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?
 A: 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.)
A: When contemplating provability, it is useful to remember that it pertains to a particular framework (usualy, a formal system). Also, truth should be argued somehow, so one also wants to rely on a framework when demonstrating the truth of a statement.   
I have not read the book, so I would not know what proof means Uncle Petros treated himself to when trying to prove Goldbach's conjecture.
To provide counterexamples to the hypothesized link between incompleteness and self-reference, I want to point out "natural" (in the sense of being of considerable interest independently of the incompleteness theorem) statements that are unprovable in Peano arithmetic, but true in its standard model, as seen through the lens of Zermelo-Fraenkel set theory. Admittedly, Peano arithmetic is probably not the natural habitat of a working mathematician; still, it is a prominent first-order theory of natural numbers.   
Both statements are combinatorial. One is an apparently slight strengthening of the finite Ramsey theorem, conditioning additionally that a certain monochromatic set, whose existence is stipulated by the theorem, should be at least of the cardinality of its least element. A well-known result of Paris and Harrington says that this stronger statement is unprovable in PA. One can argue its truth by appealing to a stronger theory, such as ZF or second-order arithmetic. 
The other example pertains to Goodstein sequences; Goodstein's theorem tells us that each such sequence eventually reaches zero; Kirby and Paris proved that this result is not a theorem of PA.
As to probability of (un)provability, that is presumably quite difficult to assess. Provability in a theory has nothing to do with whether or not the statement is "self-referential". Naturally, some of the statements that have resisted either proving or disproving for a long time may raise the question of the adequacy of the formal means employed in the attempts. 
