A Higgs Boson particle walks into a church, but the preacher says "You are a fake! A disgrace! You call yourself the 'God particle' when there is only one true God! Blasphemy! You don't exist!" The Higgs Boson replies "Well if I don't exist, how can you have mass?"
@StanShunpike : boom boom! However the Higgs mechanism is said to be responsible for only 1% of the mass of matter. And it isn't responsible for the mass of the Higgs boson. Check out Matt Strassler's article.
@Danu :-D ok got it...Unfortunately, I am afraid it could depend on it. Anyways, a lot of mathematical things may change in a world with or without uncountable choice; how many of them are physical I do not know. As I said, the algebra of CCR is one of those non-separable spaces very relevant in physics, and maybe a physical necessity for uncountable choice
@Danu You should look at the constructivst/intuitionistic mathematics then...there is a book by bishop where he reproduces a part of standard analysis by means of constructivist proofs. No axiom of choice there...just constructing mathematical objects from scratch :-D