Notre Dame Journal of Formal Logic

Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field

Patrick Reeder

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 205-214.

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1484902818

Digital Object Identifier
doi:10.1215/00294527-3839208

Mathematical Reviews number (MathSciNet)
MR3634976

Zentralblatt MATH identifier
1370.26058

Subjects
Primary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05] 13J25: Ordered rings [See also 06F25]
Secondary: 03H10: Other applications of nonstandard models (economics, physics, etc.)

Keywords
nilpotent infinitesimals nonstandard analysis

Citation

Reeder, Patrick. Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field. Notre Dame J. Formal Logic 58 (2017), no. 2, 205--214. doi:10.1215/00294527-3839208. https://projecteuclid.org/euclid.ndjfl/1484902818


Export citation

References

  • [1] Giordano, P., “Infinitesimals without logic,” Russian Journal of Mathematical Physics, vol. 17 (2010), pp. 159–91.
  • [2] Giordano, P., “The ring of Fermat reals,” Advances in Mathematics, vol. 225 (2010), pp. 2050–75.
  • [3] Goldblatt, R., Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, vol. 188 of Graduate Texts in Mathematics, Springer, New York, 1998.
  • [4] Jech, T. J., The Axiom of Choice, vol. 75 in Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973.
  • [5] Moerdijk, I., and G. E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, New York, 1991.