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	M. E. Szabo: The Collected Works of Gerhard Gentzen (Site not responding. Last check: 2007-11-07)
		Gentzen's natural deduction system in #3 [1935, see below] provides a formalization of the notion of consequence in the sense first used by Bolzano (which was introduced by Bolzano over a hundred years earlier and is analogous to Tarski's notion).
		Gentzen, like the intuitionists, attributes the antimonies to a liberal use of infinity, but does not completely reject the law of the excluded middle in infinite situations.
		Gentzen used the sharpened Hauptsatz to show the consistency of arithmetic without induction, which he acknowledged is of limited significance due to the importance of induction.
	www.andrew.cmu.edu /user/cebrown/notes/szabo.html (3057 words)

	M. E. Szabo: The Collected Works of Gerhard Gentzen (Site not responding. Last check: 2007-11-07)
		Gentzen showed in #2 [1934] that propositions of elementary number theory (the "first level" of mathematics) not involving any disjunction or existential quantifier is intuitionistically provable iff it is classically provable.
		Gentzen's natural deduction was reformulated by Beth [1965] who created "semantic tableau" as a system for studying the notion of "logical consequence".
		In #8 [1938], Gentzen gives a new proof of consistency of elementary number theory based on his sequent calculus LK (where the one in #4 was based on natural deduction).
	www.andrew.cmu.edu /%7Ecebrown/notes/szabo.html (3057 words)

	Gentzen (Site not responding. Last check: 2007-11-07)
		Gerhard Gentzen's father was a lawyer who practiced law in Bergen on the Isle of Rügen. It was there that Gerhard spent his childhood years, attending first the elementary school there, and later the Realgymnasium.
		At first Gentzen worried that it affected what he wanted to achieve on the foundations of mathematics and he withdrew what would have been his second paper after he had corrected the final proofs because of worries about the significance of Gödel's theorems.
		Gentzen was interned by the Russian forces and held in poor conditions.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Gentzen.html (1517 words)

	Gerhard Gentzen - Wikipedia
		Gerhard Gentzen (24 de noviembre de 1909 – 4 de agosto de 1945) fue un matemático y lógico alemán.
		En 1934, Gentzen introduce la noción de sistema de deducción natural para lógica clásica y lógica intuicionista.
		En 1936, Gentzen demuestra la consistencia de la teoría elemental de números.
	es.wikipedia.org /wiki/Gerhard_Gentzen (186 words)

	Gerhard Gentzen - Wikipedia
		Gentzen ist ein wichtiger Mitbegründer der modernen mathematischen Beweistheorie.
		Gentzen, der als Assistent David Hilberts in Göttingen lehrte, erhält 1943 eine Dozentur in Prag.
		Trotz Warnungen floh der wegen seiner Loyalität zu den Nazis bedrohte Gentzen bei Kriegsende nicht nach Deutschland.
	de.wikipedia.org /wiki/Gerhard_Gentzen (228 words)

	CONK! Encyclopedia: Gerhard_Gentzen (Site not responding. Last check: 2007-11-07)
		Gerhard Karl Erich Gentzen (November 24, 1909 â“ August 4, 1945) was a German mathematician and logician.
		Born in Greifswald, Germany, he died in Prague, Czechoslovakia in a prisoner of war camp, after being arrested by the Russians due to his Nazi loyalties.
		This article about a mathematician is a stub.
	www.conk.com /search/encyclopedia.cgi?q=Gerhard_Gentzen (120 words)

	Gentzen (Site not responding. Last check: 2007-11-07)
		Gerhard Gentzen was taught by Bernays, Carathéodory, Courant, Hilbert, Kneser, Landau and Weyl at the University of Göttingen.
		Gentzen remained Hilbert's assistant until 1943, although he had to undertake military service in the years 1939 until 1941.
		Gentzen's work was on logic and the foundations of mathematics.
	www.bg-rams.ac.at /intranet/Physik/history/Gentzen.html (277 words)

