Thank you for posting that part of the paper. It makes things much easier to understand.

Unfortunately I don't really see the point. I mean there's nothing wrong with looking at things with a fresh start, but your constant comparisons with "typical" Mathematics just doesn't seem to have direction. I looked at the first 10 pages or so (before I got lost in the notation) and came up with two comments:

1. As an example of your comparisons with current Mathematics: "numerical semi-lines" and "numerical points" are merely a collection of points and line segments. Those concepts are geometric in nature and have been around since Euclid.

2. At the very least your first 5 or 6 "proofs" end up being derived by introduction of new notation, which in and of itself proves nothing.

My advice:

1. Include an abstract to your paper explaining what your goals and results are.

2. Define your symbols as a part of the proof and use it to actually prove your assertions. New notation does not imply proof. This needs to be sharpened up.

3. Look more carefully at what's already out there. It's nice to say that the natural numbers are constructed by axiom, but that is not a requirement. The Peano axioms can be used to construct the natural numbers, for example.

Address at least these issues (maybe someone else will chime in with other thoughts for you) and perhaps post your revised paper. As a Forum we are not going to edit your paper for you but if someone here is willing to give you advice they are welcome to.

-Dan