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Paradoxes Of The Infinite
''Paradoxes of the Infinite'' (German title: ''Paradoxien des Unendlichen'') is a mathematical work by Bernard Bolzano on the theory of sets. It was published by a friend and student, František Přihonský, in 1851, three years after Bolzano's death. The work contained many interesting results in set theory. Bolzano expanded on the theme of Galileo's paradox, giving more examples of correspondences between the elements of an infinite set and proper subset In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...s of infinite sets. In the work he also explained the term ''Menge'', rendered in English as "set", which he had coined and used in several works since the 1830s. References * ''Paradoxes of the Infinite''; trans. by D.A.Steele; London: Routledge, 1950 * (German original) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views. Bolzano wrote in German, his native language. For the most part, his work came to prominence posthumously. Family Bolzano was the son of two pious Catholics. His father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. Career When he was ten years old, Bolzano entered the Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796. Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics. Starting in 1800, he also began studying theology, becoming a Catholic priest in 1804. He was appointed to the new chair of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
František Přihonský
František () is a masculine Czech and Slovak given name. It is a cognate of Francis, Francisco, François and Franz. It can be also surname (feminine: Františková). Notable people with the name include: Given name Arts * Frank Daniel (František Daniel) (1926–1996), Czech film director, producer and screenwriter * František Bartoš (folklorist) (1837–1906), Moravian ethnomusicologist and folklorist *František Bělský, known as Franta Belsky (1921–2000), Czech sculptor * František Bílek (1872–1941), Czech sculptor and architect * František Brikcius, Czech cellist * František Brixi (1732–1771), Czech composer * František Čáp (1913–1972), Czech film director and screenwriter * František Čelakovský (1799–1852), Czech writer and translator * František Čermák (painter) (1822–84), Czech painter * František Doucha (1810–1884), Czech literary translator and writer *František Drdla (1868–1944), Czech violinist and composer * František Drtikol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Galileo's Paradox
Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, '' Two New Sciences'', Galileo Galilei made apparently contradictory statements about the positive integers. First, a square is an integer which is the square of an integer. Some numbers are squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets. Galileo concluded that the ideas of ''less'', ''equal'', and ''greater'' apply to finite quantities but not to infinite quantities. During the nineteenth century Cantor found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Subset
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique can be seen as a consequence of univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad (region), Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Ancient Egypt, Egypt – ''Plimpton 322'' (Babylonian mathematics, Babylonian – 1900 BC),Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", ''Historia Mathematica'', 8, pp. 277–318. the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Literature
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
1851 Essays
Events January–March * January 11 – Hong Xiuquan officially begins the Taiping Rebellion in China, one of the bloodiest revolts that would lead to 20 million deaths. * January 15 – Christian Female College, modern-day Columbia College, receives its charter from the Missouri General Assembly. * January 23 – The flip of a coin, subsequently named the Portland Penny, determines whether a new city in the Oregon Territory will be named after Boston, Massachusetts, or Portland, Maine, with Portland winning. * January 28 – Northwestern University is founded in Illinois. * February 1 – ''Brandtaucher'', the oldest surviving submersible craft, sinks during acceptance trials in the German port of Kiel, but the designer, Wilhelm Bauer, and the two crew escape successfully. * February 6 – Black Thursday occurs in Australia as bushfires sweep across the state of Victoria, burning about a quarter of its area. * February 12 – Edward ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
