Ultraprodiucts and their application to non-standard analysis

Abstract

[en] The intuitive notion of infinitesimals has been used in mathematical arguments for many years. However, they are still seen as controversial. The aim of this work is to give an introduction to ultraproducts, bring the reader closer to non-standard analysis and to prove that it is rigorously defined. In order to do that, we introduce the concept of reduced product of structures, as well as we give an ultraproduct version of the compactness theorem of first-order logic. We also build the set of hyperreal numbers and give some known results of calculus using infinitesimals.