Abstract
We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the \(\omega \)-rule.
“Ihr Vorschlag, die Beweismittel nicht finit’, sondern konstruktiv’
zu nennen, hat wie eine Art Erlösung auf mich gewirkt.
Paul Lorenzen to Paul Bernays, 1947
Download chapter PDF
References
Bernays, P. (1964). On platonism in mathematics. Philosophy of mathematics, edited by Paul Benacerraf and Hilary Putnam, pages 274–286. Prentice-Hall. Lecture delivered June 18, 1934, in the cycle of Conferences internationales des sciences mathematiques organized by the University of Geneva, in the series on Mathematical Logic. Translated from the French by C. D. Parsons from L’enseignement mathematique, 1st ser., vol. 34 (1935), pp. 52–69.
Carnap, R. (1935). Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik. Monatshefte für Mathematik und Physik, 42, 163–190.
Coquand, T. (2020). Lorenzen and constructive mathematics (pp. 45–59).
Coquand, T., Lombardi, H., & Neuwirth, S. (2020). Regular entailment relations (pp. 101–122).
Coquand, T., & Stefan N. (2017). An introduction to Lorenzen’s Algebraic and logistic investigations on free lattices (1951). arXiv: 1711.06139v1 [math.LO].
Coquand, T., & Stefan N. (2020). Lorenzen’s proof of consistency for elementary number theory. History and Philosophy of Science. https://doi.org/10.1080/01445340.2020.1752034.
Feferman, S. (1986). Introductory note to [Gödel 1931c]. In S. Feferman et al. (Eds.) Kurt Gödel: Collected works, i: Publications 1929–1936 (pp. 208–213). Oxford: Oxford University Press.
Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112, 493–565.
Gentzen, G. (1969). Collected works, Edited by M. E. Szabo, North-Holland.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme i. Monatshefte für Mathematik und Physik, 38, 173–198.
Heinzmann, G. (2020). Operation and predicativity: Lorenzen’s approach to arithmetic (pp. 9–20).
Hilbert, D. (1926). Über das Unendliche. Mathematische Annalen, 95, 161–190.
Hilbert, D. (1928). Probleme der Grundlegung der Mathematik. In Atti del Congresso Internazionale dei Matematici (Bologna, 3–19 settembre 1928). Bologna: Nicola Zanichelli.
Hilbert, D. (1931a). Beweis des Tertium non datur. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (pp. 120–125). Talk given on July 17, 1931 in Göttingen.
Hilbert, D. (1931b). Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen104(1), 485–494. Talk given in December 1930 in Hamburg.
Hilbert, D. (1967). On the infinite. In van Heijenoort, J. (Ed.) From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 367–392). Harvard University Press. English translation of Hilbert 1926.
Hilbert, D., & Paul Bernays. (1939). Grundlagen der Mathematik ii. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 50. 2nd edn. 1970. Berlin: Springer.
Kahle, R. (2015). Gentzen’s theorem in context. In K. Reinhard & R. Michael (Eds.) Gentzen’s centenary: The quest for consistency (pp. 3–24). Berlin: Springer.
Kahle, R. (2020). Sehr geehrter Herr Professor!’ Proof theory in 1949 in a letter from Schütte to Bernays. In R. Kahle & M. Rathjen (Eds.) The legacy of Kurt Schütte (pp. 3–19). Berlin: Springer.
Lorenz, K. (2020). Paul Lorenzens Weg von der Mathematik zur Philosophie: Persönliche Erinnerungen (pp. 1–8).
Lorenzen, P. (1948). Grundlagen der Mathematik. In W. Süss (Ed.) Reine Mathematik, Naturforschung und Medizin in Deutschland 1939–1946 (für Deutschland bestimmte Ausgabe der fiat Review of German Science), Band 1, Teil 1 (pp. 11–22). Dietrich’sche Verlagsbuchhandlung.
Lorenzen, P. (1951a). Algebraische und logistische Untersuchungen über freie Verbände. Journal Symbolic Logic, 16, 81–106.
