What are the real numbers?

Question

I know "What are real numbers" has probably been asked before, and the answer would be "the unique complete ordered field" BUT, isn't there some subtlety going on here? In the sense that (at least the way I interpret it) when saying complete ordered field, we really mean complete as a metric space. But to define a metric space you need the notion of real numbers (in order to define a metric). So it seems like saying complete ordered field already assumes the real numbers to be constructed. The question I suppose is: is there some sort of concise way of defining real numbers, without appealing to real numbers? :) (of course, you can describe their construction vie Dedekind cuts for example, but that isn't exactly as succint - guess I'm looking for something closer to a definition rather than a construction)

Answer 1

Expanding the above comments into an answer:

There are multiple notions of completeness. When we say that $\mathbb{R}$ is the unique (up to isomorphism) complete ordered field, we are using Dedekind completeness: that is, $\mathbb{R}$ is the unique (up to isomorphism) ordered field in which every nonempty bounded-above subset has a least upper bound. This makes no reference to the real numbers, so the circularity you are concerned about does not arise. (It's separately worth noting that we can talk about metric spaces even before introducing the reals, by using "rationals-only" relations like "$d(a,b)<{p\over q}$" instead of an exact distance function, but that circumlocution is unnecessary here.)

In fact, metric completeness is a very weak notion in this context and does not suffice to characterize $\mathbb{R}$: see the discussion here.