This is the second of two videos that look at the official formal definition of a limit of a sequence, as initiated by Bolzano, Cauchy and Weierstrass. Although commonly regarded as a pillar of modern analysis, in fact this definition has serious logical problems. We state what these problems are, and then start to try to explain them. In this lecture, we go beyond the baby examples usually found in calculus and analysis texts, revealing more of the typical generality and inaccessibility of a lot of the discussion of general limits. In particular we examine an extended harmonic sequence formed by playing the Collatz game (also called the 3n+1 problem, and described at some length in my video Famous Math Problems 2: The Collatz Conjecture FMP2). We will see the essential problem is that the definition actually requires us to make an infinite number of computations to assert that a general `sequence' has a limit. Is it possible to make an infinite number of computations? No, it is not. Are we allowed to pretend that we can make an infinite number of computations? Well I suppose so, but let's not call it mathematics.
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.