Thanks for putting this discussion in the right place.
Just a tiny question or thought: has anyone explored the algebraic idea of a "cut on a cut" as per "Dedekind cut"?
A Cartesian coordinate frame is usually defined in the X and Y axes in real numbers, and the Dedekind cut might be defined in the X axis somewhere, perhaps as the square root of 2. What if the origin was defined by the X axis being a Dedekind cut in Y and the Y axis being a Dedekind cut in X? Is this completely obvious or trivial -- or wrong headed? It seems possible to explore a bunch of interesting things in this way, by seeing the lowest level of decimal point as defining the unit interval, shown in this graphic as the range from 1 to 2.
PS -- this is my current exploratory superflash regarding "struggles with the continuum" -- maybe a cool way to map the finite to the infinite (?)