In mathematics, model theory is the study of classes of mathematical structures from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components. A model of a theory is a structure. Model theory recognizes and is intimately concerned with a duality: it examines semantical elements by means of syntactical elements of a corresponding language: in a summary definition, dating from 1973, universal algebra + logic = model theory. Model theory developed during the 1990s, a more modern definition is provided by Wilfrid Hodges: model theory = algebraic geometry − fields. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, non-standard analysis. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics and computer science; the most prominent professional organization in the field of model theory is the Association for Symbolic Logic.
This article focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. However, a great deal of study has been done in such logics. Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, Robinson's development of non-standard analysis.
An important step in the evolution of classical model theory occurred with the birth of stability theory, which developed a calculus of independence and rank based on syntactical conditions satisfied by theories. During the last several decades applied model theory has merged with the more pure stability theory; the result of this synthesis is called geometric model theory in this article. An example of a proof from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields; the ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory. Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article is an informal introduction which consists of examples of the way these terms are used.
The standard signature of rings is σring =, where × and + are binary, − is unary, 0 and 1 are nullary. The standard signature of semirings is σsmr =; the standard signature of groups is σgrp =, where × is binary, −1 is unary and 1 is nullary. The standard signature of monoids is σmnd =. A ring is a σring-structure which satisfies the identities u + = + w, u + v = v + u, u + 0 = u, u + = 0, u × = × w, u × 1 = u, 1 × u = u, u × = + and × u = +. A group is a σgrp-structure which satisfies the identities u × = × w, u × 1 = u, 1 × u = u, u × u−1 = 1 and u−1 × u = 1. A monoid is 1 × u = u. A semigroup is a - structure. A magma is just a -structure; this is a efficient way to define most classes of algebraic structures, because there is the concept of σ-homomorphism, which specializes to the usual notions of homomorphism for groups, semigroups and rings. For this to work, the signature must be chosen well. Terms such as the σring-term t given by + are used to define identities t = t', but to construct free algebras.
An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities. Birkhoff's theorem states: A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, direct products. An important non-trivial tool in universal algebra are ultraproducts Π i ∈ I A i / U, where I is an infinite set indexing a system of σ-structures Ai, U is an ultrafilter on I. While model theory is considered a part of mathematical logic, universal algebra, which grew out of Alfred North Whitehead's work on abstract algeb
The coat of arms of the London Borough of Bromley is the official coat of arms of the London Borough of Bromley, granted on 20 April 1965. This London Borough was created by merging five earlier entities, the Municipal Borough of Beckenham, the Municipal Borough of Bromley, the Orpington Urban District, the Penge Urban District and a part of the Chislehurst and Sidcup Urban District; some of these had arms. Instead of combining the older arms to a new more complex design, it was decided to create all new arms for Bromley to allow for a simpler design; the pierced silver cinquefoil in the shield alludes to the five earlier authorities with its five petals, while the green field stands for this wooded and rural part of London. There are different interpretations of the acorns surrounding the cinquefoil, they may refer to the characteristic Kentish oaks, represent the seed of the new London Borough, but may stand for the many semi-rural villages in the area; the crest has crossed swords for the many military establishments within Bromley's boundaries and a scallop, present in the crest in the arms of the old Municipal Borough of Bromley and comes from the arms of the Diocese of Rochester, owner of the manor of Bromley from the reign of King Ethelbert.
