If you need actual proof...
then you are going to have to learn some logic than I have not. I've merely quoted an often-quoted fact from logic.
The result that I am quoting is a consequence of Goedel's "constructible universe" construction. The purpose of that construction was to prove that if ZF is consistent, then ZFC is consistent. A side-effect of the construction is that every arithmetical statement provable in ZFC turns out to be provable in ZF.
However I haven't personally been through that construction, so I can't give you any details other than "ask a logician". If you post what I said above on sci.math and ask for recommendations on where you can learn about the construction, it is likely that someone will have a good answer.
As for the sufficiency of integers for questions of physical interest, integers are sufficient to model classical computers. Any aspect of reality that can't be modelled with a good scientific theory and a good enough computer is unlikely to ever be amenable to scientific analysis. That is a statement of belief on my part, but it is a belief that I think most will accept.
Cheers,
Ben
