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 51
- Downloads 0
About some problem for entire functions of unbounded index in any direction
In this paper, we select a class of entire functions F(z1,…,zn) such that for any direction (b1,…,bn)∈Cn∖{0} and for every point (z01,…,z0n)∈C the function F(z01+tb1,…,z0n+tbn) is of bounded index as a function in variab...
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...
Matrix representation of Taylor’s formula for mappings in finite dimensional spaces
In this paper, the matrix representations of derivatives, differentials and Taylor’s formulas for functionals and mappings are presented. The behavior law of intermediate points in the formula’s remainder terms is found...
Automorphism groups of superextensions of groups
A family L of subsets of a set X is called {\em linked} if A∩B≠∅ for all A,B∈L. A linked family M is {\em maximal linked} if M coincides with each linked family L on X that contains M. The superextension λ(X) consists of...
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...