|
Robert Vaught
Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a mathematical logician and one of the founders of model theory. Life Vaught was a musical prodigy in his youth, in his case playing the piano. He began his university studies at , at age 16. When broke out, he enlisted into the , which assigned him to the |
Alhambra, California
Alhambra (, , ; from " Alhambra") is a city located in the western San Gabriel Valley region of Los Angeles County, California, United States, approximately eight miles from the Downtown Los Angeles Downtown Los Angeles (DTLA) contains the central business district of Los Angeles. In addition, it contains a diverse residential area of some 85,000 people, and covers . A 2013 study found that the district is home to over 500,000 jobs. It is ... civic center. It was incorporated on July 11, 1903. As of the 2020 census, the population was 82,868. The city's ZIP Codes are 91801 and 91803 (plus 91802 for P.O. boxes). History The original inhabitants of the land where Alhambra now sits are the Tongva. The Mission San Gabriel Arcángel, San Gabriel Mission was founded nearby on September 8, 1771, as part of the Spanish conquest and occupation of Alta California. The land that would later become Alhambra was part of a 300,000 acre land grant given to Manuel Nieto (soldier), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John L
John Lasarus Williams (29 October 1924 – 15 June 2004), known as John L, was a Welsh nationalist activist. Williams was born in Llangoed on Anglesey, but lived most of his life in nearby Llanfairpwllgwyngyll. In his youth, he was a keen footballer, and he also worked as a teacher. His activism started when he campaigned against the refusal of Brewer Spinks, an employer in Blaenau Ffestiniog, to permit his staff to speak Welsh. This inspired him to become a founder of Undeb y Gymraeg Fyw, and through this organisation was the main organiser of ''Sioe Gymraeg y Borth'' (the Welsh show for Menai Bridge using the colloquial form of its Welsh name).Colli John L Williams , '''', 15 June ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feferman–Vaught Theorem
Feferman–Vaught theorem in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of first-order structures to the first-order theory of elements of the structure. The theorem is considered as one of the standard results in model theory. The theorem extends the previous result of Andrzej Mostowski on direct products of theories. It generalizes (to formulas with arbitrary quantifiers) the property in universal algebra that equalities (identities) carry over to direct products of algebraic structures (which is a consequence of one direction of Birkhoff's theorem). Direct product of structures Consider a first-order logic signature ''L''. The definition of product structures takes a family of ''L''-structures \mathbf_i for i \in I for some index set ''I'' and defines the product structure \mathbf = \prod_ \mathbf_i, which is also an ''L''-structure, with all functions and rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vaught Transform
Vaught is a surname, and may refer to: * BC Vaught, drummer for Hed PE * DeAnn Vaught, member of the Arkansas House of Representatives * James B. Vaught, United States Army Lieutenant General * Johnny Vaught, American college football player * Loy Vaught, American basketball player * Robert Lawson Vaught, American mathematical logician See also * Vaughn (other) Vaughn may refer to: Places in the United States *Vaughn, California, former name of Bodfish, California *Vaughn, Montana * Vaughn, New Mexico * Vaughn, Oregon * Vaughn, Pennsylvania * Vaughn, Virginia * Vaughn, Washington Name *Vaughn (surname) ... * Big V {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vaught's Theorem
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 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...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 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saturated Structure
In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \aleph_1-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection. Definition Let ''κ'' be a finite or infinite cardinal number and ''M'' a model in some first-order language. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ... less than ''κ'', the model ''M'' realizes all Type (model theory), complete types over ''A''. The model ''M'' is called saturated if it is , ''M'', -saturated where , ''M'', denote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski–Vaught Test
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Submodel
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Washington
The University of Washington (UW, simply Washington, or informally U-Dub) is a public research university in Seattle, Washington. Founded in 1861, Washington is one of the oldest universities on the West Coast; it was established in Seattle approximately a decade after the city's founding. The university has a 703 acre main campus located in the city's University District, as well as campuses in Tacoma and Bothell. Overall, UW encompasses over 500 buildings and over 20 million gross square footage of space, including one of the largest library systems in the world with more than 26 university libraries, art centers, museums, laboratories, lecture halls, and stadiums. The university offers degrees through 140 departments, and functions on a quarter system. Washington is the flagship institution of the six public universities in Washington state. It is known for its medical, engineering, and scientific research. Washington is a member of the Association of American Universiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tulane University
Tulane University, officially the Tulane University of Louisiana, is a private university, private research university in New Orleans, Louisiana. Founded as the Medical College of Louisiana in 1834 by seven young medical doctors, it turned into a comprehensive public university as the University of Louisiana by the state legislature in 1847. The institution became private under the endowments of Paul Tulane and Josephine Louise Newcomb in 1884 and 1887. Tulane is the 9th oldest private university in the Association of American Universities. The Tulane University Law School and Tulane University Medical School are, respectively, the 12th oldest law school and 15th oldest medical school in the United States. Tulane has been a member of the Association of American Universities since 1958 and is classified among "R1: Doctoral Universities – Very high research activity". Tulane has an overall acceptance rate of 8.4%. Alumni include twelve List of governors of Louisiana, governors o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
