In this final video on the most fundamental and important problem in mathematics [which happens to be: How to model the continuum?] we tackle the seriously unfortunate developments leading to the current misunderstandings about the so-called 'real numbers'. Of course this name is a complete misnomer: they are not 'real' at all; rather they constitute a desperate attempt to enforce the existence in mathematics of objects which are actually unattainable without resorting to an infinite number of computational steps (whatever that might actually mean!)
In this video we give a bird's eye view of the various misguided attempts at establishing 'real numbers' and sketch some of the logical and technical difficulties that students are usually shielded from. The basic construction arises from Stevin's decimal numbers extended, using a dollop of wishful thinking, to arbitrary infinite decimals, not just the repeating decimals encoded by rational numbers. Understanding the difficulties with this approach is not that hard, and in essence the same kinds of problems resurface in the various variants which we also discuss: infinite sequences of nested intervals of rational numbers, monotonic and bounded sequences of rationals, Cauchy sequences of rationals, equivalence classes of Cauchy sequences, and finally the icing on the cake of irrationality: Dedekind cuts.
Students of mathematics! Listen carefully: none of these approaches work. This is the reason why not one of these 'theories' are properly laid out in front of you when you begin work in calculus or even analysis. To those who would try to convince you otherwise, via appeals to authority or numbers, name-calling, or by special pleading on behalf of all those lovely 'results' that supposedly follow from the required beliefs: ask rather for explicit examples and concrete computations.
These are the true coin in the realm of mathematics, and will not lead you astray.
This is perhaps a place to thank my many contributors, subscribers and online friends. We are on our way to a more beautiful and logically coherent mathematics, but there is a long ways to go from here to there! Your support is a big help.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .
