TITLE: Is there a version of the Archimedean property which does not presuppose the Naturals?
QUESTION [9 upvotes]: All the statements of the Archimedean property with which I am acquainted fundamentally uses ℕ -- more than as a totally ordered semi-group, really being the 'standard model' of the naturals.  It is a fundamental ingredient in showing that the reals are (up to isomorphism) the only complete totally ordered field.
In this particular result, there are two non ingredients which seem to be of a fundamentally different nature: Dedekind completeness (which is about subsets, while all the other axioms are about elements), and the Archimedean property, which pre-supposes the existence of the Naturals.  But because being 'Archimedean' already makes sense in (ordered) monoids, that is the one I am most interested in.
Crazy scenario: replace the 'naturals' in the Archimedean property with a non-standard model of the 'naturals' (call this N-Archimedean).  Now the reasoning for the argument that all Archimedean totally ordered fields are sub-fields of ℝ readily lifts, but no longer proves quite the same thing.
In other words, it seems that this unique position of ℝ is in part due to ℕ already being baked in to the question.  And thus my question: is there a proper generalization of the Archimedean property which is ``properly abstract''?

REPLY [2 votes]: Tarski showed the first-order theory of the reals is decidable.  But of course the first-order theory of the naturals isn't.  So (assuming consistency, I guess) we know you cannot construct the naturals in the reals using only first-order notions.  But we can construct the natural numbers if we are allowed to quantify on sets of reals.