Abstract
In this chapter, we propose a mathematical and epistemological study about two classical constructions of the real number system, by Dedekind (cuts) and Cantor (Cauchy sequences), and the associated proofs of its completeness. In addition, we present two contrasting constructions leaning on decimal expansions. Our analysis points out that Dedekind’s construction fosters a conceptualization of the real numbers leaning strongly on the total ordering of Q and R, while putting aside the metrical aspects. By contrast, the more intricate construction through Cauchy sequences calls on complex objects, but yields to a better understanding of the topological relationship between rational and real numbers. We argue that suitable considerations of decimal expansions and of approximation issues enable to connect and complement those two approaches. These analyses highlight the dialectical interplay between syntax and semantics, and the crucial role of the definitions of objects at play in proof and proving. The general didactical issue pertains to the potential contribution of analyzing proofs as a means for deepening the understanding of the related objects and of their ensuing conceptualization. We hypothesize that doing so with Dedekind’s cuts, Cauchy sequences and decimal expansions open paths towards improving the conceptualization of the real numbers, by taking into account the triad discreteness/density/ continuity.