There are three quite different approaches to the idea of a real number as an infinite decimal. In this lecture we look carefully at the first and most popular idea: that an infinite decimal can be defined in terms of an infinite sequence of digits appearing to the right of a decimal point, each digit chosen arbitrary and independently. There are several very attractive reasons for taking this position, which we outline. Primary among these is that the definition allows us to turn a process into a number (ostensibly!). But ultimately the idea founders on the rocks of reality: the impossibility of specifiying competely a general such number, the impossibility of defining the addition (and multiplication) of such numbers via finite algorithms, and the resulting problematic aspects of the laws of arithmetic.We look also briefly at the role of the Axiom of Choice in trying to provide an axiomatic framework for real numbers as such `infinite choice decimals'. This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure 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.