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Transfer Principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0. History An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity". Here infinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the principle of permanence. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function. In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system. Its most common use is in Abraham Robinson's nonstan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), 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 definable set, 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 stable theory, stability theory. Compared to other areas of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics. Nearly half of Robinson's papers were in applied mathematics rather than in pure mathematics. Biography He was born to a Jewish family with strong Zionist beliefs, in Waldenburg, Germany, which is now Wałbrzych, in Poland. In 1933, he emigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. Robinson was in France when the Nazis invaded during World War II, and escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, which was more detailed than theirs. While in London, he joined the Free French Air Force and contributed to the war effort by teaching himself aerodynamics a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
An Infinitesimal Approach
An, AN, aN, or an may refer to: Businesses and organizations * Airlinair (IATA airline code AN) * Alleanza Nazionale, a former political party in Italy * AnimeNEXT, an annual anime convention located in New Jersey * Anime North, a Canadian anime convention * Ansett Australia, a major Australian airline group that is now defunct (IATA designator AN) * Apalachicola Northern Railroad (reporting mark AN) 1903–2002 ** AN Railway, a successor company, 2002– * Aryan Nations, a white supremacist religious organization * Australian National Railways Commission, an Australian rail operator from 1975 until 1987 * Antonov, a Ukrainian (formerly Soviet) aircraft manufacturing and services company, as a model prefix Entertainment and media * Antv, an Indonesian television network * '' Astronomische Nachrichten'', or ''Astronomical Notes'', an international astronomy journal * '' Avisa Nordland'', a Norwegian newspaper * ''Sweet Bean'' (あん), a 2015 Japanese film also known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Jerome Keisler
Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski at Berkeley; his dissertation is ''Ultraproducts and Elementary Classes'' (1961). Following Abraham Robinson's work resolving what had long been thought to be inherent logical contradictions in the literal interpretation of Leibniz's notation that Leibniz himself had proposed, that is, interpreting "dx" as literally representing an infinitesimally small quantity, Keisler published '' Elementary Calculus: An Infinitesimal Approach'', a first-year calculus textbook conceptually centered on the use of infinitesimals, rather than the epsilon, delta approach, for developing the calculus. He is also known for extending the Henkin construction (of Leon Henkin) to what are now called Henkin–Keisler models. He is also known for the Rudin–K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperinteger
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals. Discussion The standard integer part function: :\lfloor x \rfloor is defined for all real ''x'' and equals the greatest integer not exceeding ''x''. By the transfer principle of nonstandard analysis, there exists a natural extension: :^*\! \lfloor \,\cdot\, \rfloor defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if x = ^*\! \lfloor x \rfloor. Thus the hyperintegers are the image of the integer part function on the hyperreals. Internal sets The set ^*\mathbb of all hyperintegers is an internal subset of the hyperreal line ^*\mathbb. The set of all finite hyperintegers (i.e. \mathbb itself) is not an internal subset. Elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals. Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit ''i'' is . Finite fields cannot be ordered. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields. Definitions There are two equivalent common definitions of an order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers ''x'' and ''y'', there is an integer ''n'' such that ''nx'' > ''y''. It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''On the Sphere and Cylinder''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements are ''compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Quantifier
In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and "∃" in that bounded quantifiers restrict the range of the quantified variable. The study of bounded quantifiers is motivated by the fact that determining whether a sentence with only bounded quantifiers is true is often not as difficult as determining whether an arbitrary sentence is true. Examples Examples of bounded quantifiers in the context of real analysis include: * \forall x > 0 - for all ''x'' where ''x'' is larger than 0 * \exists y 0 \quad \exists y < 0 \quad (x = y^2) - every positive number is the square of a negative number Bounded quantifiers in arithmetic Suppose that ''L'' is the language of Peano a ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false". Classical logic In classical logic, with its intended semantics, the truth values are '' true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding sem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Equivalence
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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Henkin
Leon Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the Group in Logic and the Methodology of Science',Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed. ''The Life and Work of Leon Henkin''. Springer International Publishing. pp. 3-22. . doi:10.1007/978-3-319-09719-0_11. from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defensor of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Set
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets (note that the term "language" is used in a loose sense in the above). Edward Nelson's internal set theory is an axiomatic approach t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
