We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite proof. What if this number is very large? Maybe beyond our reach?

If one proves that it is independent that eventually every even number is the sum of two primes, can we conclude that eventually every even number is the sum of two primes?