I would prefer to ask this in the math community, but that crowd is hostile toward anything hinting of philosophy. It is my contention that a construction of the real number system which begins with the most primitive concepts will begin by constructing the natural numbers beginning with the number one.
Can zero be defined without some definition of one? Can one be defined without some definition of zero?
This is an "after the fact" edit; since I've already accepted an answer. I just want to add this to save other's the trouble of bringing to my attention the Dedekind-Peano method of constructing the natural or whole numbers.
These are my current notes on the construction of the real numbers using Peano's axioms with 1 as the non-successor:
These are my current notes on constructing the real numbers using a Peano-like set of axioms with 0 as the non-successor:
The notes are guaranteed to contain errors, dubious observations, idiosyncratic notation and methods, significant redundancy, fuzzy logic, etc.