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Non-well-founded Set Theory
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until the book Non-Well-Founded Sets by Peter Aczel introduces hyperset theory in 1988. The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science ( process algebra and final semantics), linguistics and natural language semantics (situation theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-standard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definition of limit, limits rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Boffa
Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean *Maurice, Iowa, a city *Maurice, Louisiana, a village *Maurice River, a tributary of the Delaware River in New Jersey Other uses * ''Maurice'' (2015 film), a Canadian short drama film * Maurice (horse), a Thoroughbred racehorse * ''Maurice'' (novel), a 1913 novel by E. M. Forster, published in 1971 ** ''Maurice'' (1987 film), a British film based on the novel * ''Maurice'' (Shelley), a children's story by Mary Shelley *Maurice, a character from the Madagascar ''franchise'' *Maurices, an American retail clothing chain *Maurice or Maryse, a type of cooking spatula See also *Church of Saint Maurice (other) * *Maurice Debate, a 1918 debate in the British House of Commons *Maurice Lacroix, Swiss manufacturer of mechanical timepieces, clocks, and watches *Mauricie, Quebec, Canada *Moritz (other) *Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scott's Anti-foundation Axiom
Scotts or Scott's may refer to: Businesses and brands *Scott's (restaurant), in London * Scott's Food & Pharmacy, an American supermarket chain *Scotts Miracle-Gro Company, an American multinational corporation * Scott's Porage Oats, a Scottish breakfast cereal *Scotts Shipbuilding and Engineering Company, a Scottish shipbuilding company 1711–1993 Places * Scotts, Michigan, U.S. * Scotts, North Carolina, U.S. *Scotts Valley, California, U.S. Other uses * Scotts (band), a Swedish music group * " The Scotts", a 2020 song by The Scotts (Travis Scott and Kid Cudi) See also * * * Scots (other) * Scotch (other) * Scottish (other) Scottish usually refers to something of, from, or related to Scotland, including: *Scottish Gaelic, a Celtic Goidelic language of the Indo-European language family native to Scotland *Scottish English *Scottish national identity, the Scottish ide ... * Scotts Bluff National Monument, in Nebraska, U.S. {{dab, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His work on automata theory earned him the Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has also worked on modal logic, topology, and category theory. Early career He received his B.A. in Mathematics from the University of California, Berkeley, in 1954. He wrote his Ph.D. thesis on ''Convergent Sequences of Complete Theories'' under the supervision of Alonzo Church while at Princeton, and defended his thesis in 1958. Solomon Feferman (2005) writes of this period: After completing his Ph.D. studies, he moved to the University of Chicago, working as an instructor there until 1960. In 1959, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Habilitationsschrift
Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excellence in research, teaching, and further education, which usually includes a dissertation. The degree, sometimes abbreviated ''Dr. habil''. (), ''dr hab.'' (), or ''D.Sc.'' ('' Doctor of Sciences'' in Russia and some CIS countries), is often a qualification for full professorship in those countries. In German-speaking countries it allows the degree holder to bear the title ''PD'' (for ). In a number of countries there exists an academic post of docent, appointment to which often requires such a qualification. The degree conferral is usually accompanied by a public oral defence event (a lecture or a colloquium) with one or more opponents. Habilitation is usually awarded 5–15 years after a PhD degree or its equivalent. Achieving this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Specker
Ernst Paul Specker (11 February 1920, Zürich – 10 December 2011, Zürich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Specker theorem in quantum mechanics, showing that certain types of hidden-variable theories are impossible. He also proved the ordinal partition relation thereby solving a problem of Erdős. Specker received his Ph.D. in 1949 from ETH Zurich ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ran ...,. where he remained throughout his professional career. See also * Specker sequence * Baer–Specker group References External links Biography at the University of St. Andrews Ernst Specker (1920-2011) Martin Fürer, January 25, 2012. * ... [...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 Köllnische 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 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Foundations
In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Definition The well-formed formulas of NF are the standard formulas of propositional calculus with two primitive predicates equality (=) and membership (\in). NF can be presented with only two axiom schemata: * Extensionality: Two objects with the same elements are the same object; formally, 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''. * A restricted axiom schema of comprehension: \ exists for each stratified formula \phi. A formula \phi is said to be ''stratified'' if there exists a function ''f'' from pieces of \phi's syntax to the natural numbers, such that for any atomic subformula x \in y of \phi we have ''f''(''y'') = ''f''(''x'') + 1, whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willard Van Orman Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He was the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978. Quine was a teacher of logic and set theory. He was famous for his position that first-order logic is the only kind worthy of the name, and developed his own system of mathematics and set theory, known as New Foundations. In the philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability argument, an argument for the Philosophy of mathematics#Empiricism, reality of mathematical entities.Colyvan, Mark"Indispensability Arguments in the Philosophy of Mathematics" The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), Edward N. Zalta (ed.). He was the main proponent of the view that philosophy is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
