"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency." Could Goldbach's conjecture be seen as a statement that is true but not be provable within that consistent system? Every even number can be written as a sum of two primes. This seems pretty obvious if we just think of it as another axiom. And until we can find an even number that can't be written as a sum of two primes Goldbach's conjecture is as true as 2+2=4.
Almost certainly no. The only way Gödel's incompleteness theorems could possibly be used in such a proof is if you managed to first prove a lemma such as
- if there is an undecideable statement, then Goldbach's conjecture is undecideable
- any proof or disproof of Goldbach's conjecture can be modified to prove the consistency of Peano arithmetic
or other similar thing. But it strains credulity to think that one could prove such a lemma without first having a proof that Goldbach's conjecture is undecideable.
If all you really mean to ask is whether it's possible that Goldbach's conjecture is undecideable, then yes — (as far as I know) there does not exist any proof that Goldbach's conjecture is decideable, so the current state of mathematical knowledge leaves all three of the following options open:
- that Goldbach's conjecture can be proven from Peano arithmetic
- that Goldbach's conjecture can be disproven from Peano arithmetic
- that Goldbach's conjecture is undecideable in Peano arithmetic
As many other conjectures dealing with prime numbers,we can't say yet because we don't have enough understanding of prime numbers yet. There might be an easy and obvious proof once we have better understanding of prime numbers, or there might not be.
However, you can't just assume the conjecture to be true until proven false. If you do, other results will be build on it and you might end up loosing the work of many decades should the conjecture ever turn out to be false. If you are in a more practical field, you might say "it is true for all numbers I will ever encounter during my work", but this line of thought is mostly reserved for engineers, mathematicians tend not to think that way.