Sistema de números reais: intuição ou rigor?

Abstract

It is classic that some ideas present in courses of Calculus go back to Antiquity, since Archimedes, who, among other scientists, also debated the existence of the continuum. This problem was addressed by I. Newton and G.W. Leibniz, who, in the seventeenth century, developed the Calculus. But geometric intuition, as early as the eighteenth century, did not solve problems of “continuity” that presented themselves to mathematicians. B. Bolzano, A. L. Cauchy, K. Weierstrass, and R. Dedekind questioned the foundations and began the era of rigor in mathematics, whose North turns out to be the system of real numbers: a complete ordered body. A diagnosis in current books of Calculus shows us that intuition and rigor merge, whether in definitions or in demonstrations. This may create doubts for the less attentive reader. The objective is to present a scenario in which geometric intuition reveals pedagogical strategies and rigor indicates the need for mathematical precision.