Mr Blobby posted an excellent comment in my previous post, outlining in a few words Gödel's proof of incompleteness of arithmetics. This is a mind-boggling result, which essentially states the following:

There are certain statements (I would not say "theorems") in the body of Arithmetics, which are either true or false. But there is no way to find a proof for either of these two statements. Because there is no such proof.

So, let's take for example an unproven statement, Goldbach conjecture. This conjecture simply states that any even natural number (greater than 2) is a sum of two primes. Despite its apparent simplicity, this simple conjecture has no known proofs up to date! But even if there is no proof, assuming that you have infinite time (and patience) you can verify it manually by taking every even number and trying to express it as a sum of two prime numbers. But even if this statement is true, or false, there might be no proof for it after all! We don't know yet...

But let's go now a little bit further. OK, so let's assume that the Goldbach conjecture is unprovable after all. Assuming this, what if there is a proof that says that Goldbach is unprovable? What can we say about this statement?