Des nombres infiniment petits et des entiers infiniment grands mais d\'efinis \`a l'unit\'e pr\`es

TL;DR

This paper introduces a new, simple form of Non-Standard Analysis by constructing an intrinsic extension of the non-negative integers and the smallest over-field of the reals, providing a rigorous and intuitive framework.

Contribution

It presents a novel, simplified approach to Non-Standard Analysis through the explicit construction of extended number systems.

Findings

Construction of an intrinsic extension of non-negative integers

Development of the smallest over-field of reals that is continuous

Comparison with Robinson's and Conway's Non-Standard Analysis methods

Abstract

The main results of this paper are the construction, both rigourous and intuitive, of "the" intrinsic extension of the set of non negative integers N and the smallest over-field of R set which is continue (according to R.Dedekind). The aim of this article is to provide a new Non Standard Analysis, very simple, which is compared in the Introduction with A.Robinson's and J.H.Conway's