Skip to main content

Part I: A sketch of Gentzen’s life and work

  • Chapter
  • First Online:
Saved from the Cellar

Abstract

Gerhard Gentzen died on August 4, 1945, in a prison in Prague. His fellow prisoners were professors of the local German university, and there are accounts of his last days and how he was, rendered weak by lack of food, still pondering over the consistency problem of analysis. After the war, some attempts were made to find any manuscripts he might have left behind; a mythical suitcase one letter reports he had been carrying around, filled with papers with a near-proof of the consistency of analysis. Nothing was found, though, in Prague. In Göttingen, instead, there were manuscripts that were preliminary studies for published work, by the account of Arnold Schmidt. He wrote in 1948 to Gentzen’s mother that the papers would be placed and kept together with Hilbert’s papers; yet again, nothing has been found.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Incidentally, I had the occasion to ask in 2008 Dr. Ludwig Bernays why Paul Bernays didn’t request the restitution of his position after the war: “Oh, uncle Paul would never have done such a thing!”

Bibliography for Parts I and II

  • Ackermann, W. (1940a) Zur Arbeit von G. Gentzen: Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Typewritten manuscript with the imprint “Logistisches Seminar der Universität Münster i./W. Prof. Scholz.”

    Google Scholar 

  • Allert, T. (2008) The Hitler Salute. Metropolitan Books.

    Google Scholar 

  • Bernays, P. (1935a) Quelques points essentiels de la métamathématique. L’Enseignement Mathématique, vol. 34, pp. 70–95.

    Google Scholar 

  • Bernays, P. (1965) Betrachtungen zum Sequenzen-kalkul. Contributions to Logic and Methodology in Honor of J. M. Bochenski, pp. 1–44, North-Holland.

    Google Scholar 

  • Bernays, P. (1970) On the original Gentzen consistency proof for number theory. In J. Myhill et al., eds, Intuitionism and Proof Theory, pp. 409–417, North-Holland.

    Google Scholar 

  • von Boguslawski, M. (2011) Proofs, Paradoxes, and Probabilities: The Logical Turn of Philosophy in Finland. Doctoral dissertation, University of Helsinki.

    Google Scholar 

  • Brouwer, L. (1924) Beweis, dass jede volle Funktion gleichmässig stetig ist. Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, vol. 27, pp. 189–193.

    Google Scholar 

  • Brouwer, L. (1924a) Bemerkungen zum Beweise der gleichmässigen Stetigkeit voller Funktionen. Ibid., pp. 644–646.

    Google Scholar 

  • Brouwer, L. (1928) Intuitionistische Betrachtungen über den Formalismus. Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 48–52.

    Google Scholar 

  • Cavaillès, J. (1938) Méthode axiomatique et formalisme. As republished in Oeuvres completés de Philosophie des sciences, Hermann, Paris 1994.

    Google Scholar 

  • Curry, H. (1939) A note on the reduction of Gentzen’s sequent calculus LJ. Bulletin of the American Mathematical Society, vol. 45, pp. 288–293.

    Article  MathSciNet  MATH  Google Scholar 

  • Curry, H. (1950) A Theory of Formal Deducibility. Notre Dame Mathematical Lectures no. 6.

    Google Scholar 

  • Curry, H. (1977) Foundations of Mathematical Logic. Dover reprint of the 1963 original edition.

    Google Scholar 

  • van Dalen, D. (2011) The Selected Correspondence of L.E.J. Brouwer. Springer.

    Book  MATH  Google Scholar 

  • Feferman, S. (1964) Systems of predicative analysis, The Journal of Symbolic Logic, vol. 29, pp. 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  • Gentzen, G. (1932) Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsysteme. Mathematische Annalen, vol. 107, pp. 329–250.

    Article  MATH  Google Scholar 

  • Gentzen, G. (1933) Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik. Submitted for publication March 15, 1933 but withdrawn, published in Archiv für mathematische Logik, vol. 16 (1974), pp. 119–132.

    Google Scholar 

  • Gentzen, G. (1934–35) Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, vol. 39, pp. 176–210 and 405–431.

    Google Scholar 

  • Gentzen, G. (1935) Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie. First printed in Archiv für mathematische Logik, vol. 16 (1974), pp. 97–118.

    Google Scholar 

  • Gentzen, G. (1936) Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, vol. 112, pp. 493–565.

    Article  MathSciNet  MATH  Google Scholar 

  • Gentzen, G. (1936b) Die Widerspruchsfreiheit der Stufenlogik. Mathematische Zeitschrift, vol. 41, pp. 357–366.

