Gerhard Gentzen

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Gerhard Gentzen
Gerhard Gentzen in Prague, 1945.
Born	(1909-11-24)November 24, 1909 Greifswald, Germany
Died	August 4, 1945(1945-08-04) (aged 35) Prague, Czechoslovakia
Nationality	German
Fields	Mathematics
Alma mater	University of Göttingen
Doctoral advisor	Paul Bernays

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.

Life and career

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 so.^[1] Nevertheless he kept in contact with Bernays until the beginning of the Second World War. In 1935, he corresponded with Abraham Fraenkel from 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 NSDAP in 1937. In April 1939 Gentzen swore the oath of loyalty to Adolf Hitler as part of his academic appointment.^[2] From 1943 he was a teacher at the University of Prague.^[3] Under a contract from the SS Gentzen evidently worked for the V-2 project.^[4]

After the war he starved to death in Prague, after being arrested like all other Germans in Prague on May 7, 1945 and deprived of food.^[5]

Work

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" mathematics.^[6]

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.

Publications

"Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen". Mathematische Annalen. 107 (2): 329–350. 1932.
"Untersuchungen über das logische Schließen. I". Mathematische Zeitschrift. 39 (2): 176–210. 1934.
"Untersuchungen über das logische Schließen. II". Mathematische Zeitschrift. 39 (3): 405–431. 1935.
Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der Stufenlogik". Mathematische Zeitschrift 41: 357–366. doi:10.1007/BF01180425.
Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen 112: 493–565. doi:10.1007/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. Check date values in: |date= (help) (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 gegenwartige Lage in der mathematischen Grundlagenforschung". Deutsche Mathematik 3: 255–268. 1938.
"Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie". Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 4: 19–44. 1938.
Gentzen, Gerhard (1943). "Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie". Mathematische Annalen 119: 140–161. doi:10.1007/BF01564760.

Posthumous

"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:10.1007/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:10.1007/BF02015371. – Published by Paul Bernays.

Notes

↑ Menzler-Trott, p. 52.
↑ Menzler-Trott, p. 119.
↑ Gerhard Gentzen at math.muni.cz
↑ Menzler-Trott, p. 238.
↑ Menzler-Trott, p. 273 ff.
↑ Dipl.Math. Walter Tydecks, Neuere Geschichte der Mathematik in Deutschland (in German)

References

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 and Smorynski, Craig, American Mathematical Society, ISBN 978-0-8218-3550-0 - an English translation.
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 - an English translation of papers.

External links

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

Logic

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Authority control	WorldCat VIAF: 22269649 LCCN: n88637476 ISNI: 0000 0000 5512 1572 GND: 117710458 SUDOC: 066888581 BNF: cb12859876g MGP: 52969

Persondata
Name	Gentzen, Gerhard
Alternative names
Short description	German mathematician
Date of birth	November 24, 1909
Place of birth	Greifswald, Germany
Date of death	August 4, 1945
Place of death	Prague, Czechoslovakia

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