Published in ArvixMathematics, August 1, 2011, pages 1-32.
Copyright © 2011 James F. Hall and Todor D. Todorov. Available at ArxivMathematics: http://arxiv.org/abs/1109.2098.
We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent statements borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the 18th century was already a rigorous branch of mathematics – at least much more rigorous than most contemporary mathematicians prefer to believe. By advocating our particular historical point of view, we hope to provoke a discussion on the importance of mathematical rigor in mathematics and science in general. We believe that our article will be of interest for those readers who teach courses on abstract algebra, real analysis, general topology, logic and the history of mathematics.