I wrote in part:
I also think that it's enlightening to look at Tarski's axioms for geometry. [...]
But one final axiom sets the dimension; this states (in effect) that there exist n + 1 points, affinely independent and affinely spanning the space. If you label the first n points (1,0,0,...) through (...,0,0,1) and the last point the origin, then every point has a unique label, and the space becomes \R^n.
I just realised (reading my post again on the mailing list) that Tarski's axiom doesn't give the relative distances between these points, so these labels (and the later remarks about fixing them) are invalid. Of course, given Tarski's n + 1 points, you can find my n + 1 points, and even do this algorithmically (using the Gram-Schmidt process), so my other remarks hold only if you actually do this (so that preserving the points means preserving the results only after the Gram-Schmidt process has been applied). Thus, the situation is a little less elegant than I implied (but maybe that's Tarski's fault for phrasing his axiom so liberally ^_^). --Toby