[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.

[John Locke]

Sorry for the long delay, it's been a busy week.

In no particular order: Yes, the "axiom of choice" must be either true or false. So, which is it? My position is simple (may be too simple). If the AoT is true, then the reals are well ordered. But the reals are evidently not well ordered, since the subset denoted by (0,1] has no least member.

The proof of that is a pretty strong one, namely that if you offer me a candidate least member, say r, then I can offer a better candidate, namely r/2. This is the exact same strategy as we use in the proof that there is no largest prime, which no mathematician has ever believed fallacious.

Hence, everything "proved" by the AoT is dubious, including your previous example (by Vidali wasn't it?) of a set that can be shown to have measure both zero and non-zero.

As another digression, do I believe (or suspect) that there are incomparable transfinite cardinals, ie t1, t2 such that it is not the case that t1>t2 or t1=t2 or t1<t2? Again, it's not something I've ever considered relevant to Platonism as such, but I would neither be surprised if two such cardinals existed nor devastated if they did not.