This statement does not have any proof. If the statement were false then the statement would be provable, but this would contradict the statement. Therefore the statement must be true in order to avoid the contradiction. However, although the statement is true it cannot be proven, because this statement (which we now know to be true) says so. Because Gödel could translate the above statement into mathematical notation, he was able to demonstrate that there existed statements in mathematics which are true but which could never be proven to be true, so-called undecidable statements.
