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 person, this is as good as it is proven.
Edit: I have learned a lot from the answers, thank you so much!

When we say that a statement is undecidable, this is implicitly with regard to some axiom system, such as Peano arithmetic or ZFC.  The reason why statements about arithmetic can be undecidable is that there exist inequivalent models of arithmetic that obey such axiom systems (this is a consequence of the Godel completeness and incompleteness theorems, assuming of course that the axiom system being used is consistent and recursively enumerable).  For instance, the standard or "true" natural numbers obey the Peano axioms, but so do some more exotic number systems (e.g. the nonstandard natural numbers).  It is conceivable that a given statement in arithmetic, such as Goldbach's conjecture, holds for the true natural numbers but not for some other number systems that obey the Peano axioms, though personally I view this possibility as remote.  (Note that the nonstandard natural numbers satisfy exactly the same first-order statements as the standard natural numbers, by Los's theorem, but one can also construct other, weirder, models of arithmetic which have a genuinely different theory.)

So, if Goldbach's conjecture is undecidable for some given axiom system, what this would imply is that every "true" even natural number larger than 4 is the sum of two primes (otherwise we could disprove Goldbach with an argument of finite length), but that there exists a more exotic number system (larger than the true natural numbers) obeying those axioms which contains some even exotic number that is not the sum of two (exotic) primes.  (Note that the length of a proof has to be true natural number, rather than an exotic one, so the existence of an exotic counterexample cannot be directly converted to a disproof of Goldbach.) This is an unlikely scenario, but not an a priori impossible one (as one can see from the example of Goodstein's theorem or the Paris-Harrington theorem).

[Edit: here, as usual, when talking about things like the "true" natural numbers, one has to work in some external reasoning system which may be distinct from the reasoning system one is analysing the decidability of.  For instance, one may be using ZFC as one's external reasoning system to analyse what can and cannot be decided in Peano arithmetic, with the true natural numbers then being constructed by, say, the von Neumann ordinal construction combined with the axiom of infinity.  Or one could be using an informal external reasoning system, based perhaps on some Platonic beliefs about mathematical objects, that is not explicitly axiomatised.  It's best to keep the external system conceptually distinct from any internal ones being studied, as one can get hopelessly confused otherwise.]
