Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T16:42:36.171Z Has data issue: false hasContentIssue false
  • English
  • Français

KURT GÖDEL’S FIRST STEPS IN LOGIC: FORMAL PROOFS IN ARITHMETIC AND SET THEORY THROUGH A SYSTEM OF NATURAL DEDUCTION

Published online by Cambridge University Press:  25 October 2018

JAN VON PLATO
JAN VON PLATO*
Affiliation:
UNIVERSITY OF HELSINKI, FINLAND 00014 HELSINKI, FINLANDE-mail: jan.vonplato@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines.

Keywords

01A60030303F03Gödelformal proofnatural deduction
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 3 , September 2018 , pp. 319 - 335
Copyright
Copyright © The Association for Symbolic Logic 2018 

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

Adzic, M. and Dosen, K. (2016) Gödel’s Notre Dame course. The Bulletin of Symbolic Logic, vol. 22, pp. 469–481.CrossRefGoogle Scholar
Carnap, R. (1929) Abriss der Logistik. Springer.CrossRefGoogle Scholar
Carnap, R. (2000) Untersuchungen zur allgemeinen Axiomatik. Bonk, T. and Mosterin, J., eds, Wissenschaftliche Buchgesellschaft.Google Scholar
Dawson, J. (1997) Logical Dilemmas: The Life and Work of Kurt Gödel. A. K. Peters.Google Scholar
Dosen, K. and Adzic, M. (2018) Gödel’s natural deduction. Studia Logica, vol. 106, pp. 397–415.CrossRefGoogle Scholar
Gentzen, G. (1935) Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie. First printed in Archiv für mathematische Logik, vol. 16 (1974), pp. 97–118.CrossRefGoogle Scholar
Gentzen, G. (2017) Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics. With an introduction and translation by von Plato, Jan. Studies in the History of Mathematics and Physical Sciences, Springer.Google Scholar
Gödel, K. (1929) Über die Vollständigkeit des Logikkalküls. Doctoral dissertation printed in Gödel (1986), pp. 60–101.Google Scholar
Gödel, K. (1930) Vorlesung über Vollständigkeit des Funktionenkalküls. First printed in Gödel (1995), pp. 16–29.Google Scholar
Gödel, K. (1986) Collected Works, vol. 1. Oxford U. P.Google Scholar
Gödel, K. (1995) Collected Works, vol. 3. Oxford U. P.Google Scholar
Gödel, K. (2017) Logic Lectures: Gödel’s Basic Logic Course at Notre Dame . Adzic, M. and Dosen, K., eds, Logic Society, Belgrade.Google Scholar
Goldfarb, W. (2005) On Gödel’s way in: the influence of Carnap. The Bulletin of Symbolic Logic. vol. 11, pp. 185–193.CrossRefGoogle Scholar
van Heijenoort, J., ed, (1967) From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Harvard University Press.Google Scholar
Hilbert, D. (2013) David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Edited by Ewald, W. and Sieg, W.. Springer.Google Scholar
Hilbert, D. and Ackermann, W. (1928) Grundzüge der theoretischen Logik. Springer.Google Scholar
Jaśkowski, S. (1934) On the rules of supposition in formal logic. As reprinted in McCall, S., ed, Polish Logic 1920–1939, pp. 232–258, Oxford U. P. 1967.Google Scholar
Kreisel, G. (1987) Gödel’s excursions into intuitionistic logic. In Weingartner, P. and Schmetterer, L., eds, Gödel Remembered, pp. 65–186, Bibliopolis, Naples.Google Scholar
Peano, G. (1889) Arithmetices Principia, Nova Methodo Exposita. Partial English tr. in Van Heijenoort.Google Scholar
von Plato, J. (2007) In the shadows of the Löwenheim-Skolem theorem: early combinatorial analyses of mathematical proofs. The Bulletin of Symbolic Logic, vol. 13, pp. 189–225.CrossRefGoogle Scholar
von Plato, J. (2009) Gentzen’s original proof of the consistency of arithmetic revisited. In Primiero, G. and Rahman, S. (eds.) Acts of Knowledge - History, Philosophy and Logic, pp. 151–171, College Publications.Google Scholar
von Plato, J. (2017) The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age. Princeton University Press.Google Scholar
von Plato, J. (2017a) From Gentzen to Jaskowski and back: algorithmic translation of derivations between the two main systems of natural deduction. Bulletin of the Section of Logic, vol. 46, pp. 1–9.CrossRefGoogle Scholar
Russell, B. (1919) Introduction to Mathematical Philosophy. Allen and Unwin.Google Scholar
Skolem, T. (1920) Logisch-kombinatorische Untersuchungen über die Erfüllbareit oder Beweisbarkeit mathematischer Sätze, nebst einem Theoreme über dichte Mengen. As reprinted in Skolem 1970, pp. 103–136.Google Scholar
Skolem, T. (1970) Selected Works in Logic. ed. Fenstad, J. E., Universitetsforlaget, Oslo.Google Scholar
Vailati, G. (1904) A proposito d’un teorema di Teeteto e di una dimostrazione di Euclide. Rivista di filosofia e scienze affine, vol. 6. Consulted from reprint in Vailati’s Scritti, pp. 516–527, Barth, Leipzig 1911.Google Scholar
Whitehead, A. and Russell, B. (1910–13) Principia Mathematica, vols. I–III. Cambridge. Second edition 1927.Google Scholar