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HEINRICH BEHMANN’S 1921 LECTURE ON THE DECISION PROBLEM AND THE ALGEBRA OF LOGIC

  • PAOLO MANCOSU (a1) and RICHARD ZACH (a2)
Abstract
Abstract

Heinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier work—the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.

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[1]Ackermann Wilhelm, Über die Erfüllbarkeit gewisser Zählausdrücke. Mathematische Annalen, vol. 100 (1928), pp. 638649.
[2]Ackermann Wilhelm, Untersuchungen über das Eliminationsproblem der mathematischen Logik. Mathematische Annalen, vol. 110 (1934), pp. 390413.
[3]Ackermann Wilhelm, Solvable Cases of the Decision Problem, North-Holland, Amsterdam, 1954.
[4]Behmann Heinrich, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead, Dissertation, Universität Göttingen, Göttingen, 1918, 352 pp.
[5]Behmann Heinrich, Entscheidungsproblem und Algebra der Logik, unpublished manuscript dated May 10, 1921. Behmann Archive, Staatsbibliothek zu Berlin, Preußischer Kulturbesitz, Handschriftenabteilung, Nachl. 355 (Behmann), K. 9 Einh. 37, 1921.
[6]Behmann Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Mathematische Annalen, vol. 86 (1922), pp. 163229.
[7]Behmann Heinrich, Algebra der Logik und Entscheidungsproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 32 (1923), 2. Abt. pp. 6667.
[8]Bernays Paul, Beiträge zur axiomatischen Behandlung des Logik-Kalküls, Habilitationsschrift, Universität Göttingen, 1918, Bernays Nachlaß, WHS, ETH Zürich Archive, Hs 973.192. Edited in [15], pp. 222271.
[9]Bernays Paul and Schönfinkel Moses, Zum Entscheidungsproblem der mathematischen Logik. Mathematische Annalen, vol. 99 (1928), pp. 342372.
[10]Bondoni Davide, La teoria delle relazioni nell’algebra della logica schröderiana, LED Edizioni, Milan, 2007.
[11]Bondoni Davide, Peirce and Schröder on the Auflösungsproblem. Logic and Logical Philosophy, vol. 1 (2009), pp. 1531.
[12]Börger Egon, Grädel Erich, and Gurevich Yuri, The Classical Decision Problem, Springer, Berlin, 1997.
[13]Church Alonzo, A note on the Entscheidungsproblem. The Journal of Symbolic Logic, vol. 1 (1936), pp. 4041.
[14]Dreben Burton and Goldfarb Warren, The Decision Problem, Addison-Wesley, Boston, MA, 1979.
[15]Ewald William and Sieg Wilfried (editors), David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1917–1933, David Hilbert’s Lectures on the Foundations of Mathematics and Physics, 1891–1933, vol. 3, Springer, Berlin and Heidelberg, 2013.
[16]Gandy Robin, The confluence of ideas in 1936, The universal Turing machine, 2nd edition (Herken Rolf, editor), Springer, Secaucus, NJ, 1995, pp. 51102.
[17]Gödel Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I . Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198. Reprinted and translated in [18], pp. 144–195.
[18]Gödel Kurt, Collected Works, vol. 1, Oxford University Press, Oxford, 1986.
[19]Haas Gerrit and Stemmler Elke, Der Nachlass Heinrich Behmanns, 1981, Aachener Schriften zur Wissenschaftstheorie, Logik und Logikgeschichte 1.
[20]Hessenberg Gerhard, Grundbegriffe der Mengenlehre. Abhandlungen der Fries’schen Schule, Neue Folge, vol. 1 (1906), pp. 479706.
[21]Hilbert David, Logische Principien des mathematischen Denkens, Vorlesung, Sommer-Semester 1905. Lecture notes by Ernst Hellinger. Unpublished manuscript, 277 pp. Bibliothek, Mathematisches Institut, Universität Göttingen. 1905.