    Google Scholar 

  • Gentzen, G. (1938) Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4, pp. 19–44.

    MATH  Google Scholar 

  • Gentzen, G. (1943) Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Mathematische Annalen, vol. 120, pp. 140–161.

    Article  MathSciNet  MATH  Google Scholar 

  • Gentzen, G. (1954) Zusammenfassung von mehreren vollständigen Induktionen zu einer einzigen. Archiv für Mathematische Logik und Grundlagenforschung, vol. 5, pp. 81–83.

    MathSciNet  MATH  Google Scholar 

  • Gentzen, G. (1969) The Collected Papers of Gerhard Gentzen, ed. M. Szabo.

    Google Scholar 

  • Gentzen, G. (2008) The normalization of derivations. The Bulletin of Symbolic Logic, vol. 14, pp. 245–257.

    Google Scholar 

  • Glivenko, V. (1929) Sur quelques points de la logique de M. Brouwer. Academie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 15, pp. 183–188.

    Google Scholar 

  • Gödel, K. (1938) Lecture at Zilsel’s. In Gödel (1995), pp. 86–113.

    Google Scholar 

  • Gödel, K. (1986, 1995, 2003) Collected Works, vols. 1, 3, and 4. Oxford U. P.

    Google Scholar 

  • Goodstein, R.L. (1951) Constructive Formalism. Leicester U.P.

    MATH  Google Scholar 

  • Goodstein, R.L. (1958) On the nature of mathematical systems. Dialectica, vol. 12, pp. 296–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Harrop, R. (1960) Concerning formulas of the type A BC, A (Ex)B(x) in intuitionistic formal systems. The Journal of Symbolic Logic, vol. 25, pp. 27–32.

    Google Scholar 

  • Hempel, C. (2000) An intellectual autobiography. In Science, Explanation, and Rationality, ed. J. Fetzer, pp. 3–35.

    Google Scholar 

  • Herbrand, J. (1931) Sur la non-contradiction de l’arithmétique. Journal für die reine und angewandte Mathematik, vol. 166, pp. 1–8. English translation in Van Heijenoort.

    Google Scholar 

  • Hertz, P. (1923) Über Axiomensysteme für beliebige Satzsysteme. Teil II. Mathematische Annalen, vol. 89, pp. 76–102.

    Google Scholar 

  • Hertz, P. (1929) Über Axiomensysteme für beliebige Satzsysteme. Mathematische Annalen, vol. 101, pp. 457–514.

    Article  MathSciNet  MATH  Google Scholar 

  • Heyting, A. (1930) Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse, pp. 42–56.

    Google Scholar 

  • Heyting, A. (1946) On weakened quantification. The Journal of Symbolic Logic, vol. 11, pp. 119–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Jaśkowski, S. (1934) On the rules of supposition in formal logic, as reprinted in S. McCall, ed, Polish Logic 1920–1939, pp. 232–258, Oxford U. P. 1967.

    Google Scholar 

  • Kalmár, L. (1938) Über Gentzens Beweis für die Widerspruchsfreiheit der reinen Zahlentheorie. Manuscript Hs974:105 of the Bernays collection of the ETH, 21 pages.

    Google Scholar 

  • Kleene, S. (1945) On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, vol. 10, pp. 109–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S. (1952) Introduction to Metamathematics, North-Holland.

    MATH  Google Scholar 

  • Kleene, S. (1952a) Permutability of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society, vol. 10, pp. 1–26.

    Google Scholar 

  • Kreisel, G. (1976) Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt? Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 78, pp. 177–223.

    MathSciNet  MATH  Google Scholar 

  • Kreisel, G. (1987) Gödel’s excursions into intuitionistic logic. In P. Weingartner and L. Schmetterer, eds, Gödel Remembered, pp. 65–186, Bibliopolis, Naples.

    Google Scholar 

  • Mancosu, P. (1999) Between Berlin and Vienna: The immediate reception of Gödel’s incompleteness theorems. History and Philosophy of Logic vol. 20, pp. 33–45.

    Article  MathSciNet  MATH  Google Scholar 

  • Menzler-Trott, E. (2007) Logic’s Lost Genius: The Life of Gerhard Gentzen. American Mathematical Society.

    MATH  Google Scholar 

  • Moriconi, E. (2015) Early structural reasoning. Gentzen 1932. The Review of Symbolic Logic, vol. 8, pp. 662–679.

    Google Scholar 

  • Negri, S. and J. von Plato (2001) Structural Proof Theory. Cambridge.

