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!