Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-19T18:54:37.129Z Has data issue: false hasContentIssue false
  • English
  • Français

NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

Published online by Cambridge University Press:  07 February 2019

MARIA HÄMEEN-ANTTILA
MARIA HÄMEEN-ANTTILA*
Affiliation:
DEPARTMENT OF PHILOSOPHY, HISTORY AND ART STUDIES UNIVERSITY OF HELSINKI PL 24, 00014HELSINKI, FINLAND E-mail: maria.hameen-anttila@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

Keywords

03-0301A60Gerhard GentzenPaul Bernaysfinitismconstructive ordinals
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 25 , Issue 1 , March 2019 , pp. 101 - 124
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ackermann, H. R., Aus dem Briefwechsel Wilhelm Ackermanns. History and Philosophy of Logic, vol. 4 (1983), no. 1–2, pp. 181–202.CrossRefGoogle Scholar
Ackermann, W., Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen, vol. 93 (1925), pp. 1–36.CrossRefGoogle Scholar
Ackermann, W., Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen, vol. 99 (1928), no. 1, pp. 118–133.CrossRefGoogle Scholar
Ackermann, W., Zur Widerspruchsfreiheit der Zahlentheorie. Mathematische Annalen, vol. 117 (1940), pp. 162–194.CrossRefGoogle Scholar
Bellotti, L., Decoding Gentzen’s notation. History and Philosophy of Logic, vol. 39 (2018), no. 3, pp. 270–288.CrossRefGoogle Scholar
Bernays, P., Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für Deutsche Philosophie, vol. 4 (1930), pp. 326–367.Google Scholar
Bernays, P., Hilberts Untersuchungen über die Grundlagen der Arithmetik, David Hilbert – Gesammelte Abhandlungen (Hilbert, D., editor), Springer-Verlag, Berlin, 1935, pp. 196–217.Google Scholar
Bernays, P., Quelques points essentielles de la métamathématique. L’Enseignement Mathématique, vol. 34 (1935), pp. 70–94.Google Scholar
Bernays, P., Review of die Widerspruchsfreiheit der reinen Zahlentheorie by Gerhard Gentzen. The Journal for Symbolic Logic, vol. 1 (1936), no. 2, p. 75.CrossRefGoogle Scholar
Bernays, P., Sur les questions méthodologiques actuelles de la théorie hilbertienne de la démonstration, Les entretiens de Zürich sur le fondements et la méthode des sciences mathématiques, 6-9 décembre 1938 (Gonseth, F., editor), Leemann, Zürich, 1941, pp. 144–152.Google Scholar
Calude, C., Marcus, S.,and Tevy, I., The first example of a recursive function which is not primitive recursive. Historia Mathematica, vol. 6 (1979), pp. 380–384.CrossRefGoogle Scholar
Church, A., The constructive second number class. Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 224–232.CrossRefGoogle Scholar
Ewald, W. and Sieg, W. (eds.), David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933, Springer-Verlag, Heidelberg, 2013.CrossRefGoogle Scholar
Fraenkel, A., Review of Gabriel Sudan: Sur le nombre transfini ωω. Jahrbuch über die Fortschritte der Mathematik, vol. 53 (1927), p. 171.Google Scholar
Gentzen, G., Der Unendlichkeitsbegriff in der Mathematik. Semester-Berichte Münster, vol. W/S 1936/37 (1936), pp. 65–80.Google Scholar
Gentzen, G., Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, vol. 112 (1936), pp. 493–565.CrossRefGoogle Scholar
Gentzen, G., Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik. Actualites scientifiques et industrielles, vol. 535 (1937), pp. 201–205.Google Scholar
Gentzen, G., Die gegenwärtige Lage in der mathematischen Grundlagenforschung. Deutsche Mathematik, vol. 3 (1938), pp. 255–268.Google Scholar
Gentzen, G., Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Mathematische Annalen, vol. 119 (1943), pp. 140–161.CrossRefGoogle Scholar
Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, vol. 12 (1958), pp. 280–287.CrossRefGoogle Scholar
Hilbert, D., Über das Unendliche. Mathematische Annalen, vol. 95 (1926), pp. 161–190.CrossRefGoogle Scholar
Hilbert, D., Die grundlagen der mathematik. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 6 (1928), no. 1, pp. 65–85.CrossRefGoogle Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, vol. 40, Springer-Verlag, Berlin, 1934.Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, vol. 50, Springer-Verlag, Berlin, 1939.Google Scholar
Horská, A., Where is the Gödel-Point Hiding: Gentzen’s Consistency Proof of 1936 and his Representation of Constructive Ordinals, Springer International Publishing, Cham, 2014.CrossRefGoogle Scholar
Kleene, S. C., On notation for ordinal numbers. The Journal for Symbolic Logic, vol. 3 (1938), pp. 150–155.CrossRefGoogle Scholar
Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics, vol. 77 (1955), no. 3, pp. 405–428.CrossRefGoogle Scholar
Kreisel, G., Gödel’s excursions into intuitionistic logic, Gödel Remembered (Gödel, R., Weingartner, P., and Schmetterer, L., editors), Bibliopolis, Napoli, 1987, pp. 65–186.Google Scholar
Mancosu, P., The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900–1940, Oxford University Press, 2010.CrossRefGoogle Scholar
Menzler-Trott, E., Logic’s Lost Genius, American Mathematical Society, Providence, RI, 2007.Google Scholar
Moore, G. H., Hilbert on the infinite: The role of set theory in the evolution of Hilbert’s thought. Historia Mathematica, vol. 29 (2002), no. 1, pp. 40–64.CrossRefGoogle Scholar
Péter, R., Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion. Mathematische Annalen, vol. 110 (1935), no. 1, pp. 612–632.CrossRefGoogle Scholar
Péter, R., Rekursive Funktionen, Akadémiai Kiadó, Budapest, 1951.Google Scholar
Sudan, G., Über die geordneten Mengen. Buletinul de ştiinţe matematice pure şi aplicate, vol. 28 (1925), pp. 3–23.Google Scholar
Sudan, G., Sur le nombre transfini ωω. Bulletin Mathematique de la Société Roumaine des Sciences, vol. 30 (1927), pp. 11–30.Google Scholar
Szabo, M. E. (ed.), Collected Papers of Gerhard Gentzen, Studies in Logic and the Foundations of Mathematics, North-Holland, 1969.Google Scholar
Tait, W. W., Gödel on intuition and on Hilbert’s finitism, Kurt Gödel: Essays for his Centennial(Feferman, S., Parsons, C., and Simpson, S. G., editors), Lecture Notes in Logic, Cambridge University Press, Cambridge, 2010, pp. 88–108.CrossRefGoogle Scholar
von Neumann, J., Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46.CrossRefGoogle Scholar
von Plato, J., Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics, Springer, Cham, 2017.CrossRefGoogle Scholar
Zach, R., Numbers and functions in Hilbert’s finitism. Taiwanese Journal for Philosophy and History of Science, vol. 10 (1998), pp. 33–60.Google Scholar
Zach, R., The practice of finitism: Epsilon calculus and consistency proofs in Hilbert’s program. Synthese, vol. 137 (2003), no. 1, pp. 211–259.CrossRefGoogle Scholar