I read online a statement to the effect that (I'm paraphrasing):
Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers.
I am under the (qualitative) impression that this statement is true within the axioms of natural numbers themselves, so that
- if one expanded the of axioms one could prove all of the true statements that can be expressed solely in terms of natural numbers.
Note that this larger system itself is not complete and consistent.
Does Godel's incompleteness theorem have the feature that it shows that these larger systems are somehow representable with the axioms of the natural numbers?