Editing Gerhard Gentzen

{{Short description|German mathematician (1909–1945)}}
{{Use dmy dates|date=November 2021}}
{{Infobox scientist
| name              = Gerhard Gentzen
| image             = Gerhard Gentzen.jpg
| image_size        = 
| alt               = 
| caption           = Gerhard Gentzen in Prague, 1945.
| birth_date        = {{Birth date|1909|11|24|df=y}}
| birth_place       = [[Greifswald]], [[Germany]]
| death_date        = {{Death date and age|1945|8|4|1909|11|24|df=y}}
| death_place       = [[Prague]], [[Czechoslovakia]]
| death_cause = [[Starvation]]
| nationality       = [[Germany|German]]
| fields            = [[Mathematics]]
| workplaces        = 
| alma_mater        = [[University of Göttingen]]
| doctoral_advisor  = [[Paul Bernays]]
| doctoral_students = 
| known_for         = [[Gentzen's consistency proof]]<br/>[[Cut-elimination theorem|Gentzen's Hauptsatz]]<br/>{{no wrap|[[Natural deduction|Gentzen's natural deduction calculus]]}}<br/>[[Sequent calculus|Gentzen's sequent calculus]]<br/>{{no wrap|[[Gödel–Gentzen translation]]}}<br/>[[Analytic proof]]<br/>[[Ordinal analysis]]<br/>[[Proof-theoretic semantics]]
| awards            = 
}}

'''Gerhard Karl Erich Gentzen''' (24 November 1909 – 4 August 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 of starvation in a Czech prison camp in [[Prague]] in 1945.

==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.{{sfn|Menzler-Trott|2007|p=52}} 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 [[Hitler Oath|oath of loyalty]] to [[Adolf Hitler]] as part of his academic appointment.{{sfn|Menzler-Trott|2007|p=119}} From 1943 he was a teacher at the German [[Karl-Ferdinands-Universität|Charles-Ferdinand University of Prague]].{{sfn|Folta|Šišma}} Under a contract from the [[Schutzstaffel|SS]], Gentzen worked for the [[V-2]] project.{{sfn|Menzler-Trott|2007|p=238}}

Gentzen was arrested during the [[Prague uprising|citizens uprising]] against the occupying German forces on 5 May 1945. He, along with the rest of the staff of the German University in Prague, were detained in a Soviet prison camp, where he died of starvation on 4 August 1945.{{sfn|Menzler-Trott|2007|p=273 ff}}<ref>{{MacTutor Biography|id=Gentzen}}</ref>

==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.{{sfn|Tydecks|2002}}

Gentzen [[Gentzen's consistency proof|proved the consistency]] of the [[Peano axioms]] in a paper published in 1936.{{sfn|Gentzen|1936b}} In his [[Habilitation|''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. [[Kurt Gödel|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 analysis|ordinal proof theory]].

==Publications==
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|year=1932|title=Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen|journal=Mathematische Annalen|volume=107 | issue = 2 |pages=329–350|doi=10.1007/bf01448897|s2cid=119534269|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002275872}}
* {{Cite journal|author-mask=0|last=Gentzen|first=Gerhard|year=1935a|title=Untersuchungen über das logische Schließen. I|journal=[[Mathematische Zeitschrift]]|volume=39 | issue = 2 |pages=176–210|doi=10.1007/bf01201353 |s2cid=121546341|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002375508|archive-url=https://web.archive.org/web/20151224194624/http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=17178|archive-date=2015-12-24|url-status=live}}
:{{cite journal|author-mask=1|last=Gentzen|first=Gerhard|year=1964|orig-year=1935| title=Investigations into logical deduction | journal=[[American Philosophical Quarterly]] | volume=1 | number=4 | pages=249–287}}
* {{Cite journal|author-mask=0|last=Gentzen|first=Gerhard|year=1935b|title=Untersuchungen über das logische Schließen. II|journal=[[Mathematische Zeitschrift]]|volume=39 | issue = 3 |pages=405–431|doi=10.1007/bf01201363 |s2cid=186239837|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002375605|archive-url=https://archive.today/20120709063902/http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=17188|archive-date=2012-07-09|url-status=live}}
:{{cite journal|author-mask=1|last=Gentzen|first=Gerhard| year=1965|orig-year=1935| title=Investigations into logical deduction | journal=American Philosophical Quarterly | volume=2 | number=3 | pages=204–218}}
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Die Widerspruchsfreiheit der Stufenlogik|journal=Mathematische Zeitschrift|volume=41|year=1936a|pages=357–366|doi=10.1007/BF01180425|s2cid=122979277|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002376741}}
* {{Cite journal|display-authors=0|last=Gentzen|doi=10.1007/BF01565428|first=Gerhard|title=Die Widerspruchsfreiheit der reinen Zahlentheorie|journal=Mathematische Annalen|volume=112|year=1936b|pages=493–565|s2cid=122719892|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002278391}}
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Der Unendlichkeitsbegriff in der Mathematik. Vortrag, gehalten in Münster am 27. Juni 1936 am Institut von Heinrich Scholz|journal=Semester-Berichte Münster|year=1936–1937|pages=65–80|trans-title=Lecture held in Münster on 27 June 1936 at the institute of Heinrich Scholz |language=de}}
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Unendlichkeitsbegriff und Widerspruchsfreiheit der Mathematik|journal=Actualités scientifiques et industrielles|volume=535|year=1937|pages=201–205}}
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Die gegenwärtige Lage in der mathematischen Grundlagenforschung|journal=Deutsche Mathematik|volume=3|year=1938|pages=255–268}}{{sfn|Rosser|1939}}
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie|journal=Forschungen zur Logik und zur Grundlegung der Exakten Wissenschaften|volume=4|year=1938|pages=19–44}}{{sfn|Rosser|1939}}
* {{Cite journal|display-authors=0|last=Gentzen|doi=10.1007/BF01564760|first=Gerhard|title=Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie|journal=Mathematische Annalen|volume=119|year=1943|pages=140–161|s2cid=120335524|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002281287}}

