Notre Dame Journal of Formal Logic 58 (2):205-214 (2017)

Patrick Reeder
Kenyon College
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.
Keywords nonstandard analysis  nilpotent infinitesimals
Categories (categorize this paper)
DOI 10.1215/00294527-3839208
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 64,159
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

A Cauchy-Dirac Delta Function.Mikhail G. Katz & David Tall - 2013 - Foundations of Science 18 (1):107-123.
Infinitesimals: A Defense.Stacey Lynn Edgar - 1982 - Dissertation, Syracuse University
Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis.Geoffrey Hellman - 2006 - Journal of Philosophical Logic 35 (6):621-651.
Lagrange Lecture: Methodology of Numerical Computations with Infinities and Infinitesimals.Yaroslav Sergeyev - 2010 - Rendiconti Del Seminario Matematico dell'Università E Del Politecnico di Torino 68 (2):95–113.
Forcing in Nonstandard Analysis.Masanao Ozawa - 1994 - Annals of Pure and Applied Logic 68 (3):263-297.
QE Rings in Characteristic P N.Chantal Berline & Gregory Cherlin - 1983 - Journal of Symbolic Logic 48 (1):140 - 162.


Added to PP index

Total views
23 ( #476,443 of 2,454,833 )

Recent downloads (6 months)
1 ( #449,241 of 2,454,833 )

How can I increase my downloads?


My notes