You argued (without much detail, but I'll buy it) for completeness, but then snuck in ordered-ness and Archimedean-ness. As a p-adic analyst, I object to taking these latter two characteristics for granted as necessary for mathematics.
(Also, one has to be careful about the meaning of 'complete' in the uniqueness statement; i.e., it must be understood to mean complete as an ordered field (satisfying the least-upper-bound property), not just complete as a uniform space (having every Cauchy sequence converge). See Paul Sally's "Tools of the trade" (http://www.amazon.com/Tools-Trade-Paul-J-Sally/dp/0821846345) for some discussion of this.)
I was assuming that for counting, measuring,
and arithmetic, we want a field (you know,
addition, multiplication, both with an identity
element and inverses, associativity,
a distributed law, commutativity). We want
order -- that is bigger and smaller.
And likely we want Archimedean order.
So, if we was also want each nonempty set bounded
above to have a least upper bound, then
we are stucko with the usual reals.
For algebra, and in particular number theory,
already in undergraduate school one of my
math profs declared that I'm an analyst
and not an algebraist. The difference?
As that math prof explained, it:
"Analysis has an idea behind it. Algebra
is just pushing symbols around." I can
think intuitively about analysis and often
convert the intuitive ideas into good proofs;
in algebra, mostly I can't do such a thing.
I ended up writing my honors paper in
group representations -- I didn't
like such concentration on algebra.
For using strange constructions from algebra
or computer science to replace the reals,
I will have to decline to buy in based if only
on ignorance of the offering. But somehow
I suspect that the reals are the right answer!
(Also, one has to be careful about the meaning of 'complete' in the uniqueness statement; i.e., it must be understood to mean complete as an ordered field (satisfying the least-upper-bound property), not just complete as a uniform space (having every Cauchy sequence converge). See Paul Sally's "Tools of the trade" (http://www.amazon.com/Tools-Trade-Paul-J-Sally/dp/0821846345) for some discussion of this.)