quantum mechanics - Can we have non continuous models of reality? Why don't we have them?

This question is about Godel's theorem, continuity of reality and the Luvenheim-Skolem theorem.

I know that all leading physical theories assume reality is continuous. These are my questions:

1) Is reality still continuous according to string theory? what is the meaning of continuous if there is a finite size for a string? (My understanding of string theory is very limited :)

2) According to the Luvenheim-Skolem theorem, any enumerable first order logic theory (physics?) has an enumerable model. This means we must be able to have a discrete theory of physics which is as good as the continuous ones. Is this accurate?

3) Why haven't we found one? I've heard some people say that Heisenberg's formulation of quantum mechanics doesn't speak of anything continuous... is this accurate?

4) If we can find one / have found one, does this mean that our understanding of physical laws can always be reduced into a Turing machine?

5) If we can find one / have found one, how can we explain the use of real constants in physics? (such as Pi)

6) Does Godel's theorem mean that we could always "notice" that some things are true but not be able to prove them from within the universe?

Thanks.