As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is consistent.  The consistency of ZFC can only be proved in a larger system, whose consistency is unknown.

That means proofs are not like as I once used to believe: a certificate that a counterexample for a statement can not be found.  For example, in spite of the proof of Wiles, it is conceivable that someday someone can come up with integers a,b and c and n>2 such that a^n + b^n = c^n, which would mean that our axiom system happened to be inconsistent.

I would like to learn about the reasons that, in spite of Godel's thoerem, mathematicians (or you) think that proofs are still very valuable.  Why do they worry less and less each day about Godel's theorem (edit: or do they)?  

I would also appreciate references written for non-experts addressing this question. 

If you like, you can view proofs of a statement in some formal system (e.g. ZFC) as a certificate that a counterexample cannot be found without demonstrating the inconsistency of ZFC, which would be a major mathematical event, and probably one of far greater significance than whether one's given statement was true or false.  

In practice, a given proof is not going to be closely tied to a single formal system such as ZFC, but will be robust enough that it can follow from any number of reasonable sets of axioms, including those much weaker than ZFC.  Only one of these sets of axioms then needs to be consistent in order to guarantee that no counterexample would ever be found, and this is about as close to an ironclad guarantee as one can ever hope for.

But ultimately, mathematicians are not really after proofs, despite appearances; they are after understanding.  This is discussed quite well in Thurston's article "On proof and progress in mathematics".
