Here's an example of what Peter and Tim are speaking of, see the first responce to the question.
http://www-cs-students.stanford.edu/~pdoyle/quail/questions/11_15_96.html This responce is incorrect, irrespective of how many people might propogate it. Why? Godel says that we can't prove the completeness -within- the system itself (which includes Peano Arithmetic). That to prove this we -must- step outside (ie meta-arithmetic or meta-Peano). Now the assertion is that since we have demonstrated that it is not provably complete then it must be incomplete. This is incorrect. Simply because A implies B does -not- mean that ~B implies ~A. You can not prove a negative, which is what this line of reasoning is trying to do. This is why Godels proofs focus on statements that can be proved 'true' (and we ignore all the 'false' ones, if we have a false statement we look at its inverse and try to prove it instead, that 'or' in there is critical and way too many people skip right over it, it's -really- not a Boolean OR but an exclusive or in that we take true statements -or- the inverse of statements that are false, we never work with the false statements themselves). Godel further (as I referenced earlier) makes it clear that Peano is -not- necessary to his proof. -- ____________________________________________________________________ We don't see things as they are, [EMAIL PROTECTED] we see them as we are. www.ssz.com [EMAIL PROTECTED] Anais Nin www.open-forge.org --------------------------------------------------------------------
