Framge challenge: You say Goedel's theorem tells us our theory can't sort out all truth.
Consider a statement A that is an example of G's theorem. Within the system of logic at hand the statement A cannot be proved true or false.
How do you know it is a true statement? How do you know it needs to be "sorted out?"
There have been some examples of statements of this kind in logic. People found a statement they could not prove true or false. They attempted to use such tactics as assuming it true and looking for contradictions. Or assuming it false and looking for contradictions. And they have found that fortifying the original system of logic with a new hypothesis resulted in a new system of logic that was still self consistent, but larger. And, quite charmingly, is still potentially "mineable" for new hypothesis that cannot be proved true or false within the new system.
One fairly wonderful example is non-Euclidean geometry. One inroad into non-Euclidean is to assume that parallel lines can meet.
The kicker in this is, this process provides "job security" for those who study logic. G's theorem tells us there will always be more things to discover.