TITLE: Why are metrics defined as functions in $\mathbb{R}^{+}$?
QUESTION [12 upvotes]: A metric on a set $S$ is a function $d: S^2 \to \mathbb{R}^{+}$ that is symmetric, sub-additive, non-negative, and takes $(x,y)$ to $0$ iff $x=y$.
My question is: what makes $\mathbb{R}^{+}$ so special that metrics are universally defined in terms of it? Why don't we use some other totally-ordered set with a least element $m$ and an abelian operator $+$ which preserves order and under which $m$ is neutral?
To take it one step further, why aren't metrics defined, more generally, to map elements of $S^2$ to any totally-ordered set that satisfies these conditions?

REPLY [2 votes]: Why are metrics defined as functions in $\mathbb{R}^+$?
A: Well, actually metrics are not always defined as functions in $\mathbb{R}^+$.
As is pointed out in Glitch's answer, when the metric take values in the real numbers, we have Real Analysis at our disposal. And sometimes, that is all that is needed, for example, in classical Functional Analysis, as long as we consider Archimedean valued fields (equivalently, subfields of $\mathbb{C})$, the norms and metrics that are usually considered don't need to take values outside the real numbers to obtain great results.
But certainly, this is not the case if we allow the use of non-Archimedean valued fields with valuations of rank larger than 1 (taking values in ordered groups that cannot be embedded in $\mathbb{R}^+$). Although in this case, we cannot make use of real analysis, we use $p$-adic analysis (if we are using $p$-adic fields) or Levi-Civita analysis (if we are using the Levi-Civita field) or any other ultrametric analysis.
In this context, the norms and seminorms need to take values outside the real numbers (see this definition), and here the need for metrics with values in arbitrary linearly ordered sets is evident. This generalization of matrics are called "scales" and its study has far-reaching consequences for non-Archimedean Functional Analysis.
For an introduction in this area I recommend the paper:
Banach spaces over fields with a infinite rank valuation - [H.Ochsenius A., W.H.Schikhof] - 1999
After that see: Norm Hilbert spaces over Krull valued fields - [H. Ochsenius, W.H. Schikhof] - Indagationes Mathematicae, Elsevier - 2006