Notre Dame Journal of Formal Logic

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

Patrick Reeder

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

nilpotent infinitesimals nonstandard analysis


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.

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