
Re: Axiomization of Number Theory
Posted:
Jul 29, 2003 2:24 AM


David C. Ullrich wrote:
> On Mon, 28 Jul 2003 13:38:12 +0100, Robin Chapman > <rjc@ivorynospamtower.freeserve.co.uk> wrote: > >>Pete Moore wrote: >> >>> >>> "CharlieBoo" <chvol@aol.com> wrote in message >>> news://3df1e59f.0307250955.79ee3a83@posting.google.com... >>>> Hello all, >>>> >>>> I am interested in axiomizing Number Theory. >>> >>> Didn't Godel prove that that isn't possible? >>> >> >>And din't Peano actually do it? > > It takes some definitions to clarify this. Godel certainly > did prove that axiomatizing number theory is impossible, > and no Peano didn't do it. Carefully: > > Let's say "number theory" is the (firstorder) theory of > the natural numbers, ie the set of all statements > about the natural numbers in a certain formal language > that are true. Godel showed that there is no (recursive) > axiomatization of number theory  there does not > exist a recursive set of axioms such that the logical > consequences of those axioms are precisely the > true statements of number theory. ("Recursive" > just says that there is an algorithm that allows one > to determine whether or not a given statement > is an axiom. Nonrecursive "axiomatizations" exist, > for example the set of all true statements of number > theory is an axiomatization of number theory. But > nonrecursive axiomatizations are sort of useless, > since there's no way to check whether a proof is > correct, since there's no way to recognize an > "axiom" when you see one.) > > The (firstorder) Peano axioms do not do the > job.
Then take secondorder :)
 Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html "His mind has been corrupted by colours, sounds and shapes." The League of Gentlemen

