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

Published online by Cambridge University Press:  04 June 2015

PAOLO MANCOSU  and
RICHARD ZACH
PAOLO MANCOSU
Affiliation:
DEPARTMENT OF PHILOSOPHY 314 MOSES HALL #2390 UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720–2390, USAE-mail: [email protected]: http://philosophy.berkeley.edu/mancosu/
RICHARD ZACH
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALGARY 2500 UNIVERSITY DRIVE N.W. CALGARY, AB T2N 1N4, CANADAE-mail: [email protected]: http://richardzach.org/
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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.

Keywords

01A6003-03Heinrich BehmannDavid Hilbertdecision problemdecision procedurealgebra of logic
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 2 , June 2015 , pp. 164 - 187
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Ackermann, Wilhelm, Über die Erfüllbarkeit gewisser Zählausdrücke. Mathematische Annalen, vol. 100 (1928), pp. 638–649.CrossRefGoogle Scholar
Ackermann, Wilhelm, Untersuchungen über das Eliminationsproblem der mathematischen Logik. Mathematische Annalen, vol. 110 (1934), pp. 390–413.CrossRefGoogle Scholar
Ackermann, Wilhelm, Solvable Cases of the Decision Problem, North-Holland, Amsterdam, 1954.Google Scholar
Behmann, Heinrich, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead, Dissertation, Universität Göttingen, Göttingen, 1918, 352 pp.Google Scholar
Behmann, Heinrich, Entscheidungsproblem und Algebra der Logik, unpublished manuscript dated May 10, 1921.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.Google Scholar
Behmann, Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Mathematische Annalen, vol. 86 (1922), pp. 163–229.CrossRefGoogle Scholar
Behmann, Heinrich, Algebra der Logik und Entscheidungsproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 32 (1923), 2. Abt. pp. 66–67.Google Scholar
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. 222–271.Google Scholar
Bernays, Paul and Schönfinkel, Moses, Zum Entscheidungsproblem der mathematischen Logik. Mathematische Annalen, vol. 99 (1928), pp. 342–372.CrossRefGoogle Scholar
Bondoni, Davide, La teoria delle relazioni nell’algebra della logica schröderiana, LED Edizioni, Milan, 2007.Google Scholar
Bondoni, Davide, Peirce and Schröder on the Auflösungsproblem. Logic and Logical Philosophy, vol. 1 (2009), pp. 15–31.Google Scholar
Börger, Egon, Grädel, Erich, and Gurevich, Yuri, The Classical Decision Problem, Springer, Berlin, 1997.CrossRefGoogle Scholar
Church, Alonzo, A note on the Entscheidungsproblem. The Journal of Symbolic Logic, vol. 1 (1936), pp. 40–41.CrossRefGoogle Scholar
Dreben, Burton and Goldfarb, Warren, The Decision Problem, Addison-Wesley, Boston, MA, 1979.Google Scholar
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.Google Scholar
Gandy, Robin, The confluence of ideas in 1936, The universal Turing machine, 2nd edition (Herken, Rolf, editor), Springer, Secaucus, NJ, 1995, pp. 51–102.Google Scholar
Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I . Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198. Reprinted and translated in [18], pp. 144–195.CrossRefGoogle Scholar
Gödel, Kurt, Collected Works, vol. 1, Oxford University Press, Oxford, 1986.Google Scholar
Haas, Gerrit and Stemmler, Elke, Der Nachlass Heinrich Behmanns, 1981, Aachener Schriften zur Wissenschaftstheorie, Logik und Logikgeschichte 1.Google Scholar
Hessenberg, Gerhard, Grundbegriffe der Mengenlehre. Abhandlungen der Fries’schen Schule, Neue Folge, vol. 1 (1906), pp. 479–706.Google Scholar
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.Google Scholar
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.Google Scholar
Hilbert, David, Die Grundlagen der Mathematik, Abhandlungen aus dem Seminar der Hamburgischen Universität, vol. 6 (1928), pp. 65–85, Reprinted in [15], pp. 917–942. English translation in [36], pp. 464–479.CrossRefGoogle Scholar
Hilbert, David, Probleme der Grundlegung der Mathematik, Atti del congresso internazionale dei matematici. 3–10 September 1928, Bologna (Zanichelli, Nicola, editor), 1928, pp. 135–141. Reprinted in [15], pp. 954–966.Google Scholar
Hilbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, 1st ed., Springer, Berlin, 1928, Reprinted in [15], pp. 806–916.Google Scholar
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.Google Scholar
Mancosu, Paolo, Between Russell and Hilbert: Behmann on the foundations of mathematics, this Bulletin, vol. 5 (1999), no. 3, pp. 303–330.Google Scholar
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. 346–368.Google Scholar
Mancosu, Paolo, The Russellian influence on Hilbert and his school. Synthese, vol. 137 (2003), no. 1–2, pp. 59–101.CrossRefGoogle Scholar
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. 13–19.Google Scholar
Post, Emil L., On a simple class of deductive systems (abstract). Bulletin of the American Mathematical Society, vol. 27 (1921), no. 9–10, pp. 396–397.Google Scholar
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. 340–433.Google Scholar
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. 387–394.Google Scholar
Turing, Alan M., On computable numbers, with an application to the “Entscheidungsproblem”, Proceedings of the London Mathematical Society, 2nd Series, vol. 42 (1937), pp. 230–265.CrossRefGoogle Scholar
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. 617–666.Google Scholar
Jean van Heijenoort, (editor), From Frege to Gödel. A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967.Google Scholar
von Neumann, Johann, Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46.CrossRefGoogle Scholar
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).Google Scholar
Zach, Richard, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Bulletin, vol. 5 (1999), no. 3, pp. 331–366.Google Scholar