[22]Hilbert David, Prinzipien der Mathematik, Lecture notes by Paul Bernays. Winter-Semester 1917–18. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen. 1918. Edited in [15], pp. 59–221.
[23]Hilbert David, Die Grundlagen der Mathematik, Abhandlungen aus dem Seminar der Hamburgischen Universität, vol. 6 (1928), pp. 6585, Reprinted in [15], pp. 917–942. English translation in [36], pp. 464–479.
[24]Hilbert David, Probleme der Grundlegung der Mathematik, Atti del congresso internazionale dei matematici. 3–10 September 1928, Bologna (Zanichelli Nicola, editor), 1928, pp. 135141. Reprinted in [15], pp. 954–966.
[25]Hilbert David and Ackermann Wilhelm, Grundzüge der theoretischen Logik, 1st ed., Springer, Berlin, 1928, Reprinted in [15], pp. 806916.
[26]Hilbert David and Bernays Paul, Logische Grundlagen der Mathematik, Vorlesung, Winter-Semester 1922/23. Lecture notes by Paul Bernays, with handwritten notes by Hilbert. Hilbert-Nachlaß, Niedersächsische Staats- und Universitätsbibliothek, Cod. Ms. Hilbert 567. 1922/23. Edited in [15], pp. 528–564.
[27]Mancosu Paolo, Between Russell and Hilbert: Behmann on the foundations of mathematics, this Bulletin, vol. 5 (1999), no. 3, pp. 303330.
[28]Mancosu Paolo, On the constructivity of proofs. A debate among Behmann, Bernays, Gödel, and Kaufmann, Reflections on the foundations of mathematics. Essays in honor of Solomon Feferman (Sieg Wilfried, Sommer Richard, and Talcott Carolyn, editors), Lecture Notes in Logic 15, Association for Symbolic Logic, 2002, pp. 346368.
[29]Mancosu Paolo, The Russellian influence on Hilbert and his school. Synthese, vol. 137 (2003), no. 12, pp. 59101.
[30]Parsons Charles, Introductory note to Gödel’s correspondence with Heinrich Behmann, Kurt Gödel: Collected works (Feferman Solomon et al. , editors), vol. 4, Oxford University Press, Oxford, 2003, pp. 1319.
[31]Post Emil L., On a simple class of deductive systems (abstract). Bulletin of the American Mathematical Society, vol. 27 (1921), no. 910, pp. 396397.
[32]Post Emil L., Absolutely unsolvable problems and relatively undecidable propositions: Account of an anticipation, The undecidable (Davis Martin, editor), Raven Press, Hewlett, NY, 1965, pp. 340433.
[33]Thiel Christian, Gödels Anteil am Streit über Behmanns Behandlung der Antinomien, Wahrheit und Beweisbarkeit. Leben und Werk Kurt Gödels (Köhler Eckehard et al. , editors), vol. 2, Hölder-Pichler-Tempsky, Vienna, 2002, pp. 387394.
[34]Turing Alan M., On computable numbers, with an application to the “Entscheidungsproblem”, Proceedings of the London Mathematical Society, 2nd Series, vol. 42 (1937), pp. 230265.
[35]Urquhart Alasdair, Emil Post, Logic from Russell to Church (Gabbay Dov and Woods John, editors), Handbook of the History of Logic, vol. 5, North-Holland, Amsterdam, 2009, pp. 617666.
[36]Jean van Heijenoort (editor), From Frege to Gödel. A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967.
[37]von Neumann Johann, Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, vol. 26 (1927), pp. 146.
[38]Weyl Hermann, Philosophie der Mathematik und Naturwissenschaft, Oldenbourg, Munich, 1927, Augmented and revised English translation by Olaf Helmer, Philosophy of Mathematics and Natural Science (Princeton: Princeton University Press, 1949).
[39]Zach Richard, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Bulletin, vol. 5 (1999), no. 3, pp. 331366.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
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