Topic
Hyperreal number
About: Hyperreal number is a research topic. Over the lifetime, 30 publications have been published within this topic receiving 330 citations. The topic is also known as: *R & R*.
Papers
TL;DR: In this paper, the authors describe the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis.
Abstract: This paper first describes the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic (HOL). The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis (NSA). The construction of the hyperreal number system has been carried out strictly through the use of definitions to ensure that the foundations of NSA in Isabelle are sound. Mechanizing the construction has required that various number systems including the rationals and the reals be built up first. Moreover, to construct the hyperreals from the reals has required developing a theory of filters and ultrafilters and proving Zorn's lemma, an equivalent form of the axiom of choice. This paper also describes the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis. The current work provides both standard and nonstandard definitions for the various notions, and proves their equivalence in each case. To achieve this aim, systematic methods, through which sets and functions are extended to the hyperreals, are developed in the framework. The merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem-proving are highlighted throughout the exposition.
19 citations
TL;DR: In this paper it is shown that most of the difficulties which block the progress of students trying to learn analysis stem from the fact that they understand little of ordinary algebra, still they attempt this more subtle art.
Abstract: Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra,
12 citations
TL;DR: In this paper, a construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others, which can be used to construct any hyperreal field, whose universe is a set, directly out of integers.
Abstract: A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).
11 citations
TL;DR: In this paper, the idea of a nonstandard electrical network is introduced and used to reestablish Kirchhoff's laws for a fairly broad class of infinite electrical networks, and a fairly brief tutorial on infinitesimals, hyperreal numbers, and the key ideas of nonstandard analysis needed for a comprehension of this paper.
Abstract: Kirchhoff's laws fail to hold in general for infinite electrical networks. Standard calculus is simply incapable of resolving this paradox because it cannot provide the infinitesimals and more generally the hyperreal currents and voltages that such networks often require. However, nonstandard analysis can do precisely this. The idea of a nonstandard electrical network is introduced in this paper and is used to reestablish Kirchhoff's laws for a fairly broad class of infinite electrical networks. The second section herein presents a fairly brief tutorial on infinitesimals, hyperreal numbers, and the key ideas of nonstandard analysis needed for a comprehension of this paper.
9 citations
TL;DR: The hyperreal number system as discussed by the authors provides a framework for dealing with divergent series in a more comprehensive and tractable way than the real number system, which is the basis for this paper.
Abstract: Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within the real number system, but these contradictions are largely alleviated with the hyperreal number system. Hyperreal numbers provide a framework for dealing with divergent series in a more comprehensive and tractable way.
7 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2019 | 2 |
| 2018 | 2 |
| 2015 | 3 |
| 2012 | 4 |
| 2011 | 1 |