Some time ago I wrote something about axioms, it seemed really frightening then to think about them… Doesn’t matter how interesting they [axioms] are, there is still something that doesn’t feel good… And everytime a teacher wrote an axiom on the blackboard I was always asking myself: “How do they know?”.
And after all this trouble I was giving myself, I found this: “a yound Austian mathematician named Kurt Godel proved a result that was to change our view of mathematics forever. Godel’s theorem says that if you write down any consistent axiom system for some reasonably large part of mathematics, then that axiom system must be incomplete: there will always be some questions that cannot be answered on the basis of the axioms.” (Keith Devlin, “The Language of Mathematics – Making the Invisible Visible”). And I was like WHHHATTT???!!!! Are you joking?!!! This really needs to be a trick or something, because it just doesn’t seem to be natural, it doesn’t seem to be corect. But after that I thought again about Russel’s Paradox and maybe, Maybe, I can give this idea a chance. Maybe, just maybe, it feets in this hole “axiomatization game”.
So, to give you something to thing about (in fact not only me needs to really digest this problem): “In short, in the axiomatization game, the best you can do is to assume the consistency of your axioms and hope that they are rich enough to enable you to solve the problems of highest concern to you. You have to accept that you will be unable to solve all problems using your axioms; there will always be true porpositions that you cannot prove from those axioms.”(Keith Devlin, “The Language of Mathematics – Making the Invisible Visible”)