    Google Scholar 

  • Negri, S. and J. von Plato (2016) Cut elimination in sequent calculi with implicit contraction, with a conjecture on the origin of Gentzen’s altitude line construction. In D. Probst and P. Schuster, eds, Concepts of Proof in Mathematics, Philosophy, and Computer Science, pp. 269–290. De Gruyter.

    Google Scholar 

  • von Neumann, J. (1927) Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, vol. 26, pp. 1–46.

    Article  MathSciNet  MATH  Google Scholar 

  • Pinl, M. (1969–76) Kollegen in einer dunklen Zeit. Jahresbericht der Deutschen Mathematiker–Vereinigung, vol. 71 (1969), pp. 167–228. Part II, vol. 72 (1971), pp. 165–189. Part III, vol. 73 (1972), pp. 153–208. Part IV, vol. 75 (1974), pp. 166–208. Part V, vol. 77 (1976), pp. 161–164.

    Google Scholar 

  • von Plato, J. (2001) A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic, vol. 40, pp. 9–18.

    Article  MathSciNet  MATH  Google Scholar 

  • von Plato, J. (2009) Gentzen’s logic. Handbook of the History of Logic, vol. 5, pp. 667–721.

    MathSciNet  Google Scholar 

  • von Plato, J. (2012) Gentzen’s proof systems: byproducts in a program of genius. The Bulletin of Symbolic Logic, vol. 18, pp. 313–367.

    Article  MathSciNet  MATH  Google Scholar 

  • von Plato, J. (2014) From axiomatic logic to natural deduction. Studia Logica, vol. 102, pp. 1167–1184.

    Article  MathSciNet  MATH  Google Scholar 

  • von Plato, J. (2015) From Hauptsatz to Hilfssatz. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest for Consistency, pp. 89–126, Springer.

    Google Scholar 

  • Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study. Almqvist & Wicksell.

    Google Scholar 

  • Rathjen, M. (1995) Recent advances in ordinal analysis. The Bulletin of Symbolic Logic, vol. 1, pp. 468–485.

    Article  MathSciNet  MATH  Google Scholar 

  • Rathjen, M. (1999) The realm of ordinal analysis. In S. Wainer and J. Truss, eds, Sets and Proofs, pp. 219–279, Cambridge University Press.

    Google Scholar 

  • Schmidt, H. (1960) Mathematische Gesetze der Logik. Springer.

    Book  MATH  Google Scholar 

  • Scholz, H. (1940) Urteil über die Habilitationsschrift Gentzen “Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie.” Typewritten manuscript with the date 24 February 1940 and the imprint “Logistisches Seminar der Universität Münster i./W. Prof. Scholz.”

    Google Scholar 

  • Schroeder-Heister, P. (2002) Resolution and the origins of structural reasoning: early proof-theoretic ideas of Hertz and Gentzen. The Bulletin of Symbolic Logic, vol. 8, pp. 246–265.

    Article  MathSciNet  MATH  Google Scholar 

  • Siders, A. (2015) A direct Gentzen-style consistency proof for Heyting arithmetic. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest of Consistency, pp. 177–211.

    Google Scholar 

  • Siders, A. and J. von Plato (2015) Bar induction in the proof of termination of Gentzen’s reduction procedure. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest for Consistency, pp. 127–130, Springer.

    Google Scholar 

  • Skolem, T. (1937) Über die Zurückführbarkeit einiger durch Rekursionen definierter Relationen auf “arithmetische.” As reprinted in Skolem’s Selected Works in Logic, pp. 425–440, 1970.

    Google Scholar 

  • Smorynski, C. (2007) Gentzen and geometry. In Menzler-Trott (2007), pp. 281–288.

    Google Scholar 

  • Tait, W. (2010) Review of Menzler-Trott 2007. The Bulletin of Symbolic Logic, vol. 16, pp. 270–275.

    Article  MathSciNet  Google Scholar 

  • Troelstra, A. (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344, Springer.

    Google Scholar 

  • Troelstra, A. and D. van Dalen (1988) Constructivism in Mathematics. 2 vols., North-Holland.

    Google Scholar 

  • Troelstra, A. and H. Schwichtenberg (1996) Basic Proof Theory. Second edition 2000, Cambridge U. P.

    Google Scholar 

  • Wajsberg, M. (1938) Untersuchungen über den Aussagenkalkül von A. Heyting. Wiadomsci matematyczne, vol. 46, pp. 45–101.

    Google Scholar 

  • Wolfson, H. (1976) The Philosophy of the Kalam. Harvard U.P.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

von Plato, J. (2017). Part I: A sketch of Gentzen’s life and work. In: Saved from the Cellar. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-42120-9_1

Download citation

Publish with us

Policies and ethics