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Gerhard Gentzen

gerhard gentzen, gerhard gentzen sequent calculus examples
Gerhard Karl Erich Gentzen November 24, 1909 – August 4, 1945 was a German mathematician and logician He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague


  • 1 Life and career
  • 2 Work
  • 3 Publications
    • 31 Posthumous
  • 4 See also
  • 5 Notes
  • 6 References
  • 7 External links

Life and careeredit

Gentzen was a student of Paul Bernays at the University of Göttingen Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor Gentzen joined the Sturmabteilung in November 1933 although he was by no means compelled to do so1 Nevertheless he kept in contact with Bernays until the beginning of the Second World War In 1935, he corresponded with Abraham Fraenkel in Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the Chosen People" In 1935 and 1936, Hermann Weyl, head of the Göttingen mathematics department in 1933 until his resignation under Nazi pressure, made strong efforts to bring him to the Institute for Advanced Study in Princeton

Between November 1935 and 1939 he was an assistant of David Hilbert in Göttingen Gentzen joined the Nazi Party in 1937 In April 1939 Gentzen swore the oath of loyalty to Adolf Hitler as part of his academic appointment2 From 1943 he was a teacher at the University of Prague3 Under a contract from the SS Gentzen evidently worked for the V-2 project4

Gentzen was arrested during the citizens uprising against the occupying German forces on May 5, 1945 He, along with the rest of the staff of the German University in Prague was subsequently handed over to Russian forces Because of his past association with the SA, NSDAP and NSD Dozentenbund, Gentzen was detained in a prison camp, where he died of malnutrition on August 4, 194556


Gentzen's main work was on the foundations of mathematics, in proof theory, specifically natural deduction and the sequent calculus His cut-elimination theorem is the cornerstone of proof-theoretic semantics, and some philosophical remarks in his "Investigations into Logical Deduction", together with Ludwig Wittgenstein's later work, constitute the starting point for inferential role semantics

One of Gentzen's papers had a second publication in the ideological Deutsche Mathematik that was founded by Ludwig Bieberbach who promoted "Aryan" mathematics7

Gentzen proved the consistency of the Peano axioms in a paper published in 1936 In his Habilitationsschrift, finished in 1939, he determined the proof-theoretical strength of Peano arithmetic This was done by a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, within Peano arithmetic The principle can, however, be expressed in arithmetic, so that a direct proof of Gödel's incompleteness theorem followed Gödel used a coding procedure to construct an unprovable formula of arithmetic Gentzen's proof was published in 1943 and marked the beginning of ordinal proof theory


  • "Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen" Mathematische Annalen 107 2: 329–350 1932 doi:101007/bf01448897 
  • "Untersuchungen über das logische Schließen I" Mathematische Zeitschrift 39 2: 176–210 1934 doi:101007/bf01201353 
  • "Untersuchungen über das logische Schließen II" Mathematische Zeitschrift 39 3: 405–431 1935 doi:101007/bf01201363 
  • Gentzen, Gerhard 1936 "Die Widerspruchsfreiheit der Stufenlogik" Mathematische Zeitschrift 41: 357–366 doi:101007/BF01180425 
  • Gentzen, Gerhard 1936 "Die Widerspruchsfreiheit der reinen Zahlentheorie" Mathematische Annalen 112: 493–565 doi:101007/BF01565428 
  • "Der Unendlichkeitsbegriff in der Mathematik Vortrag, gehalten in Münster am 27 Juni 1936 am Institut von Heinrich Scholz" Semester-Berichte Münster: 65–80 1936–1937  Lecture hold in Münster at the institute of Heinrich Scholz on 27 June 1936
  • "Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik" Actualités scientifiques et industrielles 535: 201–205 1937 
  • "Die gegenwärtige Lage in der mathematischen Grundlagenforschung" Deutsche Mathematik 3: 255–268 1938 8
  • "Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie" Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 4: 19–44 1938 8
  • Gentzen, Gerhard 1943 "Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie" Mathematische Annalen 119: 140–161 doi:101007/BF01564760 


  • "Zusammenfassung von mehreren vollständigen Induktionen zu einer einzigen" Archiv für mathematische Logik und Grundlagenforschung 2 1: 81–93 1954 
  • Gentzen, Gerhard 1974 "Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie" Archiv für mathematische Logik und Grundlagenforschung 16 3–4: 97–118 doi:101007/BF02015370  – Published by Paul Bernays
  • Gentzen, Gerhard 1974 "Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik" Archiv für mathematische Logik und Grundlagenforschung 16 3–4: 119–132 doi:101007/BF02015371  – Published by Paul Bernays

See alsoedit

  • Biography portal
  • Logic portal
  • Bertrand Russell


  1. ^ Menzler-Trott, p 52
  2. ^ Menzler-Trott, p 119
  3. ^ Gerhard Gentzen at mathmunicz
  4. ^ Menzler-Trott, p 238
  5. ^ Menzler-Trott, p 273 ff
  6. ^ http://www-historymcsst-andacuk/Biographies/Gentzenhtml
  7. ^ DiplMath Walter Tydecks, Neuere Geschichte der Mathematik in Deutschland in German
  8. ^ a b Rosser, J Barkley 1939 "Review of Die gegenwärtige Lage in der mathematischen Grundlagenforschung Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie by Gerhard Gentzen" Bull Amer Math Soc 45: 812–813 doi:101090/S0002-9904-1939-07067-5 


  • Gentzen, Gerhard 1969, M E, Szabo, ed, Collected Papers of Gerhard Gentzen, Studies in logic and the foundations of mathematics Hardcover ed, North-Holland, ISBN 0-7204-2254-X  - English translation
  • Menzler-Trott, Eckart 2001-08-01, Gentzens Problem: Mathematische Logik im nationalsozialistischen Deutschland, Basel, Switzerland: Birkhäuser Verlag, ISBN 3-7643-6574-9 
  • Menzler-Trott, Eckart November 21, 2007, Logic's Lost Genius: The Life of Gerhard Gentzen, History of Mathematics, vol 33, translated by Griffor, Edward; Smorynski, Craig, American Mathematical Society, ISBN 978-0-8218-3550-0  - an English translation

External linksedit

  • O'Connor, John J; Robertson, Edmund F, "Gerhard Gentzen", MacTutor History of Mathematics archive, University of St Andrews 
  • Gerhard Gentzen at the Mathematics Genealogy Project

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