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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 U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herscheid
Herscheid is a municipality in the southern Märkischer Kreis, in North Rhine-Westphalia, Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu .... 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 An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinfurt
Steinfurt (; ) is a city in North Rhine-Westphalia, Germany. It is the capital of the district of Steinfurt. From –1806, it was the capital of the County of Steinfurt. Geography Steinfurt is situated north-west of Münster, North Rhine-Westphalia. Its name, roughly meaning “stony ford”, 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 characterized 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'', e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Ordinal
In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ''φ''0 is any normal function, then for any non-zero ordinal '' ...s to produce countable ordinals even for uncountable arguments, and some of which are " collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments. The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by , and is sometimes denot ... [...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,\dots a_k are hereditarily finite, then so is \. Only sets that can be built by a finite number of applications of these two rules are hereditarily finite. Representation This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets: * \ (i.e. \emptyset, the Neumann ordinal "0") * \ (i.e. \ or \, the Neumann ordinal "1") * \ * \ and then also \ (i.e. \, the Neumann ordinal "2"), * \, \ as well as \, * ... sets represented with 6 bracket pairs, e.g. \. There are six such sets * ... sets represented wi ... [...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] |
Logic
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 wor ... [...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 axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing 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 a ... [...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] |
Consistency Proof
In deductive logic, a consistent theory (mathematical logic), theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no Formula (mathematical logic), 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-form expression, 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 there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an principle of explosion, explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a sy ... [...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. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every Structure (mathematical logic), structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. Completeness theorem By Gödel's completeness theorem, the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the ' can also be viewed as asking for an algorithm to decide whether a given statement is provable 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, ... [...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 determine ... [...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 semantics, semantic truth and syntactic Provability logic, provability in first-order logic. The completeness theorem applies to any first-order Theory (mathematical logic), 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 true in all models is provable". (This does not contradict Gödel's incompleteness theorem, which is about a formula φu that is unprovable in a certain theory ''T'' but true in the "standard" model of the natural numbers: φu is false in some other, "non-standard" models of ''T''.) The completeness theorem makes a close link between model theory, which deals with what is true in different models, and proof theory, which studies what can be formally prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
