|
|
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:
>>
>>>
>>> "Charlie-Boo" <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 (first-order) 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. Non-recursive "axiomatizations" exist,
> for example the set of all true statements of number
> theory is an axiomatization of number theory. But
> non-recursive 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 (first-order) Peano axioms do not do the
> job.
Then take second-order :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"His mind has been corrupted by colours, sounds and shapes."
The League of Gentlemen
|
|
