (1979a): Oeuvres scientifiques. Collected Paper, Vol. I, Springer-Verlag, New York.Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontrealMontrealCanada

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