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Re: Axiomization of Number Theory
Posted:
Jul 30, 2003 4:12 PM
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On Wed, 30 Jul 2003 16:11:45 +0100, Robin Chapman
<rjc@ivorynospamtower.freeserve.co.uk> wrote:
>David C. Ullrich wrote:
>
>>
>>>I am interested in axiomizing Number Theory. I'm not talking about
>>>some bogus list of properties of addition and multiplication, but
>>>rather a set of formal axioms and rules of inference that allows us (a
>>>program) to derive theorems from Number Theory.
>>
>> [hmm, looking at that again it appears that we are a program.]
>>
>> Someone said Godel showed this was impossible, you said didn't
>> Peano do it,
>
>Do you really believe that Godel showed that
>"... to derive theorems from Number Theory"
>was impossible?
Well first, it's clear that the "from Number Theory"
meant "of Number Theory" - we want to derive theorems
of number theory from the formal system.
Of course Godel didn't show that it's impossible to
derive theorems of number theory from a formal system,
if by that we mean derive _some_ theorems of number
theory. Giving a formal system from which it's possible
to derive some theorems of number theory is so easy
that it seems it can't be what we really wanted (for
example, consider the formal system with Fermat's
Last Theorem as the one and only axiom, and no
inference rules. It's not hard to derive Fermat's Last
Theorem from that formal system, but it doesn't seem
very interesting.) I and the person who claimed Godel
showed "it" was impossible were assuming that "it"
was to derive _all_ the theorems of number theory
from a formal system.
If the "it" in your "didn't Peano do it" meant give
a formal system from which one could derive
_some_ theorems of number theory then yes,
of course Peano did it (but there are much easier
ways to do it, as above.)
************************
David C. Ullrich
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