===Posthumous===
* {{Cite journal|display-authors=0|last=Gentzen|first=Gerhard|title=Zusammenfassung von mehreren vollständigen Induktionen zu einer einzigen|journal=Archiv für mathematische Logik und Grundlagenforschung|language=de|volume=2 | issue = 1 |year=1954|pages=81–93|doi=10.1007/BF01969420 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002042649}}
*{{Cite book|display-authors=0|last=Gentzen|first=Gerhard|editor-last=M. E.|editor-first=Szabo|editor-link=M. E. Szabo|year=1969|title=Collected Papers of Gerhard Gentzen|publisher=North-Holland|edition=Hardcover|series=Studies in logic and the foundations of mathematics|isbn=0-7204-2254-X|url-access=registration|url=https://archive.org/details/collectedpaperso0000gent}} - (English translation).
* {{Cite journal|display-authors=0|last=Gentzen|doi=10.1007/BF02015370|first=Gerhard|title=Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie |journal=Archiv für mathematische Logik und Grundlagenforschung|volume=16|year=1974|pages=97–118|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002044366|issue=3–4|s2cid=117444881}} – Published by [[Paul Bernays]].
* {{Cite journal|display-authors=0|last=Gentzen|doi=10.1007/BF02015371|first=Gerhard|title=Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik|journal=Archiv für mathematische Logik und Grundlagenforschung|volume=16|year=1974|pages=119–132|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002044374|issue=3–4|s2cid=120131107}} – Published by [[Paul Bernays]].
* {{Cite journal|display-authors=0|last1=Gentzen|first1=Gerhard|title=The normalization of derivations  |journal=The Bulletin of Symbolic Logic|volume=14|year=2008|pages=245–257|url=https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/issue/AE2DFF90879F5A83C60567BC42FC1A85}} – Published by [[Jan von Plato]].

==See also==
{{Portal|Biography|Philosophy}}
*[[Bertrand Russell]]

==Notes==
{{Reflist|2}}

==References==
*{{Cite web|last1=Folta|first1=Jaroslav|last2=Šišma|first2=Pavel|date=|url=http://web.math.muni.cz/biografie/gerhard_gentzen.html|title=Gerhard Karl Erich Gentzen|work=Department of Mathematics and Statistics of the Faculty of Science, Masaryk University|language=cs|access-date=2025-01-06}}
*{{Cite book|last=Menzler-Trott|first=Eckart|author-link=:de:Eckart Menzler-Trott|date=1 August 2001|title=Gentzens Problem: Mathematische Logik im nationalsozialistischen Deutschland|language=de|publisher=Birkhäuser Verlag|location=Basel, Switzerland|isbn=3-7643-6574-9}}
*{{Cite book|last=Menzler-Trott|first=Eckart|author-link=Eckart Menzler-Trott|translator=Griffor, Edward|translator-link=Edward Griffor|translator2=Smorynski, Craig|translator2-link=Craig Smorynski|date=21 November 2007|title=Logic's Lost Genius: The Life of Gerhard Gentzen|publisher=American Mathematical Society|series=History of Mathematics|volume=33|isbn=978-0-8218-3550-0|url=https://www.ams.org/bookstore-getitem/item=hmath-33}} — English translation of {{harvtxt|Menzler-Trott|2001}}.
*{{cite journal|last=Rosser|first=J. Barkley|author-link=J. Barkley Rosser|title=Review of ''Die gegenwärtige Lage in der mathematischen Grundlagenforschung. Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie'' by Gerhard Gentzen|journal=Bull. Amer. Math. Soc.|year=1939|volume=45|pages=812–813|doi=10.1090/S0002-9904-1939-07067-5|doi-access=free}}
*{{cite web|url=http://www.tydecks.info/online/math_natur_deu.html#k5#k5|first=Walter|last=Tydecks|title=Neuere Geschichte der Mathematik in Deutschland|year=2002|language=de|access-date=2025-01-06}}
*{{Cite book|last=von Plato|first=Jan|date= 2017|title=Saved from the Cellar: Gerhard Gentzen' Shorthand Notes on Logic and Foundations Mathematics|publisher=Springer|isbn=978-3-319-42119-3}}

==External links==
* {{MacTutor Biography|id=Gentzen}}
* {{MathGenealogy|id=52969}}

{{Authority control}}

{{DEFAULTSORT:Gentzen, Gerhard}}
[[Category:1909 births]]
[[Category:1945 deaths]]
[[Category:People from Greifswald]]
[[Category:University of Göttingen alumni]]
[[Category:Nazi Party members]]
[[Category:Academic staff of Charles University]]
[[Category:20th-century German mathematicians]]
[[Category:German logicians]]
[[Category:People from the Province of Pomerania]]
[[Category:20th-century German philosophers]]
[[Category:Deaths by starvation]]
[[Category:Nazis who died in prison custody]]
[[Category:German people who died in Soviet detention]]