The answer is that an incomplete arithmetic axiom system could presumably by consistent, but who cares? If it is incomplete there will be true statements that it cannot prove and we are back to the platonist position! The alternative of an inconsistent system that is complete may actually be more interesting and has been explored in recent mathematics.
A great reference is "Inconsistent Mathematics" by Chris Mortensen (Kluwer 1995). -Joao Hal Finney wrote: > Jesse Mazer writes: > > Yes, a Platonist can feel as certain of the statement "the axioms of Peano > > arithmetic will never lead to a contradiction" as he is of 1+1=2, based on > > the model he has of what the axioms mean in terms of arithmetic. It's hard > > to see how non-Platonist could justify the same conviction, though, given > > Godel's results. Since many mathematicians probably would be willing to bet > > anything that the statement was true, this suggests a lot of them are at > > least closet Platonists. > > What is the status of the possibility that a given formal system such as > the one for arithmetic is inconsistent? Godel's theorem only shows that > if consistent, it is incomplete, right? Are there any proofs that formal > systems specifying arithmetic are consistent (and hence incomplete)? > > Hal Finney -- Joao Pedro Leao ::: [EMAIL PROTECTED] Harvard-Smithsonian Center for Astrophysics 1815 Massachussetts Av. , Cambridge MA 02140 Work Phone: (617)-496-7990 extension 124 VoIP Phone: (617)=384-6679 Cell-Phone: (617)-817-1800 ---------------------------------------------- "All generalizations are abusive (specially this one!)" -------------------------------------------------------
