Juergen Schmidhuber, [EMAIL PROTECTED], writes: > All formal proofs of number theory are computable from the axioms > describable by a few bits. So what about Goedel's theorem? We cannot > derive it from the axioms. But what does that mean? It just means we > can add it as an axiom. It just means the proof of Goedel's statement > requires more bits than those conveyed by the original axioms. > > In other words, Goedel used additional information. In many of the UTM's > universes his theorem will be proven by those willing to do the same.
You can also add the negation of the Goedel statement as an axiom, and get no contradictions. It is interesting to consider what the resulting axiom set expresses. You tried to capture all the essence of arithmetic in the original axioms, but when you add not-G you get a consistent formal system which has describes all the properties of numbers as we know them, but is somehow "wrong". Hal