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Gentzen
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 Soviet prison camp in Prague in 1945, having been interned as a German national after the Second World War. 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. 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 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequent Calculus
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems in a first-order language rather than conditional tautologies. Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. * Hilbert style. Every line is an unconditional tautology (or theorem). * Gentzen s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning. Motivation Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise ''Principia Mathematica''. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935. His proposals led to diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents Mathematical proof, proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as Recursive data type, inductively-defined data structures such as list (computer science), lists, boxed lists, or Tree (data structure), trees, which are constructed according to the axioms and rule of inference, rules of inference of the logical system. Consequently, proof theory is syntax (logic), syntactic in nature, in contrast to model theory, which is Formal semantics (logic), semantic in nature. Some of the major areas of proof theory include structural proof theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-elimination Theorem
The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 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 making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule. The cut rule A sequent is a logical expression relating multiple formulas, in the form , which is to be read as proves , and (as glossed by Gentzen) should be understood as equivalent to the truth-function "If (A_1 and A_2 and A_3 …) then (B_1 or B_2 or B_3 …)." Note that the left-hand side (LHS) is a conjunction (and) and the right-hand side (RHS) is a disjunction (or). The LHS may have arbitrarily many or few formulae; when the LH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof-theoretic Semantics
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...s, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference. Overview Gerhard Gentzen is the founder of proof-theoretic semantics, providing the formal basis for it in his account of cut-elimination for the sequent calculus, and some provocative philosophical remarks about locating the meaning of logical connectives in their Natural_deduction#Introduction_and_elimination, introduction rules within natural deduction. The history of proof-theoretic semantics since then has been devo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Bernays
Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. Biography Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849. Bernays spent his childhood in Berlin, and attended the Köllner Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. In 1912, the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Socialist German Lecturers League
The National Socialist German Lecturers League (''Nationalsozialistischer Deutscher Dozentenbund'', also called ''NS-Dozentenbund'' , or abbreviated ''NSDDB''), was a party organization under the NSDAP (the Nazi Party). Origin and purpose The NSDDB emerged in 1935 from the National Socialist Teachers League and was established on the basis of an order of the Deputy Führer Rudolf Hess; its purpose being, the exertion of influence on the universities and the political control of higher education. Massive influence was applied especially on appointments to staff positions.Hentschel, 1996, Appendix C; see the entry for the NSDDB District leaders had a decisive role in the acceptance of an ''Habilitationsschrift'', which was a prerequisite to attaining the rank of ''Privatdozent'' necessary to becoming a university lecturer.Hentschel, 1996, Introduction p. xxxvi ff. The expulsion of the Jewish scientists from the universities was substantially carried out by the activists of the L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl-Ferdinands-Universität
Charles University ( cs, Univerzita Karlova, UK; la, Universitas Carolina; german: Karls-Universität), also known as Charles University in Prague or historically as the University of Prague ( la, Universitas Pragensis, links=no), is the oldest and largest university in the Czech Republic. It is one of the List of oldest universities in continuous operation, oldest universities in Europe in continuous operation. Today, the university consists of 17 faculties located in Prague, Hradec Králové, and Plzeň. Charles University belongs among the top three universities in Central and Eastern Europe. It is ranked around 200–300 in the world. History Medieval university (1349–1419) The establishment of a medieval university in Prague was inspired by Holy Roman Emperor Charles IV, Holy Roman Emperor, Charles IV. He asked his friend and ally, Pope Clement VI, to do so. On 26 January 1347 the pope issued the bull establishing a university in Prague, modeled on the University of Paris, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greifswald
Greifswald (), officially the University and Hanseatic City of Greifswald (german: Universitäts- und Hansestadt Greifswald, Low German: ''Griepswoold'') is the fourth-largest city in the German state of Mecklenburg-Western Pomerania after Rostock, Schwerin and Neubrandenburg. In 2021 it surpassed Stralsund for the first time, and became the largest city in the Pomeranian part of the state. It sits on the River Ryck, at its mouth into the Danish Wiek (''Dänische Wiek''), a sub-bay of the Bay of Greifswald (''Greifswalder Bodden''), which is itself a sub-bay of the Bay of Pomerania (''Pommersche Bucht'') of the Baltic Sea. It is the seat of the district of Western Pomerania-Greifswald, and is located roughly in the middle between the two largest Pomeranian islands of Rugia (''Rügen'') and Usedom. The closest larger cities are Stralsund, Rostock, Szczecin and Schwerin. It lies west of the River Zarow, the historical cultural and linguistic boundary between West (west of the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their model theory, models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematics, metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
