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


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


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