Translator Disclaimer
2017 Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field
Patrick Reeder
Notre Dame J. Formal Logic 58(2): 205-214 (2017). DOI: 10.1215/00294527-3839208

Abstract

The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to its real part. We interpret this asymmetry to mean that the nilpotent infinitesimal values of Giordano’s ring are “smaller” than the hyperreal infinitesimals. By viewing things from the “point of view” of the hyperreals, all nilpotents are zero, whereas by viewing things from the “point of view” of Giordano’s ring, nonnilpotent, nonzero infinitesimals register as nonzero infinitesimals. This suggests that Giordano’s infinitesimals are more fine-grained.

Citation

Download Citation

Patrick Reeder. "Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field." Notre Dame J. Formal Logic 58 (2) 205 - 214, 2017. https://doi.org/10.1215/00294527-3839208

Information

Received: 10 September 2013; Accepted: 7 July 2014; Published: 2017
First available in Project Euclid: 20 January 2017

zbMATH: 1370.26058
MathSciNet: MR3634976
Digital Object Identifier: 10.1215/00294527-3839208

Subjects:
Primary: 13J25, 26E35
Secondary: 03H10

Rights: Copyright © 2017 University of Notre Dame

JOURNAL ARTICLE
10 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.58 • No. 2 • 2017
Back to Top