The supporters are a silver dragon, similar to the dragon supporters in the coat of arms of the City of London, since Bromley is now part of Greater London, the white horse of Kent from the coat of arms of the Kent County Council, for the county in which the London Borough was situated. Arms: Vert a pierced Cinqefoil Argent within an orle of Acorns Or. Crest: On a Wreath of the Colours two Swords in saltire Gules ensigned by an Escallop Or. Supporters: On the dexter side a Dragon and on the sinister side a Horse both Argent. Motto:'SERVIRE POPULO' - To serve the people
Ambly-Fleury is a commune in the Ardennes department in the Grand Est region of northern France. Ambly-Fleury is located some 10 km east by south-east of Rethel and some 23 km north-west of Vouziers. Access to the commune is by road D983 from Seuil in the west passing through the heart of the commune just south of the village and continuing east to Givry; the D45 minor road comes from near Amagne in the north-west through the village south to Mont-Laurent. Apart from the village there are the hamlets of Ambly-Haut and Fleury on the D983 to the east of the village; the commune consists of farmland. The Canal des Ardennes passes through the heart of the commune parallel to the D983 in the west continuing north-east out of the commune; the Aisne river passes through the commune from the west passing to the north of the village meandering though the commune and forming part of the northern border. The Ruisseau de Saulces Champenoises flows from the south through the commune to join the Aisne. List of Successive Mayors The Church of Saint-Pierre-aux-Liens contains a number of items which are registered as historical objects: A Statue: Saint Paul A Statue: Saint Pierre A Group Sculpture: The Nativity A Bas-relief: The Crucifixion Communes of the Ardennes department Ambly-Fleury on the old National Geographic Institute website Ambly-Fleury on Lion1906 Ambly-Fleury on Google Maps Ambly-Fleury on Géoportail, National Geographic Institute website Ambly sur Aisne and Fleury on the 1750 Cassini Map Ambly-Fleury on the INSEE website INSEE
Santa Barbara National Forest was established as the Santa Barbara Forest Reserve by the General Land Office in California on December 22, 1903 with 1,838,323 acres by consolidation of Pine Mountain and Zaka Lake and Santa Ynez Forest Reserves. It included areas of the San Rafael Mountains and Santa Ynez Mountains. After the transfer of federal forests to the U. S. Forest Service in 1905, it became a U. S. National Forest on March 4, 1907. On July 1, 1910, San Luis National Forest was added. On August 18, 1919 Monterey National Forest was added. On December 3, 1936 the name was changed to Los Padres National Forest. Forest History Society Listing of the National Forests of the United States and Their Dates Text from Davis, Richard C. ed. Encyclopedia of American Forest and Conservation History. New York: Macmillan Publishing Company for the Forest History Society, 1983. Vol. II, pp. 743-788
James Edward Bullbrook was a politician from Ontario, Canada. He was a Liberal member in the Legislative Assembly of Ontario representing the riding of Sarnia in from 1967 to 1977. Bullbrook was born in North Bay, Ontario in 1927, he attended law school and graduated in 1953. He married Joyce, in that same year, they moved to Toronto and later settled in Sarnia, Ontario. He and Joyce raised four children. Bullbrook served as a member of Sarnia Town Council, he ran for provincial office in 1967 as the Liberal candidate in the new riding of Sarnia. He defeated Progressive Conservative candidate Ralph Knox by 1,073 votes, he was re-elected in 1971 and 1975. He returned to his law practice in Sarnia, he died of a heart attack on October 27, 1978 at the age of 51. Ontario Legislative Assembly parliamentary history
Dirk Klingenberg is a former professional water polo player from Germany. Klingenberg is counted as the best water polo player in Germany in the 1990s, he started playing for the ASC Duisburg. A year he moved to the Duisburger SV 98. After 10 years he left his hometown in 1991 and moved to Berlin, where he played for the Wasserfreunde Spandau 04, he won 17 national titles during his time in Berlin. He became the top-scorer of the water polo Bundesliga twice; as a member of team Germany he competed in 190 international games. He was able to achieve his goal to participate in the Olympic Games in 1996, he was team speaker of the water polo national team in 1998, was selected for the World-All-Star-Team in 1999. Because of personal and job-related reasons, Klingenberg began to end his career in Düsseldorf, where he led the Düsseldorfer SC in the first league in 1999/2000. Achievements with Wasserfreunde Spandau German Champions: 1992, 1994, 1995, 1996, 1997, 1998, 1999 German Cup: 1992, 1994, 1995, 1996, 1997, 1999 German Super Cup: 1997Achievements with the national team European Junior Championship: 3rd place Junior World Cup: 3rd place European Championship: 7th place, 9th place, 3rd place Good-Will-Games: 2nd place World Cup: 9th place Olympic games: 9th place Besides his activities as a competitive sportsman, Klingenberg always focused on his occupational career.
From 1999 to 2008 he worked in sports marketing as the head of marketing communications and sporting events of the Deutsche Telekom AG. He founded his own agency Klingenberg Sportconsulting GmbH in 2009, worked from on as an independent consultant; the main fields of activity are strategic consulting and planning for companies and brands, as well as creative event and project management. He works as a lecturer at a media-university in Cologne, as a consultant of the water polo Team Germany. Since March 2014, the Berlin agency group Exit-Network Holding has held a stake in the Cologne-based sports marketing agency Klingenberg Sportconsulting GmbH. At the same time Dirk Klingenberg moves into the management of the subsidiary Exit-Media GmbH. After four years the contract has been terminated by mutual agreement and Dirk Klingenberg has left Exit-Sports GmbH as a partner. Since February 2018 Klingenberg has been working as an independent consultant for sponsoring and sponsorship for various clients.
Dirk Klingenberg was married to Kerstin Klingenberg. After his divorce in 2015, he moved his professional and personal focus to Berlin, his two children Zoe and Mira Klingenberg continue to live in Cologne. Klingenberg Sportsconsulting