This is a rambling collection of thoughts which are tangential to your question--I am posting in the hope that they might interest you. Let me begin by recalling that the reals can be charaterised as a totally ordered, Dedekind complete field. This is often used as the basis of an analysis course. There are various possibilities for its presentation--one can simply postulate the existence of such a structure and then bash on, or one can prove its existence based on a postulated model of Peano´s axiom. This involves extending the positive integers to the integers, then to the rationals, finally to the reals. Each of the extensions are similar in method (the first one has been expounded in the above thread), the last one being, as one would expect, rather more sophisticated (equivalence classes of Cauchy sequences of rationals).
However, there is an alternative approach which only uses the order structure and which has certain advantages. We work backwards from the reals with the following claims:
I. The real line is a totally ordered, Dedekind complete space with a countable order dense subset and without a largest or smallest element. One can also incorporate open intervals (same axioms), half open intervals, closed intervals or the positive and negative half lines by assuming the existence of least or greatest elements, in this system.
II. The rationals form a countable totally ordered space without gaps and no smallest or largest element. (A gap is a pair $x$ and $y$ with $x<y$ but with no elements between them).
III. The positive integers form a countable, Dedekind complete, totally ordered space with a smallest element but no largest one and the property that if $x<y$, there are at most a finite number of elements between them (this is not a circular argument--the notion of a finite set is internal to set theory). In order to get the integers, one simply drops the corresponding condition and assumes that there is no smallest element.
If one is not prepared simply to postulate the existence (and uniqueness) of such structures, one is then faced with two tasks--to demonstrate the consistency and categoricity of these axioms,i.e., that they have models which are are unique in the natural sense. This can be done with about the same effort as for the standard method.
I would claim that there are several advantages in these characterisations. Mathematically, the simplicity of using only one structure, the order, rather than three (order, addition and multiplication with a list of axioms). Also, these characterisations explain the magic appearance of the real numbers to measure physical quantities, much discussed in the philosophy of physics. Many (most?) natural quantities have a natural order structure but no sensible algebraic one. The classical examples are entropy and temperature (nobody would want to multiply temperatures). The question of going from the ordering "being warmer than" to a numerical value (empirical temperature, then absolute temperature) was of considerable interest to the pioneers of thermodynamics and is discussed in some detail in Maxwell´s "Theory of Heat". The corresponding difference between an ordering (preference) and a utility function is also fundamental in economics (fundamental enough to result in Debreu in being made a laureate).
Of course, once you have the reals, it is routine to then show that you can deduce the algebraic structure from the axioms. This involves some arbitrary choices and so is not natural or unique, but this a desirable property for reasons too intricate to expound here.
I will add a caveat. I am dragging these results from memory. They are certainly known but I have been unable to pinpoint explicit occurrences in the literature. Hence I have called them claims rather than results, since I haven´t sat down to write out the proofs.