Earlier quoted context omitted.
Goedel's theorem is not about nonexistance of foundation in mathematics, it's about existence of true but nonproveable statements in every nontrivial formal system. One could imagine a hypothetical stronger result - maybe every nontrivial set of axioms can actually derive p^(not p)?
Provided the axiom set is recursively enumerable. The second order Peano axioms for the natural numbers are complete.
And frankly, though not 100% proved (yet?), it seems to me very clear that any numbering and arithmetic system that contains any form of infinity is doomed to be incomplete.
So the only useful arithmetical fields that escape Godel are Zn.