Here we continue explaining why the current use of `axiomatics' to try to formulate a theory of `real numbers' is fundamentally flawed. We also clarify the layered structure of the rational numbers: we have seen these several times already in prior discussion of the Stern- Brocot tree, here we view rationals in [0,1] in terms of increasing denominators. This allows us to explain why, if you believe the usual sad story of being able to `do an infinite number of operations', we can `create' nested interval sequences (or equivalently Cauchy sequences) with no rational limit.This paves the way for introducing the much vaunted ``least upper bound property'' of `real numbers', which is a mainstay of classical analysis, and props up the modern theories of areas, integrals, infinite sums, transcendental functions, infinite products, lengths of curves and much more. The reality is that much of modern mathematics, sadly, is an elaborate dream system. Here we are slowly, slowly....waking up. Feel free to join us, it's actually quite invigorating!
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.