I no longer believe the Axiom of Choice

Particles with integer spin obey Bose-Einstein statistics. These are named for two papers, published together, by Bose and Einstein. Bose's paper described the statistics. Einstein's said that Bose's paper appeared to have an off-by-one error, but was in fact correct, because the particles could not be distinguished from each other.

It seems to me, then, that we have a physical example of a situation where the axiom of choice fails.

Therefore, using the axiom of choice as part of the foundation for all of mathematics seems to not be a wise idea. (ZFC can still be an interesting system, and consistent with itself, of course.)

Or am I missing something?