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Skolem
Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled ''Investigations on the Algebra of Logic''. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the wint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Löwenheim–Skolem Theorem
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number ''κ'' it has a model of size ''κ'', and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic. Theorem In its general form, the Löwenheim–Skolem Theorem states that for every signature ''σ'', every infinite ''σ''-structure ''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skolem–Noether Theorem
In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in his paper ''Zur Theorie der assoziativen Zahlensysteme'' (German: ''On the theory of associative number systems'') and later rediscovered by Emmy Noether. Statement In a general formulation, let ''A'' and ''B'' be simple unitary rings, and let ''k'' be the center of ''B''. The center ''k'' is a field since given ''x'' nonzero in ''k'', the simplicity of ''B'' implies that the nonzero two-sided ideal is the whole of ''B'', and hence that ''x'' is a unit. If the dimension of ''B'' over ''k'' is finite, i.e. if ''B'' is a central simple algebra of finite dimension, and ''A'' is also a ''k''-algebra, then given ''k''-algebra homomorphisms :''f'', ''g'' : ''A'' → ''B'', there exists a unit ''b'' in ''B'' such that for all ''a'' in '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axel Thue
Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-called word problem for semigroups or Thue problem, closely related to the halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for .... Ronald V. Book and Friedrich Otto, ''String-rewriting Systems'', Springer, 1993, , p. 36. His only known PhD student was Thoralf Skolem. The esoteric programming language Thue is named after him. Publications * * See also * * * * * * * * References External links Axel Thue private archiveexists at NTNU University LibrarDorabiblioteket 1863 births 1922 deaths 20th-century Norwegian mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with a Cand.Real.degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled ''Zur Theorie der algebraischen Körper'', supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo. Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Profe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norway
Norway, officially the Kingdom of Norway, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. The remote Arctic island of Jan Mayen and the archipelago of Svalbard also form part of the Kingdom of Norway. Bouvet Island, located in the Subantarctic, is a Dependencies of Norway, dependency, and not a part of the Kingdom; Norway also Territorial claims in Antarctica, claims the Antarctic territories of Peter I Island and Queen Maud Land. Norway has a population of 5.6 million. Its capital and largest city is Oslo. The country has a total area of . The country shares a long eastern border with Sweden, and is bordered by Finland and Russia to the northeast. Norway has an extensive coastline facing the Skagerrak strait, the North Atlantic Ocean, and the Barents Sea. The unified kingdom of Norway was established in 872 as a merger of Petty kingdoms of Norway, petty kingdoms and has existed continuously for years. From 1537 to 1814, Norway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's Hilbert program, attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics. History Metamathematical metatheorems about mathematics itself were or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Oslo
The University of Oslo (; ) is a public university, public research university located in Oslo, Norway. It is the List of oldest universities in continuous operation#Europe, oldest university in Norway. Originally named the Royal Frederick University, the university was established in 1811 as the de facto Norwegian continuation of Denmark-Norway's common university, the University of Copenhagen, with which it shares many traditions. It was named for King Frederick VI of Denmark and Norway, and received its current name in 1939. The university was commonly nicknamed "The Royal Frederick's" (''Det Kgl. Frederiks'') before the name change, and informally also referred to simply as ''Universitetet'' (). The university was the only university in Norway until the University of Bergen was founded in 1946. It has approximately 27,700 students and employs around 6,000 people. Its faculties include (Lutheranism, Lutheran) theology (with the Lutheran Church of Norway having been Norway's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norwegian Academy Of Science And Letters
The Norwegian Academy of Science and Letters (, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick University in Christiania was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Trondhjem had no university, but had a learned society, the Royal Norwegian Society of Sciences and Letters, established in 1760. The purpose of a learned society in Christiania was to support scientific studies and aid publication of academic papers. The idea of the Humboldt-inspired university, where independent research stood strong, had overtaken the instrumental view of a university as primarily a means to produce civil servants. The city already had societies for specific professions, for instance the Norwegian Medical Society, which was founded in 1833. However, these societies were open to both academics within medicine as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sandsvær
Sandsvær is a area in Kongsberg, Buskerud county, Norway. The area is identical to the former municipality named Sandsvær. Sandsvær is situated in the valley of Lågendalen on the Numedalslågen river, and is bordered by Kongsberg town in the north, and the county border to Vestfold and Telemark counties in the south. Historically, Sandsvær as a part of ''Sandsvær og Numedal fogderi'' had belonged to Akershus '' amt'' from 7 February 1685. In 1760 ''Sandsvær fogderi'' was moved administratively to Buskerud. The municipality was created as Kongsberg landdistrikt in 1837. The municipality existed until 1 January 1908, when it was split to form the new municipalities Ytre Sandsvær and Øvre Sandsvær. Prior to the split Sandsvær had a population of 5,709. Both municipalities were later incorporated into Kongsberg. The name The Old Norse form of the name was ''Sandshverfi''. The first element is the genitive case of ''sandr'' m 'sand Sand is a granular material compo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bergen, Norway
Bergen (, ) is a city and municipality in Vestland county on the west coast of Norway. Bergen is the second-largest city in Norway after the capital Oslo. By May 2025 the population is 294 029 according to Statistics Norway. The municipality covers and is on the peninsula of Bergenshalvøyen. The city centre and northern neighbourhoods are on Byfjorden, 'the city fjord'. The city is surrounded by mountains, causing Bergen to be called the "city of seven mountains". Many of the extra-municipal suburbs are on islands. Bergen is the administrative centre of Vestland county. The city consists of eight boroughs: Arna, Bergenhus, Fana, Fyllingsdalen, Laksevåg, Ytrebygda, Årstad, and Åsane. Trading in Bergen may have started as early as the 1020s. According to tradition, the city was founded in 1070 by King Olav Kyrre and was named Bjørgvin, 'the green meadow among the mountains'. It served as Norway's capital in the 13th century, and from the end of the 13th century b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
