Introdução à análise não standard por meio dos hiper-reais

Abstract

The primary objective of this work is to present hyperreal numbers and non-standard analysis, an alternative approach to classical analysis developed by Abraham Robinson. Through the construction of the hyperreals *R using the ultraproduct technique, it is possible to extend the set of real numbers to include infinitesimals and infinities, allowing a rigorous formalization of intuitive concepts in analysis. To enable this construction, the concepts of filters and ultrafilters are introduced, with an emphasis on non-principal ultrafilters, which play an essential role in defining the hyperreals. Non-standard analysis also allows the reformulation of fundamental notions such as limits, continuity, derivatives, and integrals using infinitesimals, without compromising mathematical rigor. Furthermore, the work discusses the properties of hyperreal numbers, such as their algebraic structure and the transfer principle.