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Schröder–Bernstein Theorem
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if and , then ; that is, and are equipotent. This is a useful feature in the ordering of cardinal numbers. The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as Cantor–Bernstein theorem, or Cantor–Schröder–Bernstein, after Georg Cantor who first published it without proof. Proof The following proof is attributed to Julius König. Assume without loss of generality that ''A'' and ''B'' are disjoint. For any ''a'' in ''A'' or ''b'' in ''B'' we can form a unique two-sided sequence of elements that are alternately in ''A'' and ''B'', by repeatedly applying f and g^ to go from ''A'' to ''B'' and g and f^ to go from ''B'' to ''A'' (where defined; the inverses f^ and g^ are understood as partial functi ... [...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] |
Another Proof
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Another or variant may refer to: * anOther or Another Magazine, culture and fashion magazine * ''Another'' (novel), a Japanese horror novel ** ''Another'' (film), a Japanese 2012 live-action film based on the novel * Another River, a river in the U.S. state of Alaska * A. N. Other, a pseudonym See also * Yet another * Indefinite pronoun * English determiners * Other (other) * Others (other) Others or The Others may refer to: Fictional characters * Others (''A Song of Ice and Fire''), supernatural creatures in the fictional world of George R. R. Martin's fantasy series ''A Song of Ice and Fire'' * Others (''Lost''), mysterious inh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alwin Reinhold Korselt
Alwin Reinhold Korselt (17 March 1864, in Mittelherwigsdorf – 4 February 1947, in Plauen) was a German mathematician. He discovered Korselt's criterion, which provides a secondary definition for Carmichael numbers and also contributed an early result in algebraic logic. Personal life The Korselts are a huge, widespread family that has been resident in the village of Mittelherwigsdorf near Zittau in Saxony (nowadays close to the Czech and Polish borders) since the early Middle Ages. Alwin Korselt was born there in 1864. After attending '' Gymnasium'' in Zittau between 1876 and 1885, he studied mathematics and physics in Leipzig (with one semester in 1886 in Freiburg im Breisgau) until 1890. After a probationary year as a teacher at Nikolaigymnasium in Leipzig he taught from 1891 to 1898 at various schools in Pirna, Dresden, Keilhau near Rudolstadt, Löbau and Meerane. Frequent changes, often between very different school types, indicate that he found it difficult to gain ground ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be renamed as the IMU Abacus Medal), the Carl Friedrich Gauss Prize, Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being List of International Congresses of Mathematicians Plenary and Invited Speakers, invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review ''CMS Notes'', vol 31, no. 3, April 1999 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ... and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was born in Saint-Affrique, Aveyron, the son of a Protestant pastor. He studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure (Paris), École normale supérieure and the École Polytechnique. He qualified in the first position for both and chose to attend the former institution in 1889. That year he also won the concours général, an annual national mathematics competition. After graduating in 1892, he placed first in the agrégati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jahresbericht Der Deutschen Mathematiker-Vereinigung
The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). It was founded in 1890 in Bremen with the set theorist Georg Cantor as first president. Founding members included Georg Cantor, Felix Klein, Walther von Dyck, David Hilbert, Hermann Minkowski, Carl Runge, Rudolf Sturm, Hermann Schubert, and Heinrich Weber. The current president of the DMV is Ilka Agricola (2021–2022). Activities In honour of its founding president, Georg Cantor, the society awards the Cantor Medal. The DMV publishes two scientific journals, the ''Jahresbericht der DMV'' and ''Documenta Mathematica''. It also publishes a quarterly magazine for its membership the ''Mitteilungen der DMV''. The annual meeting of the DMV is called the ''Jahrestagung''; the DMV traditionally meets e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Stanley Jevons
William Stanley Jevons (; 1 September 183513 August 1882) was an English economist and logician. Irving Fisher described Jevons's book ''A General Mathematical Theory of Political Economy'' (1862) as the start of the mathematical method in economics. It made the case that economics, as a science concerned with quantities, is necessarily mathematical. In so doing, it expounded upon the "final" (marginal) utility theory of value. Jevons' work, along with similar discoveries made by Carl Menger in Vienna (1871) and by Léon Walras in Switzerland (1874), marked the opening of a new period in the history of economic thought. Jevons's contribution to the marginal revolution in economics in the late 19th century established his reputation as a leading political economist and logician of the time. Jevons broke off his studies of the natural sciences in London in 1854 to work as an assayer in Sydney, where he acquired an interest in political economy. Returning to the UK in 1859, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Moritz Hartogs
Friedrich Moritz "Fritz" Hartogs (20 May 1874 Р18 August 1943) was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables. Life Hartogs was the son of the merchant Gustav Hartogs and his wife Elise Feist and grew up in Frankfurt am Main. He studied at the K̦nigliche Technische Hochschule Hannover, at the Technische Hochschule Charlottenburg, at the University of Berlin, and at the Ludwig Maximilian University of Munich, graduating with a doctorate in 1903 (supervised by Alfred Pringsheim). He did his Habilitation in 1905 and was Privatdozent and Professor in Munich (from 1910 to 1927 extraordinary professor and since 1927 ordinary professor). As a Jew, he suffered greatly under the Nazi regime: he was fired in 1935, was mistreated and briefly interned in KZ Dachau in 1938, and eventually committed suicide in 1943. Work Hartogs main work was in several complex variables where he is known for Hartogs's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method. Life Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied mathematics, physics and philosophy at the University of Berlin, the University of Halle, and the University of Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (''Untersuchungen zur Variationsrechnung''). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