Lorenzen, P. (1951b). Die Widerspruchsfreiheit der klassischen Analysis. Mathematische Zeitschrift, 54, 1–24.
Lorenzen, P. (1962). Metamathematik. Volume 25 of BI Hochschultaschenbücher. Bibliographisches Institut.
Lorenzen, P. (1965). Differential und Integral: Eine konstruktive Einführung in die klassische Analysis. Akademische Verlagsgesellschaft.
Lorenzen, P. (1968). Constructive mathematics as a philosophical problem. Compositio Mathematica20: 133–142. Also published (with identical pagination) in Logic and foundations of mathematics, edited by D. van Dalen, J. G. Dijkman, S. C. Kleene, & A. S. Troelstra. Alphen aan den Rijn: Wolters-Noordhoff Publishing.
Lorenzen, P. (2017). Algebraic and logistic investigations on free lattices. arXiv: 1710.08138. Translation by Stefan Neuwirth of Lorenzen 1951a.
Lorenzen, P. (1944/2020). Ein halbordnungstheoretischer Widerspruchsfreiheitsbeweis. History and Philosophy of Logic. From the Oskar Becker Nachlass, Philosophical Archive of the University of Konstanz, file OB 5-3b-5, edited and translated as A proof of freedom from contradiction within the theory of partial order by Stefan Neuwirth. https://doi.org/10.1080/01445340.2020.1752040
Macintyre, A. (2005). The mathematical significance of proof theory. Philosophical Transactions of the Royal Society A, 363, 2419–2435.
Martin-Löf, P. (2008). The Hilbert–Brouwer controversy resolved? In M. van Atten, P. Boldini, M. Bourdeau, & G. Heinzmann (Eds.) One hundred years of intuitionism (1907–2007): The Cerisy conference (pp. 243–256). Basel: Birkhäuser.
Menzler-Trott, E. (2007). Logic’s lost genius: The life of Gerhard Gentzen. Volume 33 of History of mathematics. American Mathematical Society.
Neuwirth, S. (2020). Lorenzen’s reshaping of Krull’s Fundamentalsatz for integral domains (1938–1953) (pp. 141–180).
Rathjen, M. (2005a). An ordinal analysis of parameter free \(\Pi ^1_2\)- comprehension. Archive for Mathematical Logic, 44(3), 263–362.
Rathjen, M. (2005b). An ordinal analysis of stability. Archive for Mathematical Logic, 44(1), 1–62.
Rosser, B. (1937). Gödel theorems for non-constructive logics. Journal of Symbolic Logic, 2(3), 129–137.
Schuster, P., & Wessel, D. (2020). Syntax for semantics: Krull’s maximal ideal theorem (pp. 75–100).
Schütte, K. (1951). Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie. Mathematische Annalen, 122, 369–389.
Schütte, K. (1953). Review of Paul Lorenzen, Die Widerspruchsfreiheit der Klassischen Analysis (Lorenzen 1951a). Journal of Symbolic Logic, 18(3), 261–262.
Schütte, Kurt. (1960). Beweistheorie. Volume 103 of Grundlehren der Mathematischen Wissenschaften. Berlin: Springer.
Schütte, K. (1977). Proof theory. Berlin: Springer.
Sundholm, G. (1983). Proof theory: A survey of the omega-rule. Ph.D. dissertation, Magdalen College, University of Oxford.
Tarski, A. (1933). Einige Betrachtungen über die Begriffe der \(\omega \)- Widerspruchsfreiheit und der \(\omega \)-Vollständigkeit. Monatshefte für Mathematik und Physik, 40, 97–112.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2021 The Author(s)
About this chapter
Cite this chapter
Kahle, R., Oitavem, I. (2021). Lorenzen Between Gentzen and Schütte. In: Heinzmann, G., Wolters, G. (eds) Paul Lorenzen -- Mathematician and Logician. Logic, Epistemology, and the Unity of Science, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-030-65824-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-65824-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-65823-6
Online ISBN: 978-3-030-65824-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)