Homomorphisms between rings with infinitesimals and infinitesimal comparisons
Journal Title: Математичні Студії - Year 2019, Vol 52, Issue 1
Abstract
We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers ∙R are smaller than any infinitesimal hyperreal number of Abraham Robinson's nonstandard extension of the real numbers ∗R. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ∙R and ∗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 ∙R while it is seen as a nonzero infinitesimal in ∗R suggests that some infinitesimals in ∗R are smaller than the infinitesimals in ∙R. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ∙R.
Authors and Affiliations
E. Bottazzi
Keywords
Related Articles
- EP ID EP673790
- DOI 10.30970/ms.52.1.3-9
- Views 78
- Downloads 0
Intertwining maps for the Weitzenbock and Chebyshev derivations
The notions of Chebyshev derivations of the first and the second kind are presented. Explicit forms of the corresponding intertwining maps are found.
Periodic words connected with the Tribonacci-Lucas numbers
We introduce periodic words that are connected with the Tribonacci-Lucas numbers and investigate their properties.
An extremal problem for volume functionals
We consider the class of ring Q-homeomorphisms with respect to p-modulus in Rn with p>n, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in...
Meromorphic function sharing three sets with its shift, q-shift, q-difference and analogous operators
Using the notion of weighted sharing of three sets we deal with the uniqueness problem of meromorphic function with its shift, q-shift, q-difference and analogous operator. A handful number of examples have been provided...
Fourier problems for parabolic equations with variable exponents of nonlinearity and time delay
The Fourier problem for nonlinear parabolic equations with variable exponents of nonlinearity and time delay is considered. The existence and uniqueness of weak solutions of the problem are investigated. Also, its a prio...