Homomorphisms between rings with infinitesimals and infinitesimal comparisons 

Author 
emanuele.bottazzi@unipv.it; emanuele.bottazzi.phd@gmail.com
University of Pavia, Italy

Abstract 
We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $\,\!^{\bullet}\mathbb{\mathbb{R}}$ are smaller than any infinitesimal hyperreal number of Abraham Robinson's nonstandard extension of the real numbers $\,\!^\ast\mathbb{R}$.
Our approach consists in the study of two canonical orderpreserving homomorphisms taking values in $\,\!^{\bullet}\mathbb{\mathbb{R}}$ and $\,\!^\ast\mathbb{R}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of ``nonnonstandard'' analysis.
The existence of a nonzero element in Henle's ring that is mapped to $0$ in $\,\!^{\bullet}\mathbb{\mathbb{R}}$ while it is seen as a nonzero infinitesimal in $\,\!^\ast\mathbb{R}$ suggests that some infinitesimals in $\,\!^\ast\mathbb{R}$ are smaller than the infinitesimals in $\,\!^{\bullet}\mathbb{\mathbb{R}}$.
We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in $\,\!^{\bullet}\mathbb{\mathbb{R}}$.

Keywords 
canonical orderpreserving homomorphism; Henle’s ring; hyperreal number; nonstandard analysis

DOI 
doi:10.30970/ms.52.1.39

Reference 
1. P. Giordano, The ring of fermat reals, Advances in Mathematics, 225 (2010), ¹4, 2050–2075.
2. P. Giordano, M.G. Katz, Two ways of obtaining infinitesimals by refining Cantor’s completion of the reals, arXiv:1109.3553 3. R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Vol. 188 of Graduate Texts in Mathematics, Springer, New York, 1998. Zbl 0911.03032. MR 1643950. DOI 10.1007/97814612 06156. 213 4. J.M. Henle, Nonnonstandard analysis: real infinitesimals, Math. Intelligencer, 21 (1999), 67–73. 5. P. Reeder, Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field, Notre Dame Journal of Formal Logic, 58 (2017), ¹2. 6. A. Robinson, Nonstandard analysis, Amsterdam: NorthHolland, 1966. 
Pages 
39

Volume 
52

Issue 
1

Year 
2019

Journal 
Matematychni Studii

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