- Jacques Herbrand
Jacques Herbrand (
February 12 ,1908 -July 27 ,1931 ) was a Frenchmathematician who was born inParis ,France and died inLa Bérarde ,Isère , France.He worked in
mathematical logic andclass field theory . He introducedrecursive function s. "Herbrand's theorem " refers to either of two completely different theorems. One is a result from his doctoral thesis inproof theory , and the other one half of theHerbrand-Ribet theorem . TheHerbrand quotient is a type ofEuler characteristic , used inhomological algebra . He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic. The proof uses the above mentioned, proof-theoretic Herbrand's Theorem.Biography
Herbrand finished his doctorate at
École Normale Supérieure in Paris underErnest Vessiot in 1929. He joined the army in October 1929, however, and so did not defend his thesis at the Sorbonne until the following year. He was awarded a Rockefeller fellowship that enabled him to study inGermany in 1931, first withJohn von Neumann inBerlin , then during June withEmil Artin inHamburg , and finally withEmmy Noether inGöttingen .He submitted his principal study of proof theory and general recursive functions "On the consistency of arithmetic" early in 1931. While the essay was under consideration,
Gödel 's "On formally undecidable sentences of "Principia Mathematica" and related systems I" announced the impossibility of formalizing within a theory that theory's consistency proof. Herbrand studied Gödel's essay and wrote an appendix to his own study explaining why Gödel's result did not contradict his own. In July of that year he was mountain-climbing in the FrenchAlps with two friends when he fell to his death in the granite mountains ofMassif des Écrins . "On the consistency of arithmetic" was published posthumously.Quotation
"Jacques Herbrand would have hated Bourbaki" said French mathematician
Claude Chevalley quoted in Michèle Chouchan "Nicolas Bourbaki Faits et légendes" Edition du choix, 1995.Bibliography
Primary literature:
*Jean van Heijenoort , 1967. "From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931". Cambridge, Mass.: Harvard Univ. Press.
** 1930. "Investigations in proof theory," 525-81.
** 1931. "On the consistency of arithmetic," 618-28.
** 1968.Jean van Heijenoort (ed.), "Jacques Herbrand, Écrits logiques". Paris: Presses Universitaires de France.
**1971. Warren David Goldfarb (transl., ed.), "Logical Writings of Jacques Herbrand" Cambridge, Mass.: Harvard University Press.ee also
*
Herbrand universe
*Herbrand base
*Herbrand interpretation
*Herbrand structure
*Herbrand theory
*Herbrand–Ribet theorem
*Herbrand quotient External links
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