	Citations: Untersuchungen uber das logische Schlieen - Gentzen (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Gentzen, "Untersuchungen uber das logische schlieen," Mathematische Zeitschrift, 39(1935) pp.
		Gerhard Gentzen, "Untersuchungen uber das logische Schlieen", Mathematische Zeitschrift 39 (1934-5), Seiten 176--210, 405--431.
		Gerhard Gentzen, Untersuchungen uber das logische Schlieen, Mathematische Zeitschrift, vol.
	citeseer.ifi.unizh.ch /context/975478/0 (3258 words)

	Citations: Investigations into logical deduction - Gentzen (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Gentzen, Untersuchungen uber das logische Schliessen, Mathematische Zeitschrift 39 (1934) 176--210, 405--431, translated in Sabo (ed.), The Collected Papers of Gerhard Gentzen as "Investigations into Logical Deduction," pages 68--131, North-Holland, Amsterdam and London, 1969.
		Gentzen, `Investigations into logical deduction' The collected papers of Gerhard Gentzen (Translated from 1934 original in German), Studies in Logic and the Foundations of Mathematics (ed.
		In its most general (propositional) form, the Gentzen Calculus prescribes rules for deriving sequents) where, are sets of propositional formulae (over a set of atomic propositional variables fx 1 ; x 2 ; x 3 ;....
	citeseer.ifi.unizh.ch /context/312/0 (4084 words)

	Science Fair Projects - Proof-theoretic semantics
		Gerhard Gentzen is the founder of proof-theoretic semantics, providing the formal basis for in his account of cut-elimination for the sequent calculus, and some provocative philosophical remarks about locating the meaning of logical connectives in their introduction rules within natural deduction.
		It is not a great exaggeration that the history of proof-theoretic semantics since then has been devoted to exploring the consequences of these ideas.
		Dag Prawitz extended Gentzen's notion of analytic proof to natural deduction, and suggested that the value of a proof in natural deduction may be understood as its normal form.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Proof-theoretic_semantics (437 words)

	Citations: The collected papers of Gerhard Gentzen - Gentzen (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		A set L of formulas being given, one may model a logic by a mapping C : 2 and for every subset X of L, C(X) is understood to be the set of all the consequences of the set X of assumptions.
		A set L of formulas being given, one may model a logic by a mapping C : 2 L Gamma 2 L and for every subset X of L, C(X) is understood to be the set of all the consequences of the set X of assumptions.
		Gentzen, G.: 1969, The Collected Papers of Gerhard Gentzen, Studies in Logic and the Foundations of Mathematics.
	sherry.ifi.unizh.ch /context/20260/0 (4071 words)

	Systeme natürlichen Schließens - Wikipedia
		Systeme (oder Kalküle) natürlichen Schließens sind ein Kalkültyp, der 1934 von Gerhard Gentzen und etwa zeitgleich von Stanisław Jaśkowski (Lemberg-Warschau-Schule) entwickelt wurde.
		Zusammen mit den (gleichfalls von Gentzen entwickelten) Sequenzenkalkülen gehören Systeme des natürlichen Schließens zur Familie der Regellogiken.
		Gerhard Gentzen: Untersuchungen über das logische Schließen, Mathematische Zeitschrift 39, 1934–1935 Nachdruck in: Karel Berka, Lothar Kreier: "Logik-Texte", Berlin (Ost) 1986.
	de.wikipedia.org /wiki/Systeme_nat%FCrlichen_Schlie%DFens (219 words)

	References for Gentzen (Site not responding. Last check: 2007-11-07)
		P Pudlak, A foreword to the translation of G Gentzen's paper : "The current state of research in the foundations of mathematics" (Czech), Pokroky Mat.
		A R Raggio, The 50th anniversary of Gentzen's thesis, in Methods and applications of mathematical logic, Campinas, 1985 (Providence, RI, 1988), 93-97.
		P Vihan, Gerhard Gentzen (1909-1945) (Czech), Pokroky Mat.
	www-history.mcs.st-and.ac.uk /history/References/Gentzen.html (126 words)

