He showed that a formal arithmetical could not be demonstrated to be consistent from within itself. His fifteen page article proved that some mathematics could not be proved – that whatever axioms were accepted in mathematics there would always be some truths that could not be validated. Then there is Neurath's Boat. Nuerath was an antifoundationalist: he believed that knowledge has no secure substructure. By way of illustrations he used a nautical simile: “We are like sailors who have to rebuild their ship on the open sea, without ever being able to dismantle it in dry dock and construct it from the best components. Pg 163
I just wanted to remember this, because even though Gödel is impenetrable to me, this make sense to me.