Gödel did another very interesting thing. He gave a proof for that 'any finite system of axioms is not sufficient to prove every result in mathematics'.

". He saw that no version of the formula game could encompass all truths, even about whole numbers. In the published version of his discovery, Gödel introduced a code by which the expressions of the formula game were each represented by a single whole number. Thus, certain statements about numbers could be seen by someone privy to the code as making assertions about the formula game as well. Gödel showed how to construct a statement P that, in this manner, asserted that a certain other statement, Q, could not be proved using the rules. Then he showed how to make Q=P, so that the statement P actually asserted that P itself is unprovable. From this it follows that P must be true and therefore, because of what it asserts, unprovable. This necessary incompleteness of the formula game was the first blow to Hilbert's work. A few weeks later, Gödel delivered the knockout punch by proving that Hilbert's ( cherished goal of proving the consistency of the formula game through restricted methods was doomed to failure."

What he apparently had started with was wanting to support David Hilbert's view that mathematics was the universe, its logic impeccable. "Hilbert's radical remedy was to create an entirely new branch of mathematics, dubbed metamathematics, which would apply mathematical methods to mathematics itself. In order to achieve this, it was first necessary to present mathematics as a 'formula game' in which the propositions of mathematics were represented as mere assemblages of symbols, and the methods of inference that led from axioms to theorems were presented as transparent rules for manipulating symbols." But he (Gödel that is) ended up finding himself doing the opposite.