Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really puzzles me. (See page 321 of http://www.ams.org/notices/200203/fea-knuth.pdf )
All the examples of such statements I have heard until now were very abstract, but this one is so concrete.
My question is that is there such an example of a statement of this sort that is proved to be unprovable, i.e., for some property P(n), the statement that "every natural number n satisfies P(n)". (In the Goldbach case P(n) would be "if n is an even number greater than 2, then there exists two primes p and q such that n = p + q".)
If Knuth is right, it would be very interesting in one sense: the negation of Goldbach is obviously provable if it is true. So if someone proves that Goldbach is not provable, then we would know that Goldbach is true. We would be sure that someone would never come up with an example that would violate the condition. For the practical man, this is as good as it is proven.
Edit: I have learned a lot from the answers, thank you so much!