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Overspill
In nonstandard analysis, a branch of mathematics, overspill (referred to as ''overflow'' by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any ''internal'' subset ''A'' of *N, if :# 1 is an element of ''A'', and :# for every element ''n'' of ''A'', ''n'' + 1 also belongs to ''A'', then :''A'' = *N If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case. The overspill principle has a number of useful consequences: * The set of standard hyperreals is not internal. * The set of bounded hyperreals is not internal. * The set of infinitesimal hyperreals is not internal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonstandard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definition of limit, limits rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...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 app ... [...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. Eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the "infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. In the 3rd century BC Archimedes used what ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microcontinuity
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is infinitely close to ''f''(''a''). Here ''x'' runs through the domain of ''f''. In formulas, this can be expressed as follows: :if x\approx a then f(x)\approx f(a). For a function ''f'' defined on \mathbb, the definition can be expressed in terms of the halo as follows: ''f'' is microcontinuous at c\in\mathbb if and only if f(hal(c))\subseteq hal(f(c)), where the natural extension of ''f'' to the hyperreals is still denoted ''f''. Alternatively, the property of microcontinuity at ''c'' can be expressed by stating that the composition \text\circ f is constant on the halo of ''c'', where "st" is the standard part function. History The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Goldblatt
__NOTOC__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His doctoral advisor was Max Cresswell. His most popular books are ''Logics of Time and Computation'' and ''Topoi: the Categorial Analysis of Logic''. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis. In 1987 he took "a trip on Einstein's train" to develop hyperbolic orthogonality, the geometry of relativity of simultaneity. He has been Coordinating Editor of The ''Journal of Symbolic Logic'' and a Managing Editor of '' Studia Logica''. He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union. In 2012 the Royal Society of New Zealand awarded him the Jones Medal for lifetime achievement i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
