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Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herscheid
Herscheid is a municipality in the southern Märkischer Kreis, in North Rhine-Westphalia, Germany. Geography Herscheid is located in the ''Ebbegebirge'' (''"Ebbe Mountains"''), a part of the Sauerland mountains. Altitudes in the municipality extend from 250m above sea level in the valley of the ''Schwarze Ahe'' up to the highest elevation, the high ''Nordhelle''. The municipality covers an area of , of which 58% is forest and 33% is used agriculturally. Most of the area is protected as a nature reserve, the ''Naturpark Ebbegebirge''. Neighbouring places * Werdohl * Plettenberg * Meinerzhagen * Lüdenscheid History The first written document mentioning ''Hertsceido'' dates from 1072 in a charter from Grafschaft Abbey of bishop Anno of Cologne. However, the settlements in what is now the municipal area probably date back to the 4th century. From the 12th century till 1753 the parish of Herscheid formed a court district. During the French occupation (1806 till 1814), the ''Mairi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinfurt
Steinfurt (; Westphalian: ''Stemmert'') is a city in North Rhine-Westphalia, Germany. It is the capital of the district of Steinfurt. From roughly 1100-1806, it was the capital of the County of Steinfurt. Geography Steinfurt is situated north-west of Münster, North Rhine-Westphalia. Its name came into being in 1975 when the two hitherto independent towns Borghorst and Burgsteinfurt amalgamated. Borghorst became a prosperous city due to its flourishing textile industry, whereas Burgsteinfurt has always rather been coined by culture and administration. Tourists of the 19th century passing Burgsteinfurt praised the city as the "Paradise of Westphalia" and "Royal Diamond" (''Königsdiamant'') because of its 75 monumental buildings and moated castle. Neighbouring municipalities Steinfurt borders Ochtrup, Wettringen, Neuenkirchen, Emsdetten, Nordwalde, Altenberge, Laer, Horstmar and Metelen. City division Steinfurt consists of ''Borghorst'' and ''Burgsteinfurt'', each with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditarily Finite Set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set. Formal definition A recursive definition of well-founded hereditarily finite sets is as follows: : ''Base case'': The empty set is a hereditarily finite set. : ''Recursion rule'': If ''a''1,...,''a''''k'' are hereditarily finite, then so is . The set \ is an example for such a hereditarily finite set and so is the empty set \emptyset=\. On the other hand, the sets \ or \ are examples of finite sets that are not ''hereditarily'' finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when = \. Discussion A symbol for the class of hereditarily finite sets is H_, standing for the cardinality of each of its member being smaller than \aleph_0. Whether H_ is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constance Reid
Constance Bowman Reid (January 3, 1918 – October 14, 2010) was the author of several biographies of mathematicians and popular books about mathematics. She received several awards for mathematical exposition. She was not a mathematician but came from a mathematical family—one of her sisters was Julia Robinson, and her brother-in-law was Raphael M. Robinson. Background and education Reid was born in St. Louis, Missouri, the daughter of Ralph Bowers Bowman and Helen (Hall) Bowman. One of her younger sisters was the mathematician Julia Robinson. The family moved to Arizona and then to San Diego when the girls were a few years old. In 1950 she married a law student, Neil D. Reid, with whom she had two children, Julia and Stewart. Reid received a Bachelor of Arts degree from San Diego State University in 1938 and a Master of Education degree from University of California, Berkeley in 1949. She worked as a teacher of English and journalism from 1939 to 1950, and as a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic 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 s ... [...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 usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithm ... [...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] |
Consistency Proof
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entscheidungsproblem
In mathematics and computer science, the ' (, ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is ''universally valid'', i.e., valid in every structure satisfying the axioms. Completeness theorem By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the ' can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the ' is impossible, assuming that the intuitive notion of " effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Completeness Theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If ''T'' is such a theory, and φ is a sentence (in the same language) and every model of ''T'' is a model of φ, then there is a (first-order) proof of φ using the statements of ''T'' as axioms. One sometimes says this as "anything universally true is provable". This does not contradict Gödel's incompleteness theorem, which shows that some formula φu is unprovable although true in the natural numbers, which are a particular model of a first-order theory describing them — φu is just false in some other model of the first-order theory being considered (such as a non-standard model of arithmetic for Peano arithmetic). It makes a close link between model theory that deals with what is true in different models, and proof theory tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
