Nonstandard Analysis Axiomatically

Author : Vladimir Kanovei,Michael Reeken
Publisher : Springer Science & Business Media
Page : 421 pages
File Size : 45,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662089989

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Nonstandard Analysis, Axiomatically by Vladimir Kanovei,Michael Reeken Pdf

In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.

Author : Abraham Robinson
Publisher : Princeton University Press
Page : 308 pages
File Size : 49,7 Mb
Release : 2016-08-11
Category : Mathematics
ISBN : 9781400884223

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Non-standard Analysis by Abraham Robinson Pdf

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body.

Author : Alain Robert
Publisher : Courier Corporation
Page : 184 pages
File Size : 40,7 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 0486432793

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Nonstandard Analysis by Alain Robert Pdf

This concise text is based on the axiomatic internal set theory approach. Theoretical topics include idealization, standardization, and transfer, real numbers and numerical functions, continuity, differentiability, and integration. Applications cover invariant means, approximation of functions, differential equations, more. Exercises, hints, and solutions. "Mathematics teaching at its best." — European Journal of Physics. 1988 edition.

Author : Vladimir Kanovei,Michael Reeken
Publisher : Unknown
Page : 428 pages
File Size : 41,8 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662089998

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Nonstandard Analysis, Axiomatically by Vladimir Kanovei,Michael Reeken Pdf

Author : John L. Bell
Publisher : Cambridge University Press
Page : 7 pages
File Size : 54,9 Mb
Release : 2008-04-07
Category : Mathematics
ISBN : 9780521887182

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A Primer of Infinitesimal Analysis by John L. Bell Pdf

A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.

Author : Imme van den Berg,Vitor Neves
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 48,7 Mb
Release : 2007-12-03
Category : Mathematics
ISBN : 9783211499054

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The Strength of Nonstandard Analysis by Imme van den Berg,Vitor Neves Pdf

This book reflects the progress made in the forty years since the appearance of Abraham Robinson’s revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

Author : Martin Väth
Publisher : Springer Science & Business Media
Page : 255 pages
File Size : 52,6 Mb
Release : 2007
Category : Mathematics
ISBN : 9783764377731

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Nonstandard Analysis by Martin Väth Pdf

This book introduces Robinson's nonstandard analysis, an application of model theory in analysis. Unlike some texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but points to some applications in more advanced analysis. The contents proceed from a discussion of the preliminaries to Nonstandard Models; Nonstandard Real Analysis; Enlargements and Saturated Models; Functionals, Generalized Limits, and Additive Measures; and finally Nonstandard Topology and Functional Analysis. No background in model theory is required, although some familiarity with analysis, topology, or functional analysis is useful. This self-contained book can be understood after a basic calculus course.

Author : Nader Vakil
Publisher : Cambridge University Press
Page : 587 pages
File Size : 54,8 Mb
Release : 2011-02-17
Category : Mathematics
ISBN : 9781107002029

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Real Analysis Through Modern Infinitesimals by Nader Vakil Pdf

A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.

Author : Nigel Cutland
Publisher : Cambridge University Press
Page : 365 pages
File Size : 53,6 Mb
Release : 1988-09-30
Category : Mathematics
ISBN : 9780521351096

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Nonstandard Analysis and Its Applications by Nigel Cutland Pdf

This textbook is an introduction to non-standard analysis and to its many applications. Non standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology. The book arises from a conference held in July 1986 at the University of Hull which was designed to provide both an introduction to the subject through introductory lectures, and surveys of the state of research. The first part of the book is devoted to the introductory lectures and the second part consists of presentations of applications of NSA to dynamical systems, topology, automata and orderings on words, the non- linear Boltzmann equation and integration on non-standard hulls of vector lattices. One of the book's attractions is that a standard notation is used throughout so the underlying theory is easily applied in a number of different settings. Consequently this book will be ideal for graduate students and research mathematicians coming to the subject for the first time and it will provide an attractive and stimulating account of the subject.

