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VON NEUMANN’S CONSISTENCY PROOF

Published online by Cambridge University Press:  14 July 2016

LUCA BELLOTTI
LUCA BELLOTTI*
Affiliation:
Department CFS (Philosophy), University of Pisa
*
*DEPARTMENT CFS (PHILOSOPHY) UNIVERSITY OF PISA VIA PAOLI 15, 56126 PISA, ITALY E-mail: luca.bellotti@unipi.it
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Abstract

We consider the consistency proof for a weak fragment of arithmetic published by von Neumann in 1927. This proof is rather neglected in the literature on the history of consistency proofs in the Hilbert school. We explain von Neumann’s proof and argue that it fills a gap between Hilbert’s consistency proofs for the so-called elementary calculus of free variables with a successor and a predecessor function and Ackermann’s consistency proof for second-order primitive recursive arithmetic. In particular, von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 3 , September 2016 , pp. 429 - 455
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Ackermann, W. (1925). Begründung des ‘Tertium non datur’ mittels der Hilbertschen Theorie des Widerspruchsfreiheits. Mathematische Annalen, 93, 1–36.Google Scholar
Ackermann, W. (1940). Zur Widerspruchsfreiheit der Zahlentheorie. Mathematische Annalen, 117, 162–194.Google Scholar
Börger, E., Grädel, E., & Gurevich, Y. (2001). The classical decision problem. Springer-Verlag.Google Scholar
Buchholz, W., Mints, G., & Tupailo, S. (1996). Epsilon substitution method for elementary analysis. Archive for Mathematical Logic, 35, 103–130.Google Scholar
Ferrante, J. & Rackoff, C. (1979). The computational complexity of logical theories. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Gentzen, G. (1934). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 176–210, 405–431.Google Scholar
Hajek, P. & Pudlák, P. (1998). The metamathematics of first-order arithmetic. Berlin: Springer-Verlag.Google Scholar
Herbrand, J. (1931). Sur la non-contradiction de l’arithmétique. Journal für die reine und angewandte Mathematik, 166, 1–8.Google Scholar
Heyting, A. (1955). Fondements des mathématiques. Paris: Gauthier-Villars.Google Scholar
Hilbert, D. (1923). Die logischen Grundlagen der Mathematik. Mathematische Annalen, 88, 151–165.Google Scholar
Hilbert, D. (1928). Probleme der Grundlegung der Mathematik. In: Atti del Congresso internazionale dei matematici, 135–141. Bologna: Zanichelli.Google Scholar
Hilbert, D. (1929). Probleme der Grundlegung der Mathematik. Mathematische Annalen, 102, 1–9.Google Scholar
Hilbert, D. (2013). David Hilbert’s lectures on the foundations of arithmetic and logic, 1917–1933, Ewald, W. and Sieg, W. (eds.). Berlin: Springer-Verlag.Google Scholar
Hilbert, D. & Bernays, P. (1923). Logische Grundlagen der Mathematik. Winter-Semester 1922–23. Lecture notes by Kneser, H.. In: Hilbert (2013), 599–635.Google Scholar
Hilbert, D. & Bernays, P. (1923a). Logische Grundlagen der Mathematik. Winter-Semester 1922–23. Lecture notes by Bernays, P., with handwritten notes by Hilbert, D.. In: Hilbert (2013), 528–549.Google Scholar
Hilbert, D. & Bernays, P. (1939). Grundlagen der Mathematik, Vol. 2. Berlin: Springer-Verlag.Google Scholar
Leśniewski, S. (1929). Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae, 14, 1–81.Google Scholar
Mancosu, P., Zach, R., & Badesa, C. (2009). The development of mathematical logic from Russell to Tarski: 1900–1935. In: Haaparanta, L. (ed.), The development of modern logic, 318–470. Oxford: Oxford University Press.Google Scholar
Moser, G. (2006). Ackermann’s substitution method (remixed). Annals of Pure and Applied Logic, 142, 1–18.Google Scholar
Moser, G. & Zach, R. (2006). The Epsilon calculus and Herbrand complexity. Studia Logica, 82, 133–155.Google Scholar
Shoenfield, J. R. (1967). Mathematical Logic. Reading, Massachusetts: Addison-Wesley.Google Scholar
Tait, W. W. (1965). The substitution method. Journal of Symbolic Logic, 30, 175–192.Google Scholar
Ulam, S. (1958). John von Neumann (1903–1957). Bulletin of the American Mathematical Society, 64, 1–49.Google Scholar
Van Heijenoort, J. (ed.) (1967). From Frege to Gödel. Cambridge, Massachusetts: Harvard University Press.Google Scholar
Von Neumann, J. (1927). Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, 26, 1–46.CrossRefGoogle Scholar
Von Neumann, J. (1931). Bemerkungen zu den Ausführungen von Herrn S. Leśniewski über meine Arbeit ‘Zur Hilbertschen Beweistheorie’. Fundamenta Mathematicae, 17, 331–334.Google Scholar
Von Neumann, J. (1961). Collected works, Vol. 1. Oxford: Pergamon Press.Google Scholar
Wang, H. (1963). A survey of mathematical logic. Amsterdam: North Holland.Google Scholar
Zach, R. (2003). The practice of finitism: Epsilon calculus and consistency proofs in Hilbert’s Program. Synthese, 137, 211–259.Google Scholar