A ring homomorphism is enough to get nonstandard analysis
type: Item
instance of: scholarly article
title: A ring homomorphism is enough to get nonstandard analysis
zbMATH Open document ID: 1062.03066
author: Vieri Benci
author: Mauro Di Nasso
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-11-25T00:00:00Z
review text: The paper contains an exposition of some fundamental facts of nonstandard analysis based on a special class of ring homomorphisms. The principal notion of the paper is a {good homomorphism}; this is a special surjective homomorphism of \(\mathbb{R}^I\) where \(I\) is a set on a field \(\mathbb{F}.\) By using incomplete good ultrafilters [\textit{H. J. Keisler}, Ann. Math. (2) 79, 338--359 (1964; Zbl 0137.00803), \textit{K. Kunen}, Trans. Am. Math. Soc. 172(1972), 299--306 (1973; Zbl 0263.02033)], it is proved that for every infinite cardinal there exists a good hyper-homomorphism \(\phi:\mathbb{R}^I\rightarrow \mathbb{F}\). It is proved that under the hypothesis \(2^k=k^+\), where \(k\) denotes the succesor cardinal for \(k,\) all hyperreal fields originating from good hyper-homomorphisms on rings \(\mathbb{R}^I\) are isomorphic as ordered fields.
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 26E35
zbMATH DE Number: 2118937
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: Leibniz transfer principle
zbMATH Keywords: hyperreal fields
zbMATH Keywords: hyper-homomorphism
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2118937