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 81
- Downloads 0
Distance between a maximum modulus point and zero set of an analytic function
Let f be an analytic function in the disk DR={z∈C:|z|≤R}, R∈(0,+∞]. We call a point w∈DR a maximum modulus point of f if |f(w)|=M(|w|,f), where M(r,f)=max{|f(z)|:|z|=r}. Denote by d(w,f) the distance between a maximum mo...
Existence of solitary traveling waves in Fermi-Pasta-Ulam system on 2D–lattice
The article deals with the Fermi–Pasta–Ulam system that describes an infinite system of particles on 2D–lattice. The main result concerns the existence of solitary traveling wave solutions. By means of critical point the...
On interpolation problem with derivative in the space of entire functions with fast-growing interpolation knots
In the paper are obtained the conditions on a sequence (bk,1;bk,2), k∈N, such that the interpolation problem g(λk)=bk,1, g′(λk)=bk,2 has a unique solution in a subspace of entire functions g that satisfy the condition ln...
On removable singularities of mappings in uniform spaces
The paper is devoted to the study of mappings of two metric spaces that distort the modulus of families of paths by analogy with the Poletskii inequality.We deal with the situation when the mapping acts in a space that a...
Periodic words connected with the Tribonacci-Lucas numbers
We introduce periodic words that are connected with the Tribonacci-Lucas numbers and investigate their properties.