For a long time I've been puzzled—and I still am—by this. Our physics is based on the real numbers (complex numbers, vectors, Banach spaces—all that is built out of real numbers). After all, there are non-standard numbers that can do everything real numbers can. So what reason do we have to think that "the" real numbers are what the world's physics is in fact based on?
I think one can use this to make a nice little argument against the possibility of us coming up with a complete physics—we have no way of telling which of the number fields is the one our world is based on.