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Class Logic
Class logic is a logic in its broad sense, whose objects are called classes. In a narrower sense, one speaks of a class logic only if classes are described by a property of their elements. This class logic is thus a generalization of set theory, which allows only a limited consideration of classes. Class logic in the strict sense The first class logic in the strict sense was created by Giuseppe Peano in 1889 as the basis for his arithmetic (Peano Axioms). He introduced the class term, which formally correctly describes classes through a property of their elements. Today the class term is denoted in the form , where A(x) is an arbitrary statement, which all class members x meet. Peano axiomatized the class term for the first time and used it fully. Gottlob Frege also tried establishing the arithmetic logic with class terms in 1893; Bertrand Russell discovered a conflict in it in 1902 which became known as Russell's paradox. As a result, it became generally known that you can not saf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
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 usual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burali-Forti
Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician, after whom the Burali-Forti paradox is named. Biography Burali-Forti was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time he discovered a theorem which Bertrand Russell later realised contradicted a previously proved result by Georg Cantor. The contradiction came to be called the Burali-Forti paradox of Cantorian set theory. He died in Turin. Books by C. Burali-Forti Analyse vectorielle générale: Applications à la mécanique et à la physique.with Roberto Marcolongo (Mattéi & co., Pavia, 1913). Corso di geometria analitico-proiettiva per gli allievi della R. Accademia Militare(G. B. Petrini di G. Gallizio, Torino, 1912). Geometria descrittiva(S. Lattes & c., Torino, 1921). Introduction à la géométrie différentielle, suivant la méthode de H. Grassmann(Gauthier-Villars, 1897). Lezioni Di Geometria Metrico-Proiettiva(Fratelli Bocca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Menne
Albert may refer to: Companies * Albert (supermarket), a supermarket chain in the Czech Republic * Albert Heijn, a supermarket chain in the Netherlands * Albert Market, a street market in The Gambia * Albert Productions, a record label * Albert Computers, Inc., a computer manufacturer in the 1980s Entertainment * ''Albert'' (1985 film), a Czechoslovak film directed by František Vláčil * ''Albert'' (2015 film), a film by Karsten Kiilerich * ''Albert'' (2016 film), an American TV movie * ''Albert'' (Ed Hall album), 1988 * "Albert" (short story), by Leo Tolstoy * Albert (comics), a character in Marvel Comics * Albert (''Discworld''), a character in Terry Pratchett's ''Discworld'' series * Albert, a character in Dario Argento's 1977 film ''Suspiria'' Military * Battle of Albert (1914), a WWI battle at Albert, Somme, France * Battle of Albert (1916), a WWI battle at Albert, Somme, France * Battle of Albert (1918), a WWI battle at Albert, Somme, France People * Albert (given n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Comprehension
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \implies A = B) or in words: : Given any set ''A'' and any set ''B'', if for every set ''X'', ''X'' is a member of ''A'' if and only if ''X'' is a member of ''B'', then ''A'' is equal to ''B''. :(It is not really essential that ''X'' here be a ''set'' — but in ZF, everything is. See Ur-elements below for when this is violated.) The converse, \forall A \, \forall B \, (A = B \implies \forall X \, (X \in A \iff X \in B) ), of this axiom follows from the substitution property of equality. Interpretation To understand this axiom, note that the clau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Set Theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments. Method A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. The words ''and'', ''or'', ''if ... then' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Oberschelp
Arnold Oberschelp (born 5 February 1932 in Recklinghausen) is a German mathematician and logician. He was for many years professor of logic and in Kiel. Life Oberschelp studied mathematics and physics at the universities of Göttingen and Münster. In Münster he received in December 1957 his doctorate in mathematical logic under Hans Hermes.Record in Kiel University's academic database In 1958 he was a research assistant at the Mathematical Institute of the Technical College of Hannover (now ) where he habilitated in mathematics in 1961. In 1968, he accepted an appointment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Barkley Rosser
John Barkley Rosser Sr. (December 6, 1907 – September 5, 1989) was an American logician, a student of Alonzo Church, and known for his part in the Church–Rosser theorem, in lambda calculus. He also developed what is now called the "Rosser sieve", in number theory. He was part of the mathematics department at Cornell University from 1936 to 1963, chairing it several times. He was later director of the Army Mathematics Research Center at the University of Wisconsin–Madison and the first director of the Communications Research Division of IDA. Rosser also authored mathematical textbooks. In 1936, he proved Rosser's trick, a stronger version of Gödel's first incompleteness theorem, showing that the requirement for ω-consistency may be weakened to consistency. Rather than using the liar paradox sentence equivalent to "I am not provable," he used a sentence that stated "For every proof of me, there is a shorter proof of my negation". In prime number theory, he proved Rosser' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