Author : Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson
Publisher : Springer Science & Business Media
Page : 392 pages
File Size : 44,5 Mb
Release : 1997-04-30
Category : Mathematics
ISBN : 079234586X

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Nonstandard Analysis by Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson Pdf

1 More than thirty years after its discovery by Abraham Robinson , the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum,as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals. However, while the range of applications is broad, certain fundamental themes re cur. The nonstandard framework allows many informal ideas (that could loosely be described as idealisation) to be made precise and tractable. For example, the real line can (in this framework) be treated simultaneously as both a continuum and a discrete set of points; and a similar dual ap proach can be used to link the notions infinite and finite, rough and smooth. This has provided some powerful tools for the research mathematician - for example Loeb measure spaces in stochastic analysis and its applications, and nonstandard hulls in Banach spaces. The achievements of NSA can be summarised under the headings (i) explanation - giving fresh insight or new approaches to established theories; (ii) discovery - leading to new results in many fields; (iii) invention - providing new, rich structures that are useful in modelling and representation, as well as being of interest in their own right. The aim of the present volume is to make the power and range of appli cability of NSA more widely known and available to research mathemati cians.

Author : Robert Goldblatt
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461206156

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Lectures on the Hyperreals by Robert Goldblatt Pdf

An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.

Author : A. E. Hurd
Publisher : Unknown
Page : 228 pages
File Size : 49,7 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662165449

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Nonstandard Analysis - Recent Developments by A. E. Hurd Pdf

Author : Vladislav Ėlievich Li︠a︡nt︠s︡e,Taras Kudryk
Publisher : Unknown
Page : 264 pages
File Size : 44,9 Mb
Release : 1997
Category : Generalized spaces
ISBN : UOM:39015055925864

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Introduction to Nonstandard Analysis by Vladislav Ėlievich Li︠a︡nt︠s︡e,Taras Kudryk Pdf

In Nonstandard Analysis (briefly NSA) there was solved the old problem of substantiation of differential and integral calculus with application of infinitesimals. (This problem seemed to be unsolvable from the times of Leibniz and Euler.) NSA has changed the face of the whole of Mathematics: it is a new mathematical outlook. It is necessary to emphasise that NSA does not object, does not contradict to the Ordinary Mathematics (briefly OM). NSA extends, supplements OM. (This means that all objects, which exist in OM, exist in NSA, too, and all statements which are true in OM retain to be true in NSA.) NSA often simplifies OM and makes it more transparent. NSA states new mathematical theorems and problems. In fact, NSA is a work of the only scientist, namely A. Robinson (1961). His approach to NSA was constructive. In this book we have chosen an axiomatic approach, due to E. Nelson (1977), which is less difficult to learn and apply. (Our exposition, contrary to that of Nelson, is not always strictly logical. Our aim is only some popularization of NSA, and not its foundations.) The text includes some own results of authors. Good supplements to this book are (D-R], [Die], [Lut], [Dav], [Alb], [Cut].

Author : Abraham Robinson
Publisher : Princeton University Press
Page : 318 pages
File Size : 44,6 Mb
Release : 1974
Category : Mathematics
ISBN : 0691044902

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Non-standard Analysis by Abraham Robinson Pdf

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body.

Author : Nigel J. Cutland,Mauro Di Nasso,David A. Ross
Publisher : Cambridge University Press
Page : 248 pages
File Size : 45,8 Mb
Release : 2017-03-30
Category : Mathematics
ISBN : 9781108621298

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Nonstandard Methods and Applications in Mathematics by Nigel J. Cutland,Mauro Di Nasso,David A. Ross Pdf

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the twenty-fifth publication in the Lecture Notes in Logic series, grew from a conference on Nonstandard Methods and Applications in Mathematics held in Pisa, Italy from 12–16 June, 2002. It contains ten peer-reviewed papers that aim to provide something more timely than a textbook, but less ephemeral than a conventional proceedings. Nonstandard analysis is one of the great achievements of modern applied mathematical logic. These articles consider the foundations of the subject, as well as its applications to pure and applied mathematics and mathematics education.