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Burali-Forti Paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book ''Principles of Mathematics'', he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name. Stated in terms of von Neumann ordinals We will prove this by a deliberational deconstruction. # Let be a set consisting of all ordinal numbers. # is transitive because for every element of (which is an ordinal number and can be any ordinal number) and every element of (i.e. under the definition of Von Neumann ordinals, for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies 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 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 theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell Paradox
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] |
Stanford Encyclopedia Of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from many academic institutions worldwide. Authors contributing to the encyclopedia give Stanford University the permission to publish the articles, but retain the copyright to those articles. Approach and history As of August 5th, 2022, the ''SEP'' has 1,774 published entries. Apart from its online status, the encyclopedia uses the traditional academic approach of most encyclopedias and academic journals to achieve quality by means of specialist authors selected by an editor or an editorial committee that is competent (although not necessarily considered specialists) in the field covered by the encyclopedia and peer review. The encyclopedia was created in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Symbolic Logic
The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zentralblatt MATH, and Scopus. Its 2009 MCQ was 0.28, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... was 0.631. External links * Mathematics journals Publications established in 1936 Multilingual journals Quarterly journals Association for Symbolic Logic academic journals Logic journals Cambridge University Press academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Historia Mathematica
''Historia Mathematica: International Journal of History of Mathematics'' is an academic journal on the history of mathematics published by Elsevier. It was established by Kenneth O. May in 1971 as the free newsletter ''Notae de Historia Mathematica'', but by its sixth issue in 1974 had turned into a full journal. The International Commission on the History of Mathematics began awarding the Montucla Prize, for the best article by an early career scholar in ''Historia Mathematica'', in 2009. The award is given every four years. Editors The editors of the journal have been: * Kenneth O. May, 1974–1977 * Joseph W. Dauben, 1977–1985 * Eberhard Knobloch, 1985–1994 * David E. Rowe, 1994–1996 * Karen Hunger Parshall, 1996–2000 * Craig Fraser and Umberto Bottazzini, 2000–2004 * Craig Fraser, 2004–2007 * Benno van Dalen, 2007–2009 * June Barrow-Green and Niccolò Guicciardini, 2010–2013 * Niccolò Guicciardini and Tom Archibald, 2013-2015 * Tom Archibald and Reinha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Science (journal)
''Philosophy of Science'' is dedicated to the furthering of studies and free discussion from diverse standpoints in the philosophy of science. It is a peer-reviewed academic journal. Official affiliations In January 1934 ''Philosophy of Science'' announced itself as the chief external expression of the Philosophy of Science Association, which seems to have been the expectation of its founder, William Malisoff. The journal is currently the official journal of the Association, which Philipp Frank and C. West Churchman formally constituted in December 1947. Publication history Malisoff, who was independently wealthy, seems to have financed the launch of ''Philosophy of Science''. Correspondingly he became its first editor. In the first issue he sought papers ranging from studies on "the analysis of meaning, definition, symbolism," in scientific theories to those on "the nature and formulation of theoretical principles" and "in the function and significance of science within various ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irving Copi
Irving Marmer Copi (; né Copilovich or Copilowish; July 28, 1917 – August 19, 2002) was an American philosopher, logician, and university textbook author. Biography Copi studied under Bertrand Russell while at the University of Chicago. In 1948 he contributed to the calculus of relations with his article using logical matrices. Copi taught at the University of Illinois, the United States Air Force Academy, Princeton University, and the Georgetown University Logic Institute, before teaching logic at the University of Michigan, 1958–69, and at the University of Hawaii at Manoa, 1969–90. Assigned to teach logic, Copi reviewed textbooks available and decided to write his own. The manuscript was split into ''Introduction to Logic'' and ''Symbolic Logic''. A reviewer noted that it had an "unusually comprehensive chapter on definition" and "The author accounts for the seductive nature of informal fallacies." The textbooks proved popular, and a reviewer of the third edition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendiconti Del Circolo Matematico Di Palermo
The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.The Mathematical Circle of Palermo . Retrieved 2011-06-19. It began accepting foreign members in 1888, and by the time of Guccia's death in 1914 it had become the foremost international mathematical society, with approximately one thousand members. However, subsequently to that time it declined in influence. Publications ''Rendiconti del Circolo Matemat ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Infinite
The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite. Cantor linked the Absolute Infinite with God, Cited as ''Cantor 1883b'' by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, . Original article. and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.''Infinity: New Research and Frontiers'' by Michael Heller and W. Hugh Woodin (2011)p. 11 Cantor's view Cantor said: Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original): The Burali-Forti paradox The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hao Wang (academic)
Hao Wang (; 20 May 1921 – 13 May 1995) was a Chinese-American logician, philosopher, mathematician, and commentator on Kurt Gödel. Biography Born in Jinan, Shandong, in the Republic of China (today in the People's Republic of China), Wang received his early education in China. He obtained a BSc degree in mathematics from the National Southwestern Associated University in 1943 and an M.A. in Philosophy from Tsinghua University in 1945, where his teachers included Feng Youlan and Jin Yuelin, after which he moved to the United States for further graduate studies. He studied logic under W.V. Quine at Harvard University, culminating in a Ph.D. in 1948. He was appointed to an assistant professorship at Harvard the same year. During the early 1950s, Wang studied with Paul Bernays in Zürich. In 1956, he was appointed Reader in the Philosophy of Mathematics at the University of Oxford. In 1959, Wang wrote on an IBM 704 computer a program that in only 9 minutes mechanically pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrestricted Comprehension
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for ''unrestricted'' comprehension, discussed below. Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory. Statement One instance of the schema is included for each formula φ in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in φ. In the formal language of set theory, the axiom schema is: :\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow x \in A \land \varphi(x, w_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
