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“Sehr geehrter Herr Professor!” Proof Theory in 1949 in a Letter from Schütte to Bernays

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The Legacy of Kurt Schütte

Abstract

We present a letter which Kurt Schütte sent in 1949 to his former de-facto PhD supervisor Paul Bernays. This letter contains an outline of the proof-theoretic methods which became standard in infinitary proof theory.

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References

  1. Wilhelm Ackermann. Letter to David Hilbert, August 23rd, 1933, Niedersächsische Staatsund Universitätsbibliothek Göttingen, Cod. Ms. D. Hilbert 1.

    Google Scholar 

  2. Oskar Becker. Mathematische Existenz. M. Niemeyer, 1927.

    Google Scholar 

  3. Wilfried Buchholz. Schütte, Kurt Wilhelm. In Neue Deutsche Biographie (NDB), volume 23, pages 653–654. Duncker & Humblot, Berlin, 2007.

    Google Scholar 

  4. Thierry Coquand and Stefan Neuwirth. An introduction to Lorenzen’s “Algebraic and logistic investigations on free lattices” (1951). arXiv: 1711.06139v1 [math.LO], 2017.

    Google Scholar 

  5. Solomon Feferman. Introductory note to [Gödel 1931c]. In S. Feferman et al., editors, Kurt Gödel: Collected works, I: Publications 1929–1936, pages 208–213. Oxford University Press, 1986.

    Google Scholar 

  6. Gerhard Gentzen. Collected Works. North-Holland, 1969. Edited by M. E. Szabo.

    Google Scholar 

  7. David Hilbert. Über das Unendliche. Mathematische Annalen, 95:161–190, 1926.

    Article  MathSciNet  Google Scholar 

  8. David Hilbert. Beweis des Tertium non datur. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pages 120–125, 1931. Talk given on July 17, 1931 in Göttingen.

    Google Scholar 

  9. David Hilbert. Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen, 104(1):485–494, 1931. Talk given in December 1930 in Hamburg.

    Google Scholar 

  10. David Hilbert. On the infinite. In Jean van Heijenoort, editor, From Frege to Gödel, pages 367–392. Harvard University Press, 1967. English translation of [7].

    Google Scholar 

  11. David Hilbert and Paul Bernays. Grundlagen der Mathematik I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 40. Springer, 1934. 2nd edition 1968.

    Google Scholar 

  12. David Hilbert and Paul Bernays. Grundlagen der Mathematik II. Die Grundlehren der mathematischenWissenschaften in Einzeldarstellungen, 50. Springer, 1939. 2nd edition 1970.

    Google Scholar 

  13. Reinhard Kahle. Gentzen’s theorem in context. In Reinhard Kahle and Michael Rathjen, editors, Gentzen’s Centenary: The quest for consistency, pages 3–24. Springer, 2015.

    Google Scholar 

  14. Reinhard Kahle and Isabel Oitavem. Lorenzen between Gentzen and Schütte. In Gerhard Heinzmann and Gereon Wolters, editors, Paul Lorenzen—Mathematician and Logician. Springer. To appear.

    Google Scholar 

  15. Paul Lorenzen. Algebraische und logistische Untersuchungen über freie Verbände. Journal of Symbolic Logic, 16(81–106), 1951.

    Google Scholar 

  16. Kurt Schütte. Untersuchungen zum Entscheidungsproblem der mathematischen Logik. Mathematische Annalen, 109:572–603, 1934.

    Article  MathSciNet  Google Scholar 

  17. Kurt Schütte. Schlußweisen-Kalküle der Prädikatenlogik. Mathematische Annalen, 122:47–65, 1950.

    Article  MathSciNet  Google Scholar 

  18. Kurt Schütte. Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie. Mathematische Annalen, 122:369–389, 1951.

    Article  MathSciNet  Google Scholar 

  19. Kurt Schütte. Beweistheoretische Untersuchung der verzweigten Analysis. Mathematische Annalen, 124:123–147, 1952.

    Article  MathSciNet  Google Scholar 

  20. Kurt Schütte. Letter to Paul Lorenzen, Göttingen, May 1st, 1950. Lorenzen-Nachlass, Philosophisches Archiv der Universität Konstanz, PL 1-1-45.

    Google Scholar 

  21. Kurt Schütte. Beweistheorie, volume 103 of Grundlehren der MathematischenWissenschaften. Springer, 1960.

    Google Scholar 

  22. Kurt Schütte. Proof theory, volume 225 of Grundlehren der Mathematischen Wissenschaften. Springer, 1977.

    Google Scholar 

  23. Kurt Schütte. Bemerkungen zur Hilbertschen Beweistheorie. Acta Borussica, 5:241–244, 1995. Reprinted and translated into English in Chapter 8 of this volume.

    Google Scholar 

  24. Kurt Schütte and Helmut Schwichtenberg. Mathematische Logik. In Winfried Scharlau, editor, Ein Jahrhundert Mathematik 1890–1990, volume 6 of Dokumente zur Geschichte der Mathematik, pages 717–740. Vieweg, 1990. Reprinted in Chapter 7 of this volume.

    Google Scholar 

  25. Alfred Tarski. Einige Betrachtungen über die Begriffe der !-Widerspruchsfreiheit und der !-Vollständigkeit. Monatshefte für Mathematik und Physik, 40:97–112, 1933.

    Article  MathSciNet  Google Scholar 

  26. Jan von Plato. Saved from the Cellar. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, 2017.

    Google Scholar 

  27. MatthiasWille. Zwischen Algebra und Erlanger Schule. Paul Lorenzens Beiträge zur Beweistheorie. In Ralf Krömer and Gregor Nickel, editors, Siegener Beiträge zur Geschichte und Philosophie der Mathematik, volume 1, pages 79–108. universi, 2013.

    Google Scholar 

  28. Mariko Yasugi and Nicholas Passell, editors. Memoirs of a Proof Theorist. World Scientific, 2003. English translation of a collection of essays written by Gaisi Takeuti.

    Google Scholar 

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Acknowledgements

We are grateful to the Hochschularchiv der ETH Zürich for the permission to reprint the letter of Schütte to Bernays (Hs. 975: 4230). I’m also thankful to Michael Rathjen who revised the English translation of the German texts. Research supported by the Portuguese Science Foundation, FCT, through the project Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and UID/MAT/00297/2013 (Centro de Matemática e Aplicações) and by the Udo-Keller-Stiftung.

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Correspondence to Reinhard Kahle .

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Kahle, R. (2020). “Sehr geehrter Herr Professor!” Proof Theory in 1949 in a Letter from Schütte to Bernays. In: Kahle, R., Rathjen, M. (eds) The Legacy of Kurt Schütte. Springer, Cham. https://doi.org/10.1007/978-3-030-49424-7_1

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