[John Locke]

With the conventional definition of "well ordered" (every non-empty subset has a least member) then the reals are not well ordered. I'm sure you know the proof as well as I do.

But I seem to have stupidly failed to grasp your point. Why is this an important question? The important question, to me, is rather the question whether the conventional definition of "well ordered" makes sense. Which has very little to do with Platonism.

[Doctor Stochastic]

It's very important for Platonism. Do the "reals" exist independently of set theory? If so, which set theory applies, with or without the axiom of choice or the continuum hypothesis? Both ways are consistent. The Platonist position generally has been (at least pre-1963) that either the axiom of choice is true or false and this is to be determined. It's possible to assume that both sets of "reals" are possible (as well as many more types of "reals".) In that case, one can ask which set of "reals" applies to physics.