In our last video we introduced polynumber (or polynomial) on-sequences. Today we consider how we might go beyond this, to introduce a wider range of sequences, to deal with more of the examples in OEIS, for example. The first attempt will involve 'exponomials": arithmetical expressions extending polynomials by allowing also exponentials, such as n^n for example. While tempting, this extension is rather problematic logically, and I explain why.Another important kind of object in this area are "recursive sequences". There are many familiar examples of such; primarily the Fibonacci numbers 1,1,2,3,5,8,13,21,... or the closely related Lucase numbers 1,2,4,7,11,18,29,...In OEIS we find many more recursively defined sequences, such as the Catalan numbers, or the Euler numbers (also called Slyvester's sequence). But what exactly is the definition of a ``recursive sequence"?? This is an embarrassing question for many mathematicians, who naively regard the term as almost self-evident, or at least only requiring a few examples and a bit of hand-waving. No, no, no! The idea is far from clear.Conclusion: without a lot of work, neither exponomials or general recursive on-sequences are available to us at this point. More work needs to be done first! In our next video we will move to a much more solid extension of polynumber on-sequences.
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.