Análisis real no estándar

Abstract

The aim of this Final Degree Project is to develop a presentation of the so-called «nonstandard real analysis», a 20th-century logical theory that builds real analysis on a non-archimedean ordered field extension $^{\ast}\mathbb {R}$ of the real numbers $\mathbb {R}$, that contains in particular infinitely small and infinitely large numbers. In Introduction, we contextualize its origin, justify our approach, comment the bibliography, detail some applications and fix some notation. In Chapter 1, we show how ultrafilters are a natural tool for the construction of $^{\ast}\mathbb {R}$. In Chapter 2, we set up all the needed facts about them. In Chapter 3, we explain the construction of $^{\ast}\mathbb {R}$ and its basics. In Part II, we deal with some topics on real analysis using our new tools, remarkably the useful characterizations of the completeness of $\mathbb {R}$ and the compactness of subsets of $\mathbb {R}$, as well as the algebraic account of limits and the possibilities of hyperfinite methods.