Incompleteness in other areas of mathematics I read in "Apostolos Doxiadis:Uncle Petros and Goldbach's conjecture" that SPOILER ALERT Uncle Petros practically stopped working on Goldbach's Conjecture when he learnt about the Incompleteness Theorem. He asked Godel if it could be decided if a particular statement is true but unprovable and Godel said he couldn't tell if even Goldbach's Conjecture is such.
But as I understand Godel's theorem, this incompleteness only applies to a realtively small number of statements which somehow refer back to themselves.
So isn't it very probable that Number Theory problems (like Goldbach's Conjecture), or conjectures from areas of mathematics not closely related to formal logic, don't belong to the group of true but unprovable statements?

What about questions asking whether some big, multi-variable polynomial equation (with integer coefficients) has an integer solution? There are equations of this sort for which there is no integer solution but the usual Zermelo-Fraenkel axioms of set theory can't prove that.  (If you add to ZFC the assumption that there is an inaccessible cardinal, then the unsolvability of some such equations becomes provable, but for other such equations you need bigger large-cardinal axioms.)
