Re: A question on GIT.

From: Herman Jurjus (h.jurjus_at_hetnet.nl)
Date: 09/09/04 

Date: Thu, 09 Sep 2004 11:46:07 +0200

peter_douglass wrote:
> "Herman Jurjus" wrote in message
> news:2q8dv6Fs5hohU1@uni-berlin.de...
>
>
>>But what does it mean that Presburger Arithmetic is consistent?
>>That FOL will not prove contradiction from it, in finitely many steps?
>>OK. Please define 'finitely many steps', without somehow using the
>>intuitive notion of N that we all currently use.
>
>
> You may not like this, but use the notion of N that is in
> Presburger Arithmetic. Do you need multiplication to do
> induction?
>
>
>>Now our intuitive N obviously satisfies PA. So i still say that, if PA
>>is inconsistent (again, bizar, but...), then apparantly we have the
>>wrong idea about N. So, imo, there would then be lots of theorems that i
>>would no longer trust, or that would become simply meaningless.
>>Decidability and consistency of Presburger Arithmetic included.
>
>
> If there is a sense in which we can speak of intuitive notion
"satisfying"
> a formalized theory, then I think our intuitive N also satisfies
> Presburger Arithmetic.
[snip]
> Perhaps you should explain what it means for an intuitive notion
> "satisfying" a theory.

Well, apparantly you know what it means for your intuitive N to
satisfy Presburger Arithmetic. Is there any axiom of PA that you
think is false in 'your' picture of N, in the same sense?

Now then, if PA is inconsistent (i'm not saying that it is, i say
_if_), wouldn't that mean that N, as you currently envision it,
is an illusion?

How would you deal with that? Would you throw away PA and keep
Presburger Arithmetic? But then the only models of your theory
would be non-standard, and would necessarily contain non-standard
numbers. Because the standard picture, with all the finite numbers
in them (no matter how astronomically large), but none of the
infinite numbers, would then be no longer existing.

Now imagine what that would do to the _meaning_ of notions like
'fol-proof', 'consistent', 'recursively decidable'. All of these
notions refer crucially to natural numbers, and especially to our
realistic picture of the 'intended model of PA'.
(Only finite proofs are allowed, only calculations with finitely
many steps are allowed. In all these cases, unfeasibly large numbers
are included, but infinite numbers (non-standard numbers), are
excluded.)

The difficulty, of course, is that it is very hard to imagine any
mathematics in which the natural number sequence does not exist,
or is not unique.
It's like assuming a contradiction and discussing the consequences. 

-- 
Cheers,
Herman Jurjus

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