Homomorphisms between rings with infinitesimals and infinitesimal comparisons

Author
E. Bottazzi
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 order-preserving 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 ``non-nonstandard'' 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 order-preserving homomorphism; Henles ring; hyperreal number; nonstandard analysis
DOI
doi:10.30970/ms.52.1.3-9
Reference
1. P. Giordano, The ring of fermat reals, Advances in Mathematics, 225 (2010), 4, 20502075.

2. P. Giordano, M.G. Katz, Two ways of obtaining infinitesimals by refining Cantors 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 10.1007/978-1-4612- 0615-6. 213

4. J.M. Henle, Non-nonstandard analysis: real infinitesimals, Math. Intelligencer, 21 (1999), 6773.

5. P. Reeder, Infinitesimal Comparisons: Homomorphisms between Giordanos Ring and the Hyperreal Field, Notre Dame Journal of Formal Logic, 58 (2017), 2.

6. A. Robinson, Non-standard analysis, Amsterdam: North-Holland, 1966.

Pages
3-9
Volume
52
Issue
1
Year
2019
Journal
Matematychni Studii
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