I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is:
Because every Archimedean ordered field is (up to unique isomorphism) a subfield of $\mathbb R$ anyway.
But is that actually true? And if it is, can it be strengthened to "every Achimedean ordered ring"? Or even semiring?
I know $\mathbb R$ is the only complete Archimedean field. But a priori, I suppose there could be non-complete examples that cannot be completed (without losing the Archimedean property).