	Hilbert's Program
		It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory.
		Proof theory in the tradition of Gentzen has analyzed axiomatic systems according to what extensions of the finitary standpoint are necessary to prove their consistency.
		Gentzen, Gerhard, 1936, "Die Widerspruchsfreiheit der reinen Zahlentheorie", Mathematische Annalen, 112: 493-565.
	plato.stanford.edu /entries/hilbert-program (7534 words)

	Gerhard Gentzen - rFind.net (Site not responding. Last check: 2007-11-07)
		Du kan också söka efter Gerhard Gentzen i andra artiklar på svenska wikipedia.
		Karl Gerhard StartsidaSenaste nyttLänkspegelInställningarKarl Gerhard, revyförfattare?, teaterdirektör, född 1891 som Karl Johnsson, död 1964.
		En oväntad röst kastade sig på tisdagen in i debatten om den tyske förbundskanslern Gerhard Schröders politiska framtid.
	www.rfind.net /info/Gerhard_Gentzen (262 words)

	Logical harmony (Site not responding. Last check: 2007-11-07)
		Logical harmony, a name coined by Sir Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system.
		The logician Gerhard Gentzen proposed that the meanings of logical connectives could be given by the rules for introducing them into discourse.
		Gentzen's idea was that having rules like this is what gives meaning to one's words, or at least to certain words.
	nba.servegame.org /en/Logical_harmony.htm (437 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Gentzen \cite{G3} published his new version of consistency proof for first order number theory in 1938.
		\end{quotation} It seems that the purpose of the third "Neue Fassung" is to make a lucid exposure of this {\em combinatorial complexity} which Gentzen discovered in {\em finite proof figures} of number theory.
		We follow in the wake of Gentzen and Takeuti.
	www.amsta.leeds.ac.uk /events/logic97/abstracts/arai.txt (292 words)

	GERHARD (Site not responding. Last check: 2007-11-07)
		Search the GERHARD Family Message Boards at Ancestry.com (if available).
		Search the GERHARD Family Resource Center at RootsWeb.com (if available).
		Find graves of people named GERHARD at Find-a-Grave.com (or add one that you know).
	www.worldhistory.com /surname/US/G/GERHARD.htm (73 words)

	Gerhard Gentzen - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-11-07)
		Gerhard Gentzen - Encyclopedia, History, Geography and Biography
		This page was last modified 17:04, 16 Apr 2005.
		The article about Gerhard Gentzen contains information related to Gerhard Gentzen, References and External link.
	www.arikah.com /encyclopedia/Gentzen (175 words)

	Gerhard Gentzen (Site not responding. Last check: 2007-11-07)
		Gerhard Gentzen (de noviembre el 24 de 1909 - de agosto el 4 de 1945) era matemático alemán y logician.
		Su teorema de la cortar-eliminacio'n es la piedra angular de la semántica prueba-teo'rica, y algunas observaciones filosóficas en sus "investigaciones en la deducción lógica", junto con el aphorism de Wittgenstein que el "significado es uso", constituyen el punto de partida para la semántica deductiva del papel.
		English version: Gerhard Gentzen Next: Condado De Allegany Up
	www.yotor.net /wiki/es/ge/Gerhard%20Gentzen.htm (171 words)

	A systematic proof theory for several modal logics (Site not responding. Last check: 2007-11-07)
		This model theory is generally given using Kripkean frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert style and conditions on the accessibility relation on frames.
		By contrast, the usual structural proof theory of modal logic, as given in Gentzen calculi, is ad-hoc.
		While we can formulate several modal logics in the sequent calculus that enjoy cut-elimination, their formalisation arises through system-by-system fine tuning to ensure that the cut-elimination holds, and the correspondence to the formulation in the Hilbert systems becomes opaque.
	www.linearity.org /cas/papers/sysptf.html (259 words)

