So, mathematically, here is no real number line

Think a minute.  Every point on  the real number line is either a combination of integer, or a recursive algorithm on integers. We have taken a collection of recursive  algorithms, over indices, and called it Newton's grammar.  A real number line cannot be drawn, except by including some algebraic rules of subdivision.  What did we say years ago? The theory of everything was all about the system building its own yardstick, the bounds on indices.