One of the standard ways of trying to establish `real numbers' is as Cauchy sequences of rational numbers, or rather as equivalence classes of such. In the next few videos we will be discussing why this attempt does NOT in fact work!In this lecture we provide an introduction to these ideas in an informal and descriptive way. In particular we visualize sequences of points in the plane, and discuss two different notions of when two sequences converge; including Cauchy convergence.We also outline the challenges that lie in wait for anyone who tries to set up arithmetic with `real numbers' in this way. It is not a surprise that this is nowhere properly done, although many students are under the mistaken impression that they have `covered' this material at some point in their analysis courses.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.