	Cut-elimination theorem - Wikipedia, the free encyclopedia
		It was originally proved by Gerhard Gentzen in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively.
		The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule.
		", which is to be read as "A, B, C, proves N, O, P", and (as glossed by Gentzen) should be understood as equivalent to the truth-function "If (A and B and C
	en.wikipedia.org /wiki/Cut-elimination_theorem (398 words)

	Gerhard Gentzen (Site not responding. Last check: 2007-11-07)
		he leaves undiscussed: Gentzen noticed that it is convenient...
		is well known that in Gentzen's calculi, with his rules...
		MAGAZINES IBM Systems Journal 9/1/2002 Sowa, J.F. Conceptual Graph Interchange Format (CGIF) maps directly to and from the display form; and the Knowledge Interchange Format (KIF) maps directly to and from the algebraic notation...
	enciclopedia.cc /Gerhard_Gentzen (292 words)

	Gerhard Gentzen - Unipedia (Site not responding. Last check: 2007-11-07)
		Gerhard Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician.
		Gerhard Gentzen Top 10 Bestselling Search: Gerhard Gentzen
		Kolmogorov, Heyting and Gentzen on the intuitionistic logical constants *.
	www.unipedia.info /Gentzen.html (182 words)

	Re: turing & gentzen (Site not responding. Last check: 2007-11-07)
		According to Premysl Vihan "The Last Months of Gerhard Gentzen in Prague", Collegium Logicum (Annals of the Kurt Godel Society) Vol 1, Gentzen was invited by Hans Rohrbach to a position at the German University in Prague after habilitating as Dozent from Gottingen and release from army service in 1942.
		Gentzen was a member of the SA since 1933 (reaching the rank of Rottenfuhrer), the NSDAP since 1937, and the NSD Dozentenbund since 1941.
		I expect the film's whereabouts, like Gentzen's grave, is unknown.
	www.philo.at /phlo/199805/msg00019.html (152 words)

	Citations: Investigations into logical deduction - Gentzen (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Terms are constructed by induction from variables, parameters, and constant symbols using function symbols in the usual way.
		Gentzen, G., "Investigations into logical deduction," In M.E. abo, editor, The Collected Papers Of Gerhard Gentzen.
		Gentzen G. Investigations into Logical Deduction, in: The Collected Work of Gerhard Gentzen, ed.
	citeseer.lcs.mit.edu /context/312/0 (5245 words)

	Consequence and Confirmation, Peter Roeper, Hugues Leblanc
		Gentzen's account of logical consequence is extended so as to become a matter of degree.
		We characterize and study two kinds of function G, where G(X,Y) takes values between 0 and 1, which represent the degree to which the set X of statements (understood conjunctively) logically implies the set Y of statements (understood disjunctively).
		[3] Gentzen, G., The Collected Papers of Gerhard Gentzen, edited and translated by M. \mboxSzabo, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1969.
	projecteuclid.org /getRecord?id=euclid.ndjfl/1040149352 (267 words)

	Fantastic Metropolis » Division by Zero (Site not responding. Last check: 2007-11-07)
		In 1936, Gerhard Gentzen provided a proof of the consistency of arithmetic, but to do it he needed to use a controversial technique known as transfinite induction.
		This technique is not among the usual methods of proof, and it hardly seemed appropriate for guaranteeing the consistency of arithmetic.
		What Gentzen had done was prove the obvious by assuming the doubtful.
	www.fantasticmetropolis.com /i/division/5 (638 words)

	Citations: Investigations into Logical Deductions in The Collected Papers of Gerhard Gentzen edited by M - Gentzen ...
		Citations: Investigations into Logical Deductions in The Collected Papers of Gerhard Gentzen edited by M - Gentzen (ResearchIndex)
		Gerhard Gentzen, Investigations into Logical Deductions in The Collected Papers of Gerhard Gentzen edited by M.
		presented such a system, called LK, and proved it to be sound and complete for first order classical logic (see also [3] In this paper, we assume that most of Gentzen s LK inference rules are available as well as several additional derived inference rules which are described in [2] In.
	citeseer.ist.psu.edu /context/747799/0 (206 words)

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