This statement is not provable.Let 's call the above statement S. So what S says is that
S is not provable
If this statement is false (meaning S is false) then the opposite statement must be true. The opposite statement is
S is provable
and it must be true. But wait a minute. If S is provable, meaning that there exists a proof of its truth, then S is true. But wait a minute again. We started by assuming that S is false. A contradiction... This means that our initial assumption was wrong and therefore S is true. But what does this mean ? This means that what it says is true, S is actually not provable. So S is true but non-provable. This is non-completeness. Our inability to reach every existing truth. Simple, elegant and yet so deep.