Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular scheme for testing it, that's great, but even the existence or non-existence of a proof regarding potential testability is wonderful.

How about something a little simpler: can we even test the Peano axioms? Are there experiments that can empirically verify theorems not provable by them?

This is a slightly fuzzy question, so to clarify what I mean, consider this: the parallel postulate produces good, useful geometry, yet beyond its inapplicability to the sphere, there's evidence to suggest that the universe is actually hyperbolic - this can be considered an experimental evidence "against" the parallel postulate in our universe.

Edit: Thanks to all the people who answered - I understand the concerns of those who don't like this questions, and I appreciate all those who answered a more modest interpretation that I should, in retrospect, have stated. That is, "Is the axiom of choice agreeable with testable theories of mathematical physics, is it completely and forever irrelevant, or is it conceivably relevant but in a way not yet known," to which I got several compelling answers indicating the former.