We introduce more general ``infinite sequences'', or on-sequences, generated by rational polynumbers, otherwise often known as rational functions: ratios of one polynomial over another. The association of a sequence to such an expression is surprisingly delicate, and requires us to look at factorizations. In particular we are naturally led to introduce a new symbol for ``infinity", corresponding to a ratio of integers with zero in the denominator, and associated with a limiting behaviour getting larger and larger as we approach a particular value.
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.