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Computable Model Theory
Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditions can be shown to exist. Computable model theory was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and Soviet Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ... during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of computability are studied as well: computability notions weaker than Turing machines are studied in automata theory, while computability notions stronger than Turing machines are studied in the field of hypercomputation. Problems A central idea in computability is that of a (computational) problem, which is a task whose computability can be explored. There are two key types of problems: * A decision problem fixes a set ''S'', which may be a set of strings, natural numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., federal district, five major unincorporated territories, nine United States Minor Outlying Islands, Minor Outlying Islands, and 326 Indian reservations. The United States is also in Compact of Free Association, free association with three Oceania, Pacific Island Sovereign state, sovereign states: the Federated States of Micronesia, the Marshall Islands, and the Palau, Republic of Palau. It is the world's List of countries and dependencies by area, third-largest country by both land and total area. It shares land borders Canada–United States border, with Canada to its north and Mexico–United States border, with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations. With a population of over 333 million, it is the List of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Australia
Australia, officially the Commonwealth of Australia, is a sovereign ''Sovereign'' is a title which can be applied to the highest leader in various categories. The word is borrowed from Old French , which is ultimately derived from the Latin , meaning 'above'. The roles of a sovereign vary from monarch, ruler or ... country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. With an area of , Australia is the largest country by area in Oceania and the world's sixth-largest country. Australia is the oldest, flattest, and driest inhabited continent, with the least fertile soils. It is a megadiverse country, and its size gives it a wide variety of landscapes and climates, with deserts in the centre, tropical Forests of Australia, rainforests in the north-east, and List of mountains in Australia, mountain ranges in the south-east. The ancestors of Aboriginal Australians began arriving from south east Asia approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a Federation, federal union of Republics of the Soviet Union, fifteen national republics; in practice, both Government of the Soviet Union, its government and Economy of the Soviet Union, its economy were highly Soviet-type economic planning, centralized until its final years. It was a one-party state governed by the Communist Party of the Soviet Union, with the city of Moscow serving as its capital as well as that of its largest and most populous republic: the Russian Soviet Federative Socialist Republic, Russian SFSR. Other major cities included Saint Petersburg, Leningrad (Russian SFSR), Kyiv, Kiev (Ukrainian Soviet Socialist Republic, Ukrainian SSR), Minsk (Byelorussian Soviet Socialist Republic, Byelorussian SSR), Tas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across eleven time zones and shares land boundaries with fourteen countries, more than any other country but China. It is the world's ninth-most populous country and Europe's most populous country, with a population of 146 million people. The country's capital and largest city is Moscow, the largest city entirely within Europe. Saint Petersburg is Russia's cultural centre and second-largest city. Other major urban areas include Novosibirsk, Yekaterinburg, Nizhny Novgorod, and Kazan. The East Slavs emerged as a recognisable group in Europe between the 3rd and 8th centuries CE. Kievan Rus' arose as a state in the 9th century, and in 988, it adopted Orthodox Christianity from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cold War
The Cold War is a term commonly used to refer to a period of Geopolitics, geopolitical tension between the United States and the Soviet Union and their respective allies, the Western Bloc and the Eastern Bloc. The term ''Cold war (term), cold war'' is used because there was no large-scale fighting directly between the two superpowers, but they each supported major regional conflicts known as proxy wars. The conflict was based around the ideological and geopolitical struggle for global influence by these two superpowers, following their temporary Allies of World War II, alliance and victory against Nazi Germany and Empire of Japan, Imperial Japan in 1945. Aside from the Nuclear arms race, nuclear arsenal development and conventional military deployment, the struggle for dominance was expressed via indirect means such as psychological warfare, propaganda campaigns, Cold War espionage, espionage, far-reaching Economic sanctions, embargoes, rivalry at sports events, and technolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vaught Conjecture
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified. Statement of the conjecture Let T be a first-order, countable, complete theory with infinite models. Let I(T, \alpha) denote the number of models of ''T'' of cardinality \alpha up to isomorphism, the spectrum of the theory T. Morley proved that if ''I''(''T'', ℵ0) is infinite then it must be ℵ0 or ℵ1 or the cardinality of the continuum. The Vaught